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A Drinfel'd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d1202702.png" />-module, (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d1202703.png" /> is an appropriate [[Ring|ring]]) over a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d1202704.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d1202705.png" /> is an exotic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d1202706.png" />-module structure on the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d1202707.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d1202708.png" />. In several regards, the concept of a Drinfel'd module is analogous to the concept of an [[Elliptic curve|elliptic curve]] (or more generally, of an irreducible [[Abelian variety|Abelian variety]]), with which it shares many features. Among the similarities between Drinfel'd modules and elliptic curves are the respective structures of torsion points, of Tate modules and of endomorphism rings, the existence of analytic "Weierstrass uniformizations" , and the moduli theories (modular varieties, modular forms; cf. also [[Modular form|Modular form]]). Many topics from the (classical and well-developed) theory of elliptic curves may be transferred to Drinfel'd modules, thereby revealing arithmetical information about the ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d1202709.png" />. On the other hand, since the mechanism of Drinfel'd modules is smoother and in some respects simpler than that of Abelian varieties, some results involving Drinfel'd modules over global function fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027010.png" /> can be proved, whose analogues over number fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027011.png" /> are far from being settled (e.g. parts of Stark's conjectures, of the Langlands conjectures, assertions about the arithmetical nature of zeta values and other questions of transcendence theory over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027012.png" />, cf. also [[L-function|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027013.png" />-function]]). The invention and basic theory as well as large parts of the deeper results about Drinfel'd modules are due to V.G. Drinfel'd [[#References|[a3]]], [[#References|[a4]]]. General references are [[#References|[a2]]], [[#References|[a10]]], [[#References|[a9]]], and [[#References|[a8]]].
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{{MSC|11G09|11F70,11R39,22E55}}
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{{TEX|done}}
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A Drinfel'd $A$-module, (where $A$ is an appropriate
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[[Ring|ring]]) over a
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[[Field|field]] $L$ of characteristic $p>0$ is an exotic $A$-module structure on the additive group $\def\Ga{\mathcal{G}_\alpha}\Ga$ over $L$. In several regards, the concept of a Drinfel'd module is analogous to the concept of an
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[[Elliptic curve|elliptic curve]] (or more generally, of an irreducible
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[[Abelian variety|Abelian variety]]), with which it shares many features. Among the similarities between Drinfel'd modules and elliptic curves are the respective structures of torsion points, of Tate modules and of endomorphism rings, the existence of analytic "Weierstrass uniformizations" , and the moduli theories (modular varieties, modular forms; cf. also
 +
[[Modular form|Modular form]]). Many topics from the (classical and well-developed) theory of elliptic curves may be transferred to Drinfel'd modules, thereby revealing arithmetical information about the ground field $L$. On the other hand, since the mechanism of Drinfel'd modules is smoother and in some respects simpler than that of Abelian varieties, some results involving Drinfel'd modules over global function fields $L$ can be proved, whose analogues over number fields $L$ are far from being settled (e.g. parts of Stark's conjectures, of the Langlands conjectures, assertions about the arithmetical nature of zeta values and other questions of transcendence theory over $L$, cf. also
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[[L-function|$L$-function]]). The invention and basic theory as well as large parts of the deeper results about Drinfel'd modules are due to V.G. Drinfel'd
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{{Cite|Dr}},
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{{Cite|Dr2}}. General references are
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{{Cite|DeHu}},
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{{Cite|GoHaRoGrHa}},
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{{Cite|}}, and
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{{Cite|GePuReGeWoScGo}}.
  
 
==Algebraic theory.==
 
==Algebraic theory.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027014.png" /> be any field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027015.png" />, with algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027016.png" />. The endomorphism ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027017.png" /> of the additive group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027018.png" /> is the ring of additive polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027019.png" />, i.e., of polynomials satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027020.png" />, whose (non-commutative) multiplication is defined by insertion. Then
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Let $L$ be any field of characteristic $p>0$, with algebraic closure $\overline{L}$. The endomorphism ring $\def\End{\textrm{End}}\End_L(\Ga)$ of the additive group scheme $\Ga/L$ is the ring of additive polynomials $f(x)\in L[x]$, i.e., of polynomials satisfying $f(x+y)=f(x)+f(y)$, whose (non-commutative) multiplication is defined by insertion. Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027021.png" /></td> </tr></table>
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$$\End_L(\Ga)=\{\sum a_i\tau_p^i | a_i\in L\} = L\{\tau_p\}$$
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is the twisted polynomial ring in $\tau_p=x^p$ with commutation rule $\tau_p \cdot a = a^p \cdot \tau_p$ for $a\in L$ and unit element $\tau_p^0 = x$. Fix a power $q=p^f$ of $p$. If $L$ contains the field $\F_q$ with $q$ elements, one sets $\tau=\tau_p^f = x^q$ and $L\{\tau\}$ for the subring of $\F_q$-linear polynomials in $\End_L(\Ga)$. For any $\F_q$-algebra $A$, an $A$-module structure on $\Ga/L$ is given by a morphism $\def\phi{\varphi}\phi$ of $\F_q$-algebras from $A$ to $L\{\tau\}$.
  
is the twisted polynomial ring in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027022.png" /> with commutation rule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027023.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027024.png" /> and unit element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027025.png" />. Fix a power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027027.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027028.png" /> contains the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027029.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027030.png" /> elements, one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027032.png" /> for the subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027033.png" />-linear polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027034.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027035.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027036.png" />, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027037.png" />-module structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027038.png" /> is given by a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027039.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027040.png" />-algebras from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027041.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027042.png" />.
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Fix a (smooth, projective, geometrically connected)
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[[Algebraic curve|algebraic curve]] $\def\C{\mathcal{C}}\C$ over $\F_q$ and a place $\infty$ of $\C$; let $K$ be its function field and $A$ the affine ring of $\C\setminus\{\infty\}$. (Here, "places" , or "primes" , are closed points of $\C$, the set of normalized valuations on $K$; cf. also
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[[Norm on a field|Norm on a field]].) Hence $K$ is a function field in one variable over $\F_q$ and $A$ is its subring of elements regular away from $\infty$. Put $\def\deg{\mathrm{deg}}\deg:A\to \Z$ for the associated degree function: $\deg\; a = \dim_{\F_q} A/(a)$ if $a\ne 0$. Let $L$ be a field equipped with a structure $\def\g{\gamma}\g:A\to L$ of an $A$-algebra. Then $L$ is either an extension of $K$ or of some $\def\fp{\mathfrak{p}}A/\fp$, where $\fp$ is a
 +
[[Maximal ideal|maximal ideal]]. One writes $\def\char{\mathrm{char}}\char_A(L) = \infty$ in the former and $\char_A(L) = \fp$ in the latter case. A Drinfel'd $A$-module of rank $r$ over $L$ (briefly, an $r$-Drinfel'd module over $L$) is a morphism of $\F_q$-algebras
  
Fix a (smooth, projective, geometrically connected) [[Algebraic curve|algebraic curve]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027043.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027044.png" /> and a place "∞" of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027045.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027046.png" /> be its function field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027047.png" /> the affine ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027048.png" />. (Here, "places" , or "primes" , are closed points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027049.png" />, the set of normalized valuations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027050.png" />; cf. also [[Norm on a field|Norm on a field]].) Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027051.png" /> is a function field in one variable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027053.png" /> is its subring of elements regular away from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027054.png" />. Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027055.png" /> for the associated degree function: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027056.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027057.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027058.png" /> be a field equipped with a structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027059.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027060.png" />-algebra. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027061.png" /> is either an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027062.png" /> or of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027063.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027064.png" /> is a [[Maximal ideal|maximal ideal]]. One writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027065.png" /> in the former and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027066.png" /> in the latter case. A Drinfel'd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027068.png" />-module of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027069.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027070.png" /> (briefly, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027072.png" />-Drinfel'd module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027073.png" />) is a morphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027074.png" />-algebras
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$$\phi : A \to L\{\tau\},$$
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027075.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027076.png" /></td> </tr></table>
 
  
 +
$$a\mapsto\phi_a = \sum_{i=0}^{r\;\deg\;a} l_i(a)\tau^i$$
 
subject to:
 
subject to:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027077.png" />; and
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i) $l_0(a)=\gamma(a)$; and
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027078.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027079.png" />. It supplies the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027080.png" /> of each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027081.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027082.png" /> with the structure of an abstract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027083.png" />-module. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027084.png" /> of Drinfel'd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027085.png" />-modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027086.png" /> is some element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027087.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027088.png" /> that satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027089.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027090.png" />. Similarly, one defines iso-, endo- and automorphisms.
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ii) $l_{r\;\deg\;a}\ne 0$ for $a\in A$. It supplies the additive group $L'$ of each $L$-algebra $L'$ with the structure of an abstract $A$-module. A morphism $u:\phi\to\phi'$ of Drinfel'd $A$-modules over $L$ is some element $u$ of $L\{\tau\}$ that satisfies $\phi'_a\cdot u = u\cdot \phi_a$ for $a\in A$. Similarly, one defines iso-, endo- and automorphisms.
  
