Difference between revisions of "Drinfel'd module"
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− | <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" '' | + | <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" ''Doc. Math., J. DMV '', '''5''' (2000) pp. 365-444 {{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> |
Revision as of 21:49, 13 April 2012
A Drinfel'd -module, (where
is an appropriate ring) over a field
of characteristic
is an exotic
-module structure on the additive group
over
. 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
. 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
can be proved, whose analogues over number fields
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
, cf. also
-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 [a3], [a4]. General references are [a2], [a10], [a9], and [a8].
Algebraic theory.
Let be any field of characteristic
, with algebraic closure
. The endomorphism ring
of the additive group scheme
is the ring of additive polynomials
, i.e., of polynomials satisfying
, whose (non-commutative) multiplication is defined by insertion. Then
![]() |
is the twisted polynomial ring in with commutation rule
for
and unit element
. Fix a power
of
. If
contains the field
with
elements, one sets
and
for the subring of
-linear polynomials in
. For any
-algebra
, an
-module structure on
is given by a morphism
of
-algebras from
to
.
Fix a (smooth, projective, geometrically connected) algebraic curve over
and a place "∞" of
; let
be its function field and
the affine ring of
. (Here, "places" , or "primes" , are closed points of
, the set of normalized valuations on
; cf. also Norm on a field.) Hence
is a function field in one variable over
and
is its subring of elements regular away from
. Put
for the associated degree function:
if
. Let
be a field equipped with a structure
of an
-algebra. Then
is either an extension of
or of some
, where
is a maximal ideal. One writes
in the former and
in the latter case. A Drinfel'd
-module of rank
over
(briefly, an
-Drinfel'd module over
) is a morphism of
-algebras
![]() |
![]() |
subject to:
i) ; and
ii) for
. It supplies the additive group
of each
-algebra
with the structure of an abstract
-module. A morphism
of Drinfel'd
-modules over
is some element
of
that satisfies
for
. Similarly, one defines iso-, endo- and automorphisms.
The standard example of is given by a rational function field
,
being the usual place at infinity,
. In that case, a Drinfel'd
-module
of rank
over
is given by
, where the
may be arbitrarily chosen in
(
). More generally, if
is generated over
by
,
is given by the
that in
must satisfy the same relations as do the
in
. Writing down a Drinfel'd module amounts to solving a complicated system of polynomial equations over
. For example, let
with
. From
,
, one obtains for a
-Drinfel'd module
over
:
with
,
. Using computation rules in
, one solves for
,
,
,
, which yields the unique (up to isomorphism)
-Drinfel'd module
over
. That
is unique and even definable over
corresponds to the fact that the class number of
equals
.
Let with
and let
be an
-Drinfel'd module over
. The polynomial
has degree
in
, whence has
different roots in
if
. This implies that the
-module of
-torsion points
of
is isomorphic with
. Similar, but more complicated assertions hold if one considers
-torsion points (
a not necessarily principal ideal of
) and if
divides
. A level-
structure on
is the choice of an isomorphism
of abstract
-modules (with some modification if
divides
).
The definitions of Drinfel'd modules, their morphisms, torsion points, and level structures generalize to arbitrary -schemes
(instead of
, which corresponds to the case above; cf. also Scheme). Intuitively, an
-Drinfel'd module over
is a family of
-Drinfel'd modules varying continuously over
. Let
be a non-vanishing ideal. On the category of
-schemes
, there is the contravariant functor
that to each
associates the set of isomorphism classes of
-Drinfel'd modules over
provided with a level-
structure. If
has at least two prime divisors (such
are admissible),
is representable by a moduli scheme
. In other words,
-morphisms from
to
correspond one-to-one to isomorphism classes of
-Drinfel'd modules over
with a level-
structure. The various
and
are equipped with actions of the finite groups
and related by morphisms
if
. Taking quotients, this allows one to define coarse moduli schemes
even for non-admissible ideals
, 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
to
and objects of the moduli problem holds only if
is the spectrum of an algebraically closed field. If
is admissible,
is affine, smooth, of finite type and of dimension
over
. Furthermore, for
, the morphisms from
to
are finite and flat, and even étale outside the support of
(cf. also Affine morphism; Flat morphism; Etale morphism). As an example, take
, and let
be algebraically closed. Two
-Drinfel'd modules
and
over
, given through the coefficients
and
(
) of
and
, are isomorphic if and only if there exists a
such that
. Hence
, the moduli scheme attached to the trivial ideal
of
, is the open subscheme defined by
of
, where the multiplicative group acts diagonally through
. If
,
with the "modular invariant"
, the
being regarded as indeterminates.
Analytic theory.
Let be the completion of
at
, with normalized absolute value
and complete algebraic closure
. Then
is the smallest field extension of
which is complete with respect to
and algebraically closed. For such fields, there is a reasonable function theory and analytic geometry [a1].
