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{{MSC|11G09|11F70,11R39,22E55}}
{{TEX|done}}

A Drinfel'd $A$-module, (where $A$ is an appropriate
[[Ring|ring]]) over a
[[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
[[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 $L$. 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|$L$-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
{{Cite|Dr}},
{{Cite|Dr2}}. General references are
{{Cite|DeHu}},
{{Cite|GoHaRoGrHa}},
{{Cite|}}, and
{{Cite|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|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|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

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

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

i) $l_0(a)=a$; 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|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|category]] of $A$-schemes $S$, there is the contravariant
[[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
[[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.==
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 $A$-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.,

$$\End(\phi) = \End(\L) = \{c\in C| c\L\subset \L\}$$
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
{{Cite|2}} (cf. also
[[Elliptic curve|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|$l$-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 $K$.===
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 $K$
{{Cite|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
{{Cite|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 $K$
{{Cite|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
{{Cite|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
{{Cite|3}};

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
{{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
{{Cite|RüTi}}.

====References====
{|
|-
|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}}
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