The standard example of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027091.png" /> is given by a rational function field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027093.png" /> being the usual place at infinity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027094.png" />. In that case, a Drinfel'd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027095.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027096.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027097.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027098.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d12027099.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270100.png" /> may be arbitrarily chosen in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270101.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270102.png" />). More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270103.png" /> is generated over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270104.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270106.png" /> is given by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270107.png" /> that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270108.png" /> must satisfy the same relations as do the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270109.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270110.png" />. Writing down a Drinfel'd module amounts to solving a complicated system of polynomial equations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270111.png" />. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270112.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270113.png" />. From <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270115.png" />, one obtains for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270116.png" />-Drinfel'd module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270117.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270118.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270119.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270121.png" />. Using computation rules in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270122.png" />, one solves for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270126.png" />, which yields the unique (up to isomorphism) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270127.png" />-Drinfel'd module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270128.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270129.png" />. That <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270130.png" /> is unique and even definable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270131.png" /> corresponds to the fact that the class number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270132.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270133.png" />.
+
The standard example of $(K,\infty,A)$ is given by a rational function field $K=\F_q(T),$, $\infty$ being the usual place at infinity, $A=\F_q[T]$. In that case, a Drinfel'd $A$-module $\phi$ of rank $r$ over $L$ is given by $\phi_T = \g(T)\tau^0 + l_1\tau+\cdots+l_r\tau^r$, where the $l_i=l_i(T)$ may be arbitrarily chosen in $L$ ($l_r\ne 0$). More generally, if $A$ is generated over $\F_q$ by $\{a_1,\dots,a_n\}$, $\phi$ is given by the $\phi_{a_i}$ that in $L\{\tau\}$ must satisfy the same relations as do the $a_i$ in $A$. Writing down a Drinfel'd module amounts to solving a complicated system of polynomial equations over $L$. For example, let $A=\F_2[U,V]$ with $V^2+V = U^3+U+1$. From $\deg\; U = 2$, $\deg\;V = 3$, one obtains for a $1$-Drinfel'd module $\phi$ over $L=\def\K{\overline{K}}\K=\textrm{ algebraic closure}(K)$: $\phi_V^2 + \phi_V = \phi_U^3+\phi_U+1$ with $\phi_U = UX+aX^2+bX^4$, $\phi_V = VX+cX^2+dX^4+eX^6$. Using computation rules in $\K\{\tau\}$, one solves for $a=U^2+U$, $b=e=1$, $c=V^2+V$, $d=U(V^2+V)$, which yields the unique (up to isomorphism) $1$-Drinfel'd module $\phi$ over $\K$. That $\phi$ is unique and even definable over $K$ corresponds to the fact that the class number of $A$ equals $1$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270134.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270135.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270136.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270137.png" />-Drinfel'd module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270138.png" />. The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270139.png" /> has degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270140.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270141.png" />, whence has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270142.png" /> different roots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270143.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270144.png" />. This implies that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270145.png" />-module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270147.png" />-torsion points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270148.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270149.png" /> is isomorphic with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270150.png" />. Similar, but more complicated assertions hold if one considers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270151.png" />-torsion points (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270152.png" /> a not necessarily principal ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270153.png" />) and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270154.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270155.png" />. A level-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270156.png" /> structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270157.png" /> is the choice of an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270158.png" /> of abstract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270159.png" />-modules (with some modification if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270160.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270161.png" />).
+
Let $a\in A$ with $\deg\; a>0$ and let $\phi$ be an $r$-Drinfel'd module over $L$. The polynomial $\phi_a(X)$ has degree $q^{r\;\deg\;a}$ in $X$, whence has $q^{r\;\deg\;a}$ different roots in $\def\L{\overline{L}}\L$ if $(a,\char_A(L)) = 1$. This implies that the $A$-module of $a$-torsion points ${}_a\phi(\L) = \{x\in\L | \phi_a(x) = 0 \}$ of $\phi$ is isomorphic with $(A/(a))^r$. Similar, but more complicated assertions hold if one considers $\def\fa{\mathfrak{a}}\fa$-torsion points ($\fa$ a not necessarily principal ideal of $A$) and if $\char_A(L)$ divides $\fa$. A level-$\fa$ structure on $\phi$ is the choice of an isomorphism $\alpha(A/\fa)^r\stackrel{\simeq}{\to}{}_\fa\phi(L)$ of abstract $A$-modules (with some modification if $\char_A(L)$ divides $\fa$).
  
