In
mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
over a ring of functions on a curve over a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of
complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
theory. A shtuka (also called F-sheaf or chtouca) is a sort of generalization of a Drinfeld module, consisting roughly of a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it.
Drinfeld modules were introduced by , who used them to prove the
Langlands conjectures
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic nu ...
for GL
2 of an
algebraic function field
In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebrai ...
in some special cases. He later invented shtukas and used shtukas of rank 2 to prove
the remaining cases of the Langlands conjectures for GL
2.
Laurent Lafforgue
Laurent Lafforgue (; born 6 November 1966) is a French mathematician. He has made outstanding contributions to Langlands' program in the fields of number theory and analysis, and in particular proved the Langlands conjectures for the automorphism ...
proved the Langlands conjectures for GL
''n'' of a function field by studying the
moduli stack
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such space ...
of shtukas of rank ''n''.
"Shtuka" is a Russian word штука meaning "a single copy", which comes from the German noun “Stück”, meaning “piece, item, or unit". In Russian, the word "shtuka" is also used in slang for a thing with known properties, but having no name in a speaker's mind.
Drinfeld modules
The ring of additive polynomials
We let
be a field of characteristic
. The ring
is defined to be the ring of ''noncommutative'' (or twisted)
polynomials over
, with the multiplication given by
:
The element
can be thought of as a
Frobenius element
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism m ...
: in fact,
is a left module over
, with elements of
acting as multiplication and
acting as the Frobenius endomorphism of
. The ring
can also be thought of as the ring of all (absolutely) additive polynomials
:
in