An -lattice is a finitely generated (thus projective)
-submodule
of
that has finite intersection with each ball in
. With
is associated its exponential function
, defined as the everywhere convergent infinite product
(
). It is a surjective,
-linear and
-periodic function that for each
satisfies a functional equation
with some
. The rule
defines a ring homomorphism from
to
, in fact, a Drinfel'd
-module of rank
,
being the projective rank of the
-module
. Each
-Drinfel'd module over
is so obtained, and
yields an equivalence of the category of lattices of projective rank
with the category of
-Drinfel'd modules over
. (A morphism of lattices
is some
such that
.) The description of
through the lattice
is called the Weierstrass uniformization. From
, one can read off many of the properties of
. E.g.,
![]() |
and (
). For
, there result bijections between the finite sets of:
a) classes of rank-one -lattices in
, up to scalars;
b) ideal classes of , i.e.,
;
c) isomorphism classes of rank-one Drinfel'd modules over , i.e.,
. For
, let
be the analytic subspace
![]() |
of . Note that
, which is the Drinfel'd upper half-plane. The set (in fact,
-analytic space) of
-valued points of the moduli scheme
may now be described as a finite union
of quotients of
by subgroups
of
commensurable with
, in much the same way as one usually describes the moduli of elliptic curves over
. In the standard example
,
,
, one obtains the
-analytic isomorphisms
, where
. The left-hand mapping associates with
the Drinfel'd module
with
, and the right-hand mapping is given by the modular invariant
. Writing
, the coefficients
and
become functions in
, in fact, modular forms for
of respective weights
and
. Moduli problems with non-trivial level structures correspond to subgroups
of
, i.e., to modular curves
, which are ramified covers of the above. As "classically" these curves may be studied function-theoretically via the modular forms for
. The same holds, more or less, for more general base rings than
and for higher ranks
than
. Quite generally, the moduli schemes
encode essential parts of the arithmetic of
and
, as will be demonstrated by the examples below.
Applications.
Explicit Abelian class field theory of
.
Adjoining torsion points of rank-one Drinfel'd modules results in Abelian extensions of the base field. Applying this to the -Drinfel'd module
defined by
(the so-called Carlitz module) yields all the Abelian extensions of
that are tamely ramified at
, similar to cyclotomic extensions of the field
of rationals. This also works for general base rings
with class numbers
; here the situation resembles the theory of complex multiplication of elliptic curves [a11] (cf. also Elliptic curve).
Langlands conjectures in characteristic
.
The moduli scheme is equipped with an action of
(where
is the ring of finite adèles of
). It is a major problem to determine the representation type of the
-adic cohomology modules
, 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
and Galois representations (cf. also Galois theory). In particular, the local Langlands correspondence for
over a local field of equal characteristic may so be proved [a4], [a13], [a12].
Modularity conjecture over
.
As a special case of the previous subsection, the Galois representations associated to elliptic curves over may be found in
. This leads to a Shimura–Taniyama–Weil correspondence between elliptic curves over
with split multiplicative reduction at
, isogeny factors of dimension one of Jacobians of certain Drinfel'd modular curves and (effectively calculable) automorphic Hecke eigenforms over
[a7].
Cohomology of arithmetic groups.
Invariants like Betti numbers, numbers of cusps, Euler–Poincaré-characteristics of subgroups of
are related to the geometry of the moduli scheme
. In some cases (e.g.,
), these invariants may be determined using the theory of Drinfel'd modular forms [a5].
Arithmetic of division algebras.
Exploiting the structure of endomorphism rings of Drinfel'd modules over finite -fields and using knowledge of the moduli schemes, one can find formulas for class and type numbers of central division algebras over
[a6].
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 [a14].
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 [a17];
D. Goss has developed a theory of -valued zeta- and
-functions for Drinfel'd modules and similar objects [a9];
R. Pink has proved an analogue of the Tate conjecture (cf. also Tate conjectures) for Drinfel'd modules [a15];
H.-G. Rück and U. Tipp have proved a Gross–Zagier-type formula for heights of Heegner points on Drinfel'd modular curves [a16].
References
[a1] | S. Bosch, U. Güntzer, R. Remmert, "Non-Archimedean analysis" , Springer (1984) MR0746961 Zbl 0539.14017 |
[a2] | P. Deligne, D. Husemöller, "Survey of Drinfel'd modules" Contemp. Math. , 67 (1987) pp. 25–91 MR902591 Zbl 0627.14026 |
[a3] | V.G. Drinfel'd, "Elliptic modules" Math. USSR Sb. , 23 (1976) pp. 561–592 Zbl 0386.20022 Zbl 0363.20038 Zbl 0321.14014 |
[a4] | V.G. Drinfel'd, "Elliptic modules II" Math. USSR Sb. , 31 (1977) pp. 159–170 Zbl 0386.20022 Zbl 0363.20038 |
[a5] | E.-U. Gekeler, "Drinfeld modular curves" , Lecture Notes Math. , 1231 , Springer (1986) MR0874338 MR0827352 Zbl 0607.14020 Zbl 0599.14032 |
[a6] | E.-U. Gekeler, "On the arithmetic of some division algebras" Comment. Math. Helvetici , 67 (1992) pp. 316–333 MR1161288 Zbl 0753.11025 |
[a7] | E.-U. Gekeler, M. Reversat, "Jacobians of Drinfeld modular curves" J. Reine Angew. Math. , 476 (1996) pp. 27–93 MR1401696 Zbl 0848.11029 |
[a8] | "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 |
[a9] | D. Goss, "Basic structures of function field arithmetic" , Springer (1996) MR1423131 Zbl 0874.11004 |
[a10] | "The arithmetic of function fields" D. Goss (ed.) D. Hayes (ed.) M. Rosen (ed.) , W. de Gruyter (1992) MR1196508 Zbl 0771.00031 |
[a11] | 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 |
[a12] | G. Laumon, "Cohomology of Drinfeld modular varieties I,II" , Cambridge Univ. Press (1996/7) |
[a13] | G. Laumon, M. Rapoport, U. Stuhler, "![]() |
[a14] | 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 |
[a15] | R. Pink, "The Mumford–Tate conjecture for Drinfeld modules" Publ. RIMS Kyoto Univ. , 33 (1997) pp. 393–425 MR1474696 Zbl 0895.11025 |
[a16] | 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 |
[a17] | J. Yu, "Transcendence and Drinfeld modules" Invent. Math. , 83 (1986) pp. 507–517 MR0827364 Zbl 0644.12005 Zbl 0586.12010 |
Drinfel'd module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Drinfel%27d_module&oldid=23817