The definitions of Drinfel'd modules, their morphisms, torsion points, and level structures generalize to arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270162.png" />-schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270163.png" /> (instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270164.png" />, which corresponds to the case above; cf. also [[Scheme|Scheme]]). Intuitively, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270165.png" />-Drinfel'd module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270166.png" /> is a family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270167.png" />-Drinfel'd modules varying continuously over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270168.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270169.png" /> be a non-vanishing ideal. On the [[Category|category]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270170.png" />-schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270171.png" />, there is the contravariant [[Functor|functor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270172.png" /> that to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270173.png" /> associates the set of isomorphism classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270174.png" />-Drinfel'd modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270175.png" /> provided with a level-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270176.png" /> structure. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270177.png" /> has at least two prime divisors (such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270178.png" /> are admissible), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270179.png" /> is representable by a moduli scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270180.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270181.png" />-morphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270182.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270183.png" /> correspond one-to-one to isomorphism classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270184.png" />-Drinfel'd modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270185.png" /> with a level-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270186.png" /> structure. The various <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270187.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270188.png" /> are equipped with actions of the finite groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270189.png" /> and related by morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270190.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270191.png" />. Taking quotients, this allows one to define coarse moduli schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270192.png" /> even for non-admissible ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270193.png" />, and for more general moduli problems, e.g., the problem "rank-r Drinfel'd A-modules with a point of order a A" . For such coarse moduli schemes, the above bijection between morphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270194.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270195.png" /> and objects of the moduli problem holds only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270196.png" /> is the spectrum of an [[Algebraically closed field|algebraically closed field]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270197.png" /> is admissible, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270198.png" /> is affine, smooth, of finite type and of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270199.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270200.png" />. Furthermore, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270201.png" />, the morphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270202.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270203.png" /> are finite and flat, and even étale outside the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270204.png" /> (cf. also [[Affine morphism|Affine morphism]]; [[Flat morphism|Flat morphism]]; [[Etale morphism|Etale morphism]]). As an example, take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270205.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270206.png" /> be algebraically closed. Two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270207.png" />-Drinfel'd modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270209.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270210.png" />, given through the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270211.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270212.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270213.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270214.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270215.png" />, are isomorphic if and only if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270216.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270217.png" />. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270218.png" />, the moduli scheme attached to the trivial ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270219.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270220.png" />, is the open subscheme defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270221.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270222.png" />, where the multiplicative group acts diagonally through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270223.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270224.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270225.png" /> with the "modular invariant" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270226.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270227.png" /> being regarded as indeterminates.
+
The definitions of Drinfel'd modules, their morphisms, torsion points, and level structures generalize to arbitrary $A$-schemes $S$ (instead of $S=\def\Spec{\mathrm{Spec}\;}\Spec L$, which corresponds to the case above; cf. also
 +
[[Scheme|Scheme]]). Intuitively, an $r$-Drinfel'd module over $S$ is a family of $r$-Drinfel'd modules varying continuously over $S$. Let $\fa\subset A$ be a non-vanishing ideal. On the
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[[Category|category]] of $A$-schemes $S$, there is the contravariant
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[[Functor|functor]] $\def\M{\mathcal{M}}\M^r(\fa)$ that to each $S$ associates the set of isomorphism classes of $r$-Drinfel'd modules over $S$ provided with a level-$\fa$ structure. If $\fa$ has at least two prime divisors (such $\fa$ are admissible), $\M^r(\fa)$ is representable by a moduli scheme $M^r(\fa)$. In other words, $A$-morphisms from $S$ to $M^r(\fa)$ correspond one-to-one to isomorphism classes of $r$-Drinfel'd modules over $S$ with a level-$\fa$ structure. The various $\M^r(\fa)$ and $M^r(\fa)$ are equipped with actions of the finite groups $\def\GL{\textrm{GL}}\GL(r,A/\fa)$ and related by morphisms $\def\fb{\mathfrak{b}}M^r(\fb)\to M^r(\fa)$ if $\fa|\fb$. Taking quotients, this allows one to define coarse moduli schemes $M^r(\fa)$ even for non-admissible ideals $\fa$, and for more general moduli problems, e.g., the problem ``rank-r Drinfel'd A-modules with a point of order a A'' . For such coarse moduli schemes, the above bijection between morphisms from $S$ to $M^r(\fa)$ and objects of the moduli problem holds only if $S$ is the spectrum of an
 +
[[Algebraically closed field|algebraically closed field]]. If $\fa$ is admissible, $M^r(\fa)$ is affine, smooth, of finite type and of dimension $r-1$ over $A$. Furthermore, for $\fa|\fb$, the morphisms from $M^r(\fb)$ to $M^r(\fa)$ are finite and flat, and even étale outside the support of $\fb\fa^{-1}$ (cf. also
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[[Affine morphism|Affine morphism]];
 +
[[Flat morphism|Flat morphism]];
 +
[[Etale morphism|Etale morphism]]). As an example, take $A=\F_q[T]$, and let $L$ be algebraically closed. Two $r$-Drinfel'd modules $\phi$ and $\phi'$ over $L$, given through the coefficients $l_i$ and $l_i'$ ($1\le i\le r$) of $\phi_T$ and $\phi_T'$, are isomorphic if and only if there exists a $c\in L^*$ such that $l_i' = c^{1-q^i}l_i$. Hence $\M^r((1))$, the moduli scheme attached to the trivial ideal $(1)$ of $A$, is the open subscheme defined by $l_r\ne 0$ of $\def\G{\mathcal{G}}\Spec A[l_1,\dots,l_r]/\G_m$, where the multiplicative group acts diagonally through $c(l_1,\dots,l_r) = (\dots,c^{1-q^i}l_i,\dots)$. If $r=2$, $M^2((1)) = \Spec A[j]$ with the ``modular invariant'' $j=l_1^{q+1}/l_2$, the $l_i$ being regarded as indeterminates.
  
 
==Analytic theory.==
 
==Analytic theory.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270228.png" /> be the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270229.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270230.png" />, with normalized absolute value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270231.png" /> and complete algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270232.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270233.png" /> is the smallest field extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270234.png" /> which is complete with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270235.png" /> and algebraically closed. For such fields, there is a reasonable function theory and analytic geometry [[#References|[a1]]].
+
Let $K_\infty$ be the completion of $K$ at $\infty$, with normalized absolute value $|\;.\;|_\infty$ and complete algebraic closure $C$. Then $C$ is the smallest field extension of $K$ which is complete with respect to $|\;.\;|_\infty$ and algebraically closed. For such fields, there is a reasonable function theory and analytic geometry
 
+
{{Cite|BoGüRe}}.
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270237.png" />-lattice is a finitely generated (thus projective) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270238.png" />-submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270239.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270240.png" /> that has finite intersection with each ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270241.png" />. With <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270242.png" /> is associated its exponential function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270243.png" />, defined as the everywhere convergent [[Infinite product|infinite product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270244.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270245.png" />). It is a surjective, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270246.png" />-linear and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270247.png" />-periodic function that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270248.png" /> satisfies a functional equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270249.png" /> with some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270250.png" />. The rule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270251.png" /> defines a ring homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270252.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270253.png" />, in fact, a Drinfel'd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270254.png" />-module of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270255.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270256.png" /> being the projective rank of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270257.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270258.png" />. Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270259.png" />-Drinfel'd module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270260.png" /> is so obtained, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270261.png" /> yields an equivalence of the category of lattices of projective rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270262.png" /> with the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270263.png" />-Drinfel'd modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270264.png" />. (A morphism of lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270265.png" /> is some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270266.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270267.png" />.) The description of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270268.png" /> through the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270269.png" /> is called the Weierstrass uniformization. From <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270270.png" />, one can read off many of the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270271.png" />. E.g.,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270272.png" /></td> </tr></table>
 
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270273.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270274.png" />). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270275.png" />, there result bijections between the finite sets of:
+
An $A$-lattice is a finitely generated (thus projective) $A$-submodule $\def\L{\Lambda}\L$ of $C$ that has finite intersection with each ball in $C$. With $\L$ is associated its exponential function $e_\L:C\to C$, defined as the everywhere convergent
 +
[[Infinite product|infinite product]] $e_\L(z) = z \prod(1-z/\lambda)$ ($0\ne \lambda\in \L$). It is a surjective, $\F_q$-linear and $\L$-periodic function that for each $a\in A$ satisfies a functional equation $e_\L(az) = \phi_a^\L(e_\L(z))$ with some $\phi_a^\L\in C\{\tau\}$. The rule $a\mapsto\phi_a^\L$ defines a ring homomorphism from $A$ to $C\{\tau\}$, in fact, a Drinfel'd $A$-module of rank $r$, $r$ being the projective rank of the $A$-module $\L$. Each $r$-Drinfel'd module over $C$ is so obtained, and $\L\mapsto\phi^\L$ yields an equivalence of the category of lattices of projective rank $r$ with the category of $r$-Drinfel'd modules over $C$. (A morphism of lattices $c:\L\mapsto\L'$ is some $c\in C$ such that $c\L\subset\L'$.) The description of $\phi=\phi^\L$ through the lattice $\L$ is called the Weierstrass uniformization. From $\L$, one can read off many of the properties of $\phi$. E.g.,
  
a) classes of rank-one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270276.png" />-lattices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270277.png" />, up to scalars;
+
$$\End(\phi) = \End(\L) = \{c\in C| c\L\subset \L\}$$
 +
and ${}_a\phi(C)=a^{-1}\L/\L \cong (A/(a))^r$ ($0\ne a\in A$). For $r=1$, there result bijections between the finite sets of:
  
b) ideal classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270278.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270279.png" />;
+
a) classes of rank-one $A$-lattices in $C$, up to scalars;
  
c) isomorphism classes of rank-one Drinfel'd modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270280.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270281.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270282.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270283.png" /> be the analytic subspace
+
b) ideal classes of $A$, i.e., $\def\Pic{\textrm{Pic}}\Pic(A)$;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270284.png" /></td> </tr></table>
+
c) isomorphism classes of rank-one Drinfel'd modules over $C$, i.e., $M((1))(C)$. For $r\ge 2$, let $\def\O{\Omega}\O^r$ be the analytic subspace
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270285.png" />. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270286.png" />, which is the Drinfel'd upper half-plane. The set (in fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270287.png" />-analytic space) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270288.png" />-valued points of the moduli scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270289.png" /> may now be described as a finite union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270290.png" /> of quotients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270291.png" /> by subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270292.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270293.png" /> commensurable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270294.png" />, in much the same way as one usually describes the moduli of elliptic curves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270295.png" />. In the standard example <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270296.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270297.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270298.png" />, one obtains the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270299.png" />-analytic isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270300.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270301.png" />. The left-hand mapping associates with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270302.png" /> the Drinfel'd module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270303.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270304.png" />, and the right-hand mapping is given by the modular invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270305.png" />. Writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270306.png" />, the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270308.png" /> become functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270309.png" />, in fact, modular forms for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270310.png" /> of respective weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270311.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270312.png" />. Moduli problems with non-trivial level structures correspond to subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270313.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270314.png" />, i.e., to modular curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270315.png" />, which are ramified covers of the above. As "classically" these curves may be studied function-theoretically via the modular forms for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270316.png" />. The same holds, more or less, for more general base rings than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270317.png" /> and for higher ranks <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270318.png" /> than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270319.png" />. Quite generally, the moduli schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270320.png" /> encode essential parts of the arithmetic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270321.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270322.png" />, as will be demonstrated by the examples below.
+
$$\{(w_1:\dots:w_r) | w_i\in C,\; K_\infty \textrm{-linearly independent}\}$$
 +
of $\def\P{\mathbb{P}}\P^{r-1}(C)$. Note that $\O^2 = \{(\omega_1/\omega_2)\} = C-K_\infty$, which is the Drinfel'd upper half-plane. The set (in fact, $C$-analytic space) of $C$-valued points of the moduli scheme $M^r(\fa)$ may now be described as a finite union $\bigcup\Gamma_i\backslash \O^r$ of quotients of $\O^r$ by subgroups $\Gamma_i$ of $\GL(r,K)$ commensurable with $\GL(r,A$, in much the same way as one usually describes the moduli of elliptic curves over $\mathbb{C}$. In the standard example $A=F_q[T]$, $r=2$, $\fa = (1)$, one obtains the $C$-analytic isomorphisms $\Gamma_i\backslash \O^2 \stackrel{\simeq}{\to}\M^2((1))(C)\stackrel{\simeq}{\to} C$, where $\Gamma=\GL(2,A)$. The left-hand mapping associates with $z\in \O^2$ the Drinfel'd module $\phi^\L$ with $\L=Az+A$, and the right-hand mapping is given by the modular invariant $j$. Writing $\phi_T^\L = T\tau^0+g(z)\tau+\Delta(z)\tau^2$, the coefficients $g$ and $\Delta$ become functions in $z\in\O$, in fact, modular forms for $\Gamma$ of respective weights $q-1$ and $q^2-1$. Moduli problems with non-trivial level structures correspond to subgroups $\Gamma'$ of $\Gamma$, i.e., to modular curves $\Gamma'\backslash\O^2$, which are ramified covers of the above. As ``classically'' these curves may be studied function-theoretically via the modular forms for $\Gamma'$. The same holds, more or less, for more general base rings than $A=F_q[T]$ and for higher ranks $r$ than $r=2$. Quite generally, the moduli schemes $M^r(\fa)$ encode essential parts of the arithmetic of $A$ and $K$, as will be demonstrated by the examples below.
  
 
==Applications.==
 
==Applications.==
  
  
===Explicit Abelian class field theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270323.png" />.===
+
===Explicit Abelian class field theory of $K$.===
Adjoining torsion points of rank-one Drinfel'd modules results in Abelian extensions of the base field. Applying this to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270324.png" />-Drinfel'd module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270325.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270326.png" /> (the so-called Carlitz module) yields all the Abelian extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270327.png" /> that are tamely ramified at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270328.png" />, similar to cyclotomic extensions of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270329.png" /> of rationals. This also works for general base rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270330.png" /> with class numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270331.png" />; here the situation resembles the theory of complex multiplication of elliptic curves [[#References|[a11]]] (cf. also [[Elliptic curve|Elliptic curve]]).
+
Adjoining torsion points of rank-one Drinfel'd modules results in Abelian extensions of the base field. Applying this to the $1$-Drinfel'd module $\phi:A=\F_q[t]\to K\{\tau\}$ defined by $\phi_T=T\tau^0+\tau$ (the so-called Carlitz module) yields all the Abelian extensions of $K=\F_q(T)$ that are tamely ramified at $\infty$, similar to cyclotomic extensions of the field $\Q$ of rationals. This also works for general base rings $A$ with class numbers $>1$; here the situation resembles the theory of complex multiplication of elliptic curves
 +
{{Cite|2}} (cf. also
 +
[[Elliptic curve|Elliptic curve]]).
  
===Langlands conjectures in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270332.png" />.===
+
===Langlands conjectures in characteristic $p$.===
The moduli scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270333.png" /> is equipped with an action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270334.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270335.png" /> is the ring of finite adèles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270336.png" />). It is a major problem to determine the representation type of the [[L-adic-cohomology|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270337.png" />-adic cohomology]] modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270338.png" />, i.e., to express them in terms of automorphic representations. This can partially be achieved and leads to (local or global) reciprocity laws between representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270339.png" /> and Galois representations (cf. also [[Galois theory|Galois theory]]). In particular, the local Langlands correspondence for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270340.png" /> over a local field of equal characteristic may so be proved [[#References|[a4]]], [[#References|[a13]]], [[#References|[a12]]].
+
The moduli scheme $\displaystyle\lim_\leftarrow M^r(\fa)$ is equipped with an action of $\GL(r,\mathfrak{A}_{K,f})$ (where $\mathfrak{A}_{K,f}$ is the ring of finite adèles of $K$). It is a major problem to determine the representation type of the
 +
[[L-adic-cohomology|$l$-adic cohomology]] modules $H_C^i(M^r\times\overline{K},\Q_l)$, i.e., to express them in terms of automorphic representations. This can partially be achieved and leads to (local or global) reciprocity laws between representations of $\GL(r)$ and Galois representations (cf. also
 +
[[Galois theory|Galois theory]]). In particular, the local Langlands correspondence for $\GL(r)$ over a local field of equal characteristic may so be proved
 +
{{Cite|Dr2}},
 +
{{Cite|LaRaSt}},
 +
{{Cite|La}}.
  
===Modularity conjecture over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270341.png" />.===
+
===Modularity conjecture over $K$.===
As a special case of the previous subsection, the Galois representations associated to elliptic curves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270342.png" /> may be found in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270343.png" />. This leads to a Shimura–Taniyama–Weil correspondence between elliptic curves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270344.png" /> with split multiplicative reduction at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270345.png" />, isogeny factors of dimension one of Jacobians of certain Drinfel'd modular curves and (effectively calculable) automorphic Hecke eigenforms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270346.png" /> [[#References|[a7]]].
+
As a special case of the previous subsection, the Galois representations associated to elliptic curves over $K$ may be found in $H_C^1(M^2\times\overline{K},\Q_l)$. This leads to a Shimura–Taniyama–Weil correspondence between elliptic curves over $K$ with split multiplicative reduction at $\infty$, isogeny factors of dimension one of Jacobians of certain Drinfel'd modular curves and (effectively calculable) automorphic Hecke eigenforms over $K$
 +
{{Cite|GeRe}}.
  
 
===Cohomology of arithmetic groups.===
 
===Cohomology of arithmetic groups.===
Invariants like Betti numbers, numbers of cusps, Euler–Poincaré-characteristics of subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270347.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270348.png" /> are related to the geometry of the moduli scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270349.png" />. In some cases (e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270350.png" />), these invariants may be determined using the theory of Drinfel'd modular forms [[#References|[a5]]].
+
Invariants like Betti numbers, numbers of cusps, Euler–Poincaré-characteristics of subgroups $\Gamma'$ of $\Gamma = \GL(r,A)$ are related to the geometry of the moduli scheme $\Gamma'\backslash\O^r$. In some cases (e.g., $r=2$), these invariants may be determined using the theory of Drinfel'd modular forms
 +
{{Cite|Ge}}.
  
 
===Arithmetic of division algebras.===
 
===Arithmetic of division algebras.===
Exploiting the structure of endomorphism rings of Drinfel'd modules over finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270351.png" />-fields and using knowledge of the moduli schemes, one can find formulas for class and type numbers of central division algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270352.png" /> [[#References|[a6]]].
+
Exploiting the structure of endomorphism rings of Drinfel'd modules over finite $A$-fields and using knowledge of the moduli schemes, one can find formulas for class and type numbers of central division algebras over $K$
 +
{{Cite|Ge2}}.
  
 
==Curves with many rational points.==
 
==Curves with many rational points.==
Drinfel'd modules provide explicit constructions of algebraic curves over finite fields with predictable properties. In particular, curves with many rational points compared to their genera may be tailored [[#References|[a14]]].
+
Drinfel'd modules provide explicit constructions of algebraic curves over finite fields with predictable properties. In particular, curves with many rational points compared to their genera may be tailored
 +
{{Cite|NiXi}}.
  
 
Other features and deep results in the field that definitely should be mentioned are the following:
 
Other features and deep results in the field that definitely should be mentioned are the following:
  
the transcendence theory of Drinfel'd modules, their periods, and special values of exponential lattice functions, mainly created by J. Yu [[#References|[a17]]];
+
the transcendence theory of Drinfel'd modules, their periods, and special values of exponential lattice functions, mainly created by J. Yu
 +
{{Cite|3}};
  
D. Goss has developed a theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270353.png" />-valued zeta- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270354.png" />-functions for Drinfel'd modules and similar objects [[#References|[a9]]];
+
D. Goss has developed a theory of $C$-valued zeta- and $L$-functions for Drinfel'd modules and similar objects
 +
{{Cite|}};
  
R. Pink has proved an analogue of the Tate conjecture (cf. also [[Tate conjectures|Tate conjectures]]) for Drinfel'd modules [[#References|[a15]]];
+
R. Pink has proved an analogue of the Tate conjecture (cf. also
 +
[[Tate conjectures|Tate conjectures]]) for Drinfel'd modules
 +
{{Cite|Pi}};
  
H.-G. Rück and U. Tipp have proved a Gross–Zagier-type formula for heights of Heegner points on Drinfel'd modular curves [[#References|[a16]]].
+
H.-G. Rück and U. Tipp have proved a Gross–Zagier-type formula for heights of Heegner points on Drinfel'd modular curves
 +
{{Cite|RüTi}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Bosch, U. Güntzer, R. Remmert, "Non-Archimedean analysis" , Springer (1984) {{MR|0746961}} {{ZBL|0539.14017}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Deligne, D. Husemöller, "Survey of Drinfel'd modules" ''Contemp. Math.'' , '''67''' (1987) pp. 25–91 {{MR|902591}} {{ZBL|0627.14026}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> V.G. Drinfel'd, "Elliptic modules" ''Math. USSR Sb.'' , '''23''' (1976) pp. 561–592 {{MR|}} {{ZBL|0386.20022}} {{ZBL|0363.20038}} {{ZBL|0321.14014}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> V.G. Drinfel'd, "Elliptic modules II" ''Math. USSR Sb.'' , '''31''' (1977) pp. 159–170 {{MR|}} {{ZBL|0386.20022}} {{ZBL|0363.20038}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E.-U. Gekeler, "Drinfeld modular curves" , ''Lecture Notes Math.'' , '''1231''' , Springer (1986) {{MR|0874338}} {{MR|0827352}} {{ZBL|0607.14020}} {{ZBL|0599.14032}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> E.-U. Gekeler, "On the arithmetic of some division algebras" ''Comment. Math. Helvetici'' , '''67''' (1992) pp. 316–333 {{MR|1161288}} {{ZBL|0753.11025}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> E.-U. Gekeler, M. Reversat, "Jacobians of Drinfeld modular curves" ''J. Reine Angew. Math.'' , '''476''' (1996) pp. 27–93 {{MR|1401696}} {{ZBL|0848.11029}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> "Drinfeld modules, modular schemes and applications" E.-U. Gekeler (ed.) M. van der Put (ed.) M. Reversat (ed.) J. van Geel (ed.) , World Sci. (1997) {{MR|1630594}} {{ZBL|0897.00023}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> D. Goss, "Basic structures of function field arithmetic" , Springer (1996) {{MR|1423131}} {{ZBL|0874.11004}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> "The arithmetic of function fields" D. Goss (ed.) D. Hayes (ed.) M. Rosen (ed.) , W. de Gruyter (1992) {{MR|1196508}} {{ZBL|0771.00031}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> D. Hayes, "Explicit class field theory in global function fields" , ''Studies Algebra and Number Th.'' , ''Adv. Math.'' , '''16''' (1980) pp. 173–217 {{MR|0535766}} {{ZBL|0476.12010}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> G. Laumon, "Cohomology of Drinfeld modular varieties I,II" , Cambridge Univ. Press (1996/7)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> G. Laumon, M. Rapoport, U. Stuhler, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120270/d120270355.png" />-elliptic sheaves and the Langlands correspondence" ''Invent. Math.'' , '''113''' (1993) pp. 217–338 {{MR|1228127}} {{ZBL|0809.11032}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> H. Niederreiter, C. Xing, "Cyclotomic function fields, Hilbert class fields, and global function fields with many rational places" ''Acta Arith.'' , '''79''' (1997) pp. 59–76 {{MR|1438117}} {{ZBL|0891.11057}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> R. Pink, "The Mumford–Tate conjecture for Drinfeld modules" ''Publ. RIMS Kyoto Univ.'' , '''33''' (1997) pp. 393–425 {{MR|1474696}} {{ZBL|0895.11025}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> H.-G. Rück, U. Tipp, "Heegner points and L-series of automorphic cusp forms of Drinfeld type" ''Preprint Essen'' (1998) {{MR|1787948}} {{ZBL|1012.11039}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> J. Yu, "Transcendence and Drinfeld modules" ''Invent. Math.'' , '''83''' (1986) pp. 507–517 {{MR|0827364}} {{ZBL|0644.12005}} {{ZBL|0586.12010}} </TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|2}}||valign="top"| D. Hayes, "Explicit class field theory in global function fields", ''Studies Algebra and Number Th.'', ''Adv. Math.'', '''16''' (1980) pp. 173–217 {{MR|0535766}} {{ZBL|0476.12010}}
 +
|-
 +
|valign="top"|{{Ref|3}}||valign="top"| J. Yu, "Transcendence and Drinfeld modules" ''Invent. Math.'', '''83''' (1986) pp. 507–517 {{MR|0827364}} {{ZBL|0644.12005}} {{ZBL|0586.12010}}
 +
|-
 +
|valign="top"|{{Ref|BoGüRe}}||valign="top"| S. Bosch, U. Güntzer, R. Remmert, "Non-Archimedean analysis", Springer (1984) {{MR|0746961}} {{ZBL|0539.14017}}
 +
|-
 +
|valign="top"|{{Ref|DeHu}}||valign="top"| P. Deligne, D. Husemöller, "Survey of Drinfel'd modules" ''Contemp. Math.'', '''67''' (1987) pp. 25–91 {{MR|902591}} {{ZBL|0627.14026}}
 +
|-
 +
|valign="top"|{{Ref|Dr}}||valign="top"| V.G. Drinfel'd, "Elliptic modules" ''Math. USSR Sb.'', '''23''' (1976) pp. 561–592 {{MR|}} {{ZBL|0386.20022}} {{ZBL|0363.20038}} {{ZBL|0321.14014}}
 +
|-
 +
|valign="top"|{{Ref|Dr2}}||valign="top"| V.G. Drinfel'd, "Elliptic modules II" ''Math. USSR Sb.'', '''31''' (1977) pp. 159–170 {{MR|}} {{ZBL|0386.20022}} {{ZBL|0363.20038}}
 +
|-
 +
|valign="top"|{{Ref|Ge}}||valign="top"| E.-U. Gekeler, "Drinfeld modular curves", ''Lecture Notes Math.'', '''1231''', Springer (1986) {{MR|0874338}} {{MR|0827352}} {{ZBL|0607.14020}} {{ZBL|0599.14032}}
 +
|-
 +
|valign="top"|{{Ref|Ge2}}||valign="top"| E.-U. Gekeler, "On the arithmetic of some division algebras" ''Comment. Math. Helvetici'', '''67''' (1992) pp. 316–333 {{MR|1161288}} {{ZBL|0753.11025}}
 +
|-
 +
|valign="top"|{{Ref|GePuReGeWoScGo}}||valign="top"| "Drinfeld modules, modular schemes and applications" E.-U. Gekeler (ed.) M. van der Put (ed.) M. Reversat (ed.) J. van Geel (ed.), World Sci. (1997) {{MR|1630594}} {{ZBL|0897.00023}}
 +
|-
 +
|valign="top"|{{Ref|GeRe}}||valign="top"| E.-U. Gekeler, M. Reversat, "Jacobians of Drinfeld modular curves" ''J. Reine Angew. Math.'', '''476''' (1996) pp. 27–93 {{MR|1401696}} {{ZBL|0848.11029}}
 +
|-
 +
|valign="top"|{{Ref|GoHaRoGrHa}}||valign="top"| "The arithmetic of function fields" D. Goss (ed.) D. Hayes (ed.) M. Rosen (ed.), W. de Gruyter (1992) {{MR|1196508}} {{ZBL|0771.00031}}
 +
|-
 +
|valign="top"|{{Ref|La}}||valign="top"| G. Laumon, "Cohomology of Drinfeld modular varieties I,II", Cambridge Univ. Press (1996/7)
 +
|-
 +
|valign="top"|{{Ref|LaRaSt}}||valign="top"| G. Laumon, M. Rapoport, U. Stuhler, "$\mathcal{D}$-elliptic sheaves and the Langlands correspondence" ''Invent. Math.'', '''113''' (1993) pp. 217–338 {{MR|1228127}} {{ZBL|0809.11032}}
 +
|-
 +
|valign="top"|{{Ref|NiXi}}||valign="top"| H. Niederreiter, C. Xing, "Cyclotomic function fields, Hilbert class fields, and global function fields with many rational places" ''Acta Arith.'', '''79''' (1997) pp. 59–76 {{MR|1438117}} {{ZBL|0891.11057}}
 +
|-
 +
|valign="top"|{{Ref|Pi}}||valign="top"| R. Pink, "The Mumford–Tate conjecture for Drinfeld modules" ''Publ. RIMS Kyoto Univ.'', '''33''' (1997) pp. 393–425 {{MR|1474696}} {{ZBL|0895.11025}}
 +
|-
 +
|valign="top"|{{Ref|RüTi}}||valign="top"| H.-G. Rück, U. Tipp, "Heegner points and L-series of automorphic cusp forms of Drinfeld type" ''Doc. Math., J. DMV '', '''5''' (2000) pp. 365-444 {{MR|1787948}} {{ZBL|1012.11039}}
 +
|-
 +
|}

Latest revision as of 20:11, 14 April 2012

2020 Mathematics Subject Classification: Primary: 11G09 Secondary: 11F7011R3922E55 [MSN][ZBL]

A Drinfel'd $A$-module, (where $A$ is an appropriate ring) over a field $L$ of characteristic $p>0$ is an exotic $A$-module structure on the additive group $\def\Ga{\mathcal{G}_\alpha}\Ga$ over $L$. In several regards, the concept of a Drinfel'd module is analogous to the concept of an elliptic curve (or more generally, of an irreducible Abelian variety), with which it shares many features. Among the similarities between Drinfel'd modules and elliptic curves are the respective structures of torsion points, of Tate modules and of endomorphism rings, the existence of analytic "Weierstrass uniformizations" , and the moduli theories (modular varieties, modular forms; cf. also Modular form). Many topics from the (classical and well-developed) theory of elliptic curves may be transferred to Drinfel'd modules, thereby revealing arithmetical information about the ground field $L$. On the other hand, since the mechanism of Drinfel'd modules is smoother and in some respects simpler than that of Abelian varieties, some results involving Drinfel'd modules over global function fields $L$ can be proved, whose analogues over number fields $L$ are far from being settled (e.g. parts of Stark's conjectures, of the Langlands conjectures, assertions about the arithmetical nature of zeta values and other questions of transcendence theory over $L$, cf. also $L$-function). The invention and basic theory as well as large parts of the deeper results about Drinfel'd modules are due to V.G. Drinfel'd [Dr], [Dr2]. General references are [DeHu], [GoHaRoGrHa], , and [GePuReGeWoScGo].

Algebraic theory.

Let $L$ be any field of characteristic $p>0$, with algebraic closure $\overline{L}$. The endomorphism ring $\def\End{\textrm{End}}\End_L(\Ga)$ of the additive group scheme $\Ga/L$ is the ring of additive polynomials $f(x)\in L[x]$, i.e., of polynomials satisfying $f(x+y)=f(x)+f(y)$, whose (non-commutative) multiplication is defined by insertion. Then

$$\End_L(\Ga)=\{\sum a_i\tau_p^i | a_i\in L\} = L\{\tau_p\}$$ is the twisted polynomial ring in $\tau_p=x^p$ with commutation rule $\tau_p \cdot a = a^p \cdot \tau_p$ for $a\in L$ and unit element $\tau_p^0 = x$. Fix a power $q=p^f$ of $p$. If $L$ contains the field $\F_q$ with $q$ elements, one sets $\tau=\tau_p^f = x^q$ and $L\{\tau\}$ for the subring of $\F_q$-linear polynomials in $\End_L(\Ga)$. For any $\F_q$-algebra $A$, an $A$-module structure on $\Ga/L$ is given by a morphism $\def\phi{\varphi}\phi$ of $\F_q$-algebras from $A$ to $L\{\tau\}$.

Fix a (smooth, projective, geometrically connected) algebraic curve $\def\C{\mathcal{C}}\C$ over $\F_q$ and a place $\infty$ of $\C$; let $K$ be its function field and $A$ the affine ring of $\C\setminus\{\infty\}$. (Here, "places" , or "primes" , are closed points of $\C$, the set of normalized valuations on $K$; cf. also Norm on a field.) Hence $K$ is a function field in one variable over $\F_q$ and $A$ is its subring of elements regular away from $\infty$. Put $\def\deg{\mathrm{deg}}\deg:A\to \Z$ for the associated degree function: $\deg\; a = \dim_{\F_q} A/(a)$ if $a\ne 0$. Let $L$ be a field equipped with a structure $\def\g{\gamma}\g:A\to L$ of an $A$-algebra. Then $L$ is either an extension of $K$ or of some $\def\fp{\mathfrak{p}}A/\fp$, where $\fp$ is a maximal ideal. One writes $\def\char{\mathrm{char}}\char_A(L) = \infty$ in the former and $\char_A(L) = \fp$ in the latter case. A Drinfel'd $A$-module of rank $r$ over $L$ (briefly, an $r$-Drinfel'd module over $L$) is a morphism of $\F_q$-algebras

$$\phi : A \to L\{\tau\},$$

$$a\mapsto\phi_a = \sum_{i=0}^{r\;\deg\;a} l_i(a)\tau^i$$ subject to:

i) $l_0(a)=\gamma(a)$; and

ii) $l_{r\;\deg\;a}\ne 0$ for $a\in A$. It supplies the additive group $L'$ of each $L$-algebra $L'$ with the structure of an abstract $A$-module. A morphism $u:\phi\to\phi'$ of Drinfel'd $A$-modules over $L$ is some element $u$ of $L\{\tau\}$ that satisfies $\phi'_a\cdot u = u\cdot \phi_a$ for $a\in A$. Similarly, one defines iso-, endo- and automorphisms.

The standard example of $(K,\infty,A)$ is given by a rational function field $K=\F_q(T),$, $\infty$ being the usual place at infinity, $A=\F_q[T]$. In that case, a Drinfel'd $A$-module $\phi$ of rank $r$ over $L$ is given by $\phi_T = \g(T)\tau^0 + l_1\tau+\cdots+l_r\tau^r$, where the $l_i=l_i(T)$ may be arbitrarily chosen in $L$ ($l_r\ne 0$). More generally, if $A$ is generated over $\F_q$ by $\{a_1,\dots,a_n\}$, $\phi$ is given by the $\phi_{a_i}$ that in $L\{\tau\}$ must satisfy the same relations as do the $a_i$ in $A$. Writing down a Drinfel'd module amounts to solving a complicated system of polynomial equations over $L$. For example, let $A=\F_2[U,V]$ with $V^2+V = U^3+U+1$. From $\deg\; U = 2$, $\deg\;V = 3$, one obtains for a $1$-Drinfel'd module $\phi$ over $L=\def\K{\overline{K}}\K=\textrm{ algebraic closure}(K)$: $\phi_V^2 + \phi_V = \phi_U^3+\phi_U+1$ with $\phi_U = UX+aX^2+bX^4$, $\phi_V = VX+cX^2+dX^4+eX^6$. Using computation rules in $\K\{\tau\}$, one solves for $a=U^2+U$, $b=e=1$, $c=V^2+V$, $d=U(V^2+V)$, which yields the unique (up to isomorphism) $1$-Drinfel'd module $\phi$ over $\K$. That $\phi$ is unique and even definable over $K$ corresponds to the fact that the class number of $A$ equals $1$.

Let $a\in A$ with $\deg\; a>0$ and let $\phi$ be an $r$-Drinfel'd module over $L$. The polynomial $\phi_a(X)$ has degree $q^{r\;\deg\;a}$ in $X$, whence has $q^{r\;\deg\;a}$ different roots in $\def\L{\overline{L}}\L$ if $(a,\char_A(L)) = 1$. This implies that the $A$-module of $a$-torsion points ${}_a\phi(\L) = \{x\in\L | \phi_a(x) = 0 \}$ of $\phi$ is isomorphic with $(A/(a))^r$. Similar, but more complicated assertions hold if one considers $\def\fa{\mathfrak{a}}\fa$-torsion points ($\fa$ a not necessarily principal ideal of $A$) and if $\char_A(L)$ divides $\fa$. A level-$\fa$ structure on $\phi$ is the choice of an isomorphism $\alpha(A/\fa)^r\stackrel{\simeq}{\to}{}_\fa\phi(L)$ of abstract $A$-modules (with some modification if $\char_A(L)$ divides $\fa$).

The definitions of Drinfel'd modules, their morphisms, torsion points, and level structures generalize to arbitrary $A$-schemes $S$ (instead of $S=\def\Spec{\mathrm{Spec}\;}\Spec L$, which corresponds to the case above; cf. also Scheme). Intuitively, an $r$-Drinfel'd module over $S$ is a family of $r$-Drinfel'd modules varying continuously over $S$. Let $\fa\subset A$ be a non-vanishing ideal. On the category of $A$-schemes $S$, there is the contravariant functor $\def\M{\mathcal{M}}\M^r(\fa)$ that to each $S$ associates the set of isomorphism classes of $r$-Drinfel'd modules over $S$ provided with a level-$\fa$ structure. If $\fa$ has at least two prime divisors (such $\fa$ are admissible), $\M^r(\fa)$ is representable by a moduli scheme $M^r(\fa)$. In other words, $A$-morphisms from $S$ to $M^r(\fa)$ correspond one-to-one to isomorphism classes of $r$-Drinfel'd modules over $S$ with a level-$\fa$ structure. The various $\M^r(\fa)$ and $M^r(\fa)$ are equipped with actions of the finite groups $\def\GL{\textrm{GL}}\GL(r,A/\fa)$ and related by morphisms $\def\fb{\mathfrak{b}}M^r(\fb)\to M^r(\fa)$ if $\fa|\fb$. Taking quotients, this allows one to define coarse moduli schemes $M^r(\fa)$ even for non-admissible ideals $\fa$, and for more general moduli problems, e.g., the problem ``rank-r Drinfel'd A-modules with a point of order a A . For such coarse moduli schemes, the above bijection between morphisms from $S$ to $M^r(\fa)$ and objects of the moduli problem holds only if $S$ is the spectrum of an algebraically closed field. If $\fa$ is admissible, $M^r(\fa)$ is affine, smooth, of finite type and of dimension $r-1$ over $A$. Furthermore, for $\fa|\fb$, the morphisms from $M^r(\fb)$ to $M^r(\fa)$ are finite and flat, and even étale outside the support of $\fb\fa^{-1}$ (cf. also Affine morphism; Flat morphism; Etale morphism). As an example, take $A=\F_q[T]$, and let $L$ be algebraically closed. Two $r$-Drinfel'd modules $\phi$ and $\phi'$ over $L$, given through the coefficients $l_i$ and $l_i'$ ($1\le i\le r$) of $\phi_T$ and $\phi_T'$, are isomorphic if and only if there exists a $c\in L^*$ such that $l_i' = c^{1-q^i}l_i$. Hence $\M^r((1))$, the moduli scheme attached to the trivial ideal $(1)$ of $A$, is the open subscheme defined by $l_r\ne 0$ of $\def\G{\mathcal{G}}\Spec A[l_1,\dots,l_r]/\G_m$, where the multiplicative group acts diagonally through $c(l_1,\dots,l_r) = (\dots,c^{1-q^i}l_i,\dots)$. If $r=2$, $M^2((1)) = \Spec A[j]$ with the ``modular invariant $j=l_1^{q+1}/l_2$, the $l_i$ being regarded as indeterminates.

Analytic theory.

Let $K_\infty$ be the completion of $K$ at $\infty$, with normalized absolute value $|\;.\;|_\infty$ and complete algebraic closure $C$. Then $C$ is the smallest field extension of $K$ which is complete with respect to $|\;.\;|_\infty$ and algebraically closed. For such fields, there is a reasonable function theory and analytic geometry [BoGüRe].

An $A$-lattice is a finitely generated (thus projective) $A$-submodule $\def\L{\Lambda}\L$ of $C$ that has finite intersection with each ball in $C$. With $\L$ is associated its exponential function $e_\L:C\to C$, defined as the everywhere convergent infinite product $e_\L(z) = z \prod(1-z/\lambda)$ ($0\ne \lambda\in \L$). It is a surjective, $\F_q$-linear and $\L$-periodic function that for each $a\in A$ satisfies a functional equation $e_\L(az) = \phi_a^\L(e_\L(z))$ with some $\phi_a^\L\in C\{\tau\}$. The rule $a\mapsto\phi_a^\L$ defines a ring homomorphism from $A$ to $C\{\tau\}$, in fact, a Drinfel'd $A$-module of rank $r$, $r$ being the projective rank of the $A$-module $\L$. Each $r$-Drinfel'd module over $C$ is so obtained, and $\L\mapsto\phi^\L$ yields an equivalence of the category of lattices of projective rank $r$ with the category of $r$-Drinfel'd modules over $C$. (A morphism of lattices $c:\L\mapsto\L'$ is some $c\in C$ such that $c\L\subset\L'$.) The description of $\phi=\phi^\L$ through the lattice $\L$ is called the Weierstrass uniformization. From $\L$, one can read off many of the properties of $\phi$. E.g.,

$$\End(\phi) = \End(\L) = \{c\in C| c\L\subset \L\}$$ and ${}_a\phi(C)=a^{-1}\L/\L \cong (A/(a))^r$ ($0\ne a\in A$). For $r=1$, there result bijections between the finite sets of:

a) classes of rank-one $A$-lattices in $C$, up to scalars;

b) ideal classes of $A$, i.e., $\def\Pic{\textrm{Pic}}\Pic(A)$;

c) isomorphism classes of rank-one Drinfel'd modules over $C$, i.e., $M((1))(C)$. For $r\ge 2$, let $\def\O{\Omega}\O^r$ be the analytic subspace

$$\{(w_1:\dots:w_r) | w_i\in C,\; K_\infty \textrm{-linearly independent}\}$$ of $\def\P{\mathbb{P}}\P^{r-1}(C)$. Note that $\O^2 = \{(\omega_1/\omega_2)\} = C-K_\infty$, which is the Drinfel'd upper half-plane. The set (in fact, $C$-analytic space) of $C$-valued points of the moduli scheme $M^r(\fa)$ may now be described as a finite union $\bigcup\Gamma_i\backslash \O^r$ of quotients of $\O^r$ by subgroups $\Gamma_i$ of $\GL(r,K)$ commensurable with $\GL(r,A$, in much the same way as one usually describes the moduli of elliptic curves over $\mathbb{C}$. In the standard example $A=F_q[T]$, $r=2$, $\fa = (1)$, one obtains the $C$-analytic isomorphisms $\Gamma_i\backslash \O^2 \stackrel{\simeq}{\to}\M^2((1))(C)\stackrel{\simeq}{\to} C$, where $\Gamma=\GL(2,A)$. The left-hand mapping associates with $z\in \O^2$ the Drinfel'd module $\phi^\L$ with $\L=Az+A$, and the right-hand mapping is given by the modular invariant $j$. Writing $\phi_T^\L = T\tau^0+g(z)\tau+\Delta(z)\tau^2$, the coefficients $g$ and $\Delta$ become functions in $z\in\O$, in fact, modular forms for $\Gamma$ of respective weights $q-1$ and $q^2-1$. Moduli problems with non-trivial level structures correspond to subgroups $\Gamma'$ of $\Gamma$, i.e., to modular curves $\Gamma'\backslash\O^2$, which are ramified covers of the above. As ``classically these curves may be studied function-theoretically via the modular forms for $\Gamma'$. The same holds, more or less, for more general base rings than $A=F_q[T]$ and for higher ranks $r$ than $r=2$. Quite generally, the moduli schemes $M^r(\fa)$ encode essential parts of the arithmetic of $A$ and $K$, as will be demonstrated by the examples below.

Applications.

Explicit Abelian class field theory of $K$.

Adjoining torsion points of rank-one Drinfel'd modules results in Abelian extensions of the base field. Applying this to the $1$-Drinfel'd module $\phi:A=\F_q[t]\to K\{\tau\}$ defined by $\phi_T=T\tau^0+\tau$ (the so-called Carlitz module) yields all the Abelian extensions of $K=\F_q(T)$ that are tamely ramified at $\infty$, similar to cyclotomic extensions of the field $\Q$ of rationals. This also works for general base rings $A$ with class numbers $>1$; here the situation resembles the theory of complex multiplication of elliptic curves [2] (cf. also Elliptic curve).

Langlands conjectures in characteristic $p$.

The moduli scheme $\displaystyle\lim_\leftarrow M^r(\fa)$ is equipped with an action of $\GL(r,\mathfrak{A}_{K,f})$ (where $\mathfrak{A}_{K,f}$ is the ring of finite adèles of $K$). It is a major problem to determine the representation type of the $l$-adic cohomology modules $H_C^i(M^r\times\overline{K},\Q_l)$, i.e., to express them in terms of automorphic representations. This can partially be achieved and leads to (local or global) reciprocity laws between representations of $\GL(r)$ and Galois representations (cf. also Galois theory). In particular, the local Langlands correspondence for $\GL(r)$ over a local field of equal characteristic may so be proved [Dr2], [LaRaSt], [La].

Modularity conjecture over $K$.

As a special case of the previous subsection, the Galois representations associated to elliptic curves over $K$ may be found in $H_C^1(M^2\times\overline{K},\Q_l)$. This leads to a Shimura–Taniyama–Weil correspondence between elliptic curves over $K$ with split multiplicative reduction at $\infty$, isogeny factors of dimension one of Jacobians of certain Drinfel'd modular curves and (effectively calculable) automorphic Hecke eigenforms over $K$ [GeRe].

Cohomology of arithmetic groups.

Invariants like Betti numbers, numbers of cusps, Euler–Poincaré-characteristics of subgroups $\Gamma'$ of $\Gamma = \GL(r,A)$ are related to the geometry of the moduli scheme $\Gamma'\backslash\O^r$. In some cases (e.g., $r=2$), these invariants may be determined using the theory of Drinfel'd modular forms [Ge].

Arithmetic of division algebras.

Exploiting the structure of endomorphism rings of Drinfel'd modules over finite $A$-fields and using knowledge of the moduli schemes, one can find formulas for class and type numbers of central division algebras over $K$ [Ge2].

Curves with many rational points.

Drinfel'd modules provide explicit constructions of algebraic curves over finite fields with predictable properties. In particular, curves with many rational points compared to their genera may be tailored [NiXi].

Other features and deep results in the field that definitely should be mentioned are the following:

the transcendence theory of Drinfel'd modules, their periods, and special values of exponential lattice functions, mainly created by J. Yu [3];

D. Goss has developed a theory of $C$-valued zeta- and $L$-functions for Drinfel'd modules and similar objects

R. Pink has proved an analogue of the Tate conjecture (cf. also Tate conjectures) for Drinfel'd modules [Pi];

H.-G. Rück and U. Tipp have proved a Gross–Zagier-type formula for heights of Heegner points on Drinfel'd modular curves [RüTi].

References

[2] D. Hayes, "Explicit class field theory in global function fields", Studies Algebra and Number Th., Adv. Math., 16 (1980) pp. 173–217 MR0535766 Zbl 0476.12010
[3] J. Yu, "Transcendence and Drinfeld modules" Invent. Math., 83 (1986) pp. 507–517 MR0827364 Zbl 0644.12005 Zbl 0586.12010
[BoGüRe] S. Bosch, U. Güntzer, R. Remmert, "Non-Archimedean analysis", Springer (1984) MR0746961 Zbl 0539.14017
[DeHu] P. Deligne, D. Husemöller, "Survey of Drinfel'd modules" Contemp. Math., 67 (1987) pp. 25–91 MR902591 Zbl 0627.14026
[Dr] V.G. Drinfel'd, "Elliptic modules" Math. USSR Sb., 23 (1976) pp. 561–592 Zbl 0386.20022 Zbl 0363.20038 Zbl 0321.14014
[Dr2] V.G. Drinfel'd, "Elliptic modules II" Math. USSR Sb., 31 (1977) pp. 159–170 Zbl 0386.20022 Zbl 0363.20038
[Ge] E.-U. Gekeler, "Drinfeld modular curves", Lecture Notes Math., 1231, Springer (1986) MR0874338 MR0827352 Zbl 0607.14020 Zbl 0599.14032
[Ge2] E.-U. Gekeler, "On the arithmetic of some division algebras" Comment. Math. Helvetici, 67 (1992) pp. 316–333 MR1161288 Zbl 0753.11025
[GePuReGeWoScGo] "Drinfeld modules, modular schemes and applications" E.-U. Gekeler (ed.) M. van der Put (ed.) M. Reversat (ed.) J. van Geel (ed.), World Sci. (1997) MR1630594 Zbl 0897.00023
[GeRe] E.-U. Gekeler, M. Reversat, "Jacobians of Drinfeld modular curves" J. Reine Angew. Math., 476 (1996) pp. 27–93 MR1401696 Zbl 0848.11029
[GoHaRoGrHa] "The arithmetic of function fields" D. Goss (ed.) D. Hayes (ed.) M. Rosen (ed.), W. de Gruyter (1992) MR1196508 Zbl 0771.00031
[La] G. Laumon, "Cohomology of Drinfeld modular varieties I,II", Cambridge Univ. Press (1996/7)
[LaRaSt] G. Laumon, M. Rapoport, U. Stuhler, "$\mathcal{D}$-elliptic sheaves and the Langlands correspondence" Invent. Math., 113 (1993) pp. 217–338 MR1228127 Zbl 0809.11032
[NiXi] H. Niederreiter, C. Xing, "Cyclotomic function fields, Hilbert class fields, and global function fields with many rational places" Acta Arith., 79 (1997) pp. 59–76 MR1438117 Zbl 0891.11057
[Pi] R. Pink, "The Mumford–Tate conjecture for Drinfeld modules" Publ. RIMS Kyoto Univ., 33 (1997) pp. 393–425 MR1474696 Zbl 0895.11025
[RüTi] H.-G. Rück, U. Tipp, "Heegner points and L-series of automorphic cusp forms of Drinfeld type" Doc. Math., J. DMV , 5 (2000) pp. 365-444 MR1787948 Zbl 1012.11039
How to Cite This Entry:
Drinfel'd module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Drinfel%27d_module&oldid=23817
This article was adapted from an original article by E.-U. Gekeler (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article