Drinfeld module

Drinfeld modules were introduced by Drinfeld (1974), who used them to prove the Langlands conjectures for GL₂ of an algebraic function field in some special cases. He later invented shtukas and used shtukas of rank 2 to prove the remaining cases of the Langlands conjectures for GL₂. Laurent Lafforgue proved the Langlands conjectures for GL_n of a function field by studying the moduli stack 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 ${\displaystyle L}$ be a field of characteristic ${\displaystyle p>0}$. The ring ${\displaystyle L\{\tau \}}$ is defined to be the ring of noncommutative (or twisted) polynomials ${\displaystyle a_0+a_1\tau +a_2{\tau }^2+\cdots }$ over ${\displaystyle L}$, with the multiplication given by

${\displaystyle \tau a=a^p\tau ,a\in L.}$

The element ${\displaystyle \tau }$ can be thought of as a Frobenius element: in fact, ${\displaystyle L}$ is a left module over ${\displaystyle L\{\tau \}}$, with elements of ${\displaystyle L}$ acting as multiplication and ${\displaystyle \tau }$ acting as the Frobenius endomorphism of ${\displaystyle L}$. The ring ${\displaystyle L\{\tau \}}$ can also be thought of as the ring of all (absolutely) additive polynomials

${\displaystyle a_{0}x+a_1{x}^p+a_2{x}^{p^2}+\cdots =a_0{\tau }^0+a_1\tau +a_2{\tau }^2+\cdots }$

in ${\displaystyle L[x]}$, where a polynomial ${\displaystyle f}$ is called additive if ${\displaystyle f(x+y)=f(x)+f(y)}$ (as elements of ${\displaystyle L[x,y]}$). The ring of additive polynomials is generated as an algebra over ${\displaystyle L}$ by the polynomial ${\displaystyle \tau =x^p}$. The multiplication in the ring of additive polynomials is given by composition of polynomials, not by multiplication of commutative polynomials, and is not commutative.

Definition of Drinfeld modules

Let F be an algebraic function field with a finite field of constants and fix a place ${\displaystyle \mathrm{\infty }}$ of F. Define A to be the ring of elements in F that are regular at every place except possibly ${\displaystyle \mathrm{\infty }}$. In particular, A is a Dedekind domain and it is discrete in F (with the topology induced by ${\displaystyle \mathrm{\infty }}$). For example, we may take A to be the polynomial ring ${\displaystyle F_q[t]}$. Let L be a field equipped with a ring homomorphism ${\displaystyle \iota :A\to L}$.

A Drinfeld A-module over L is a ring homomorphism ${\displaystyle \varphi :A\to L\{\tau \}}$ whose image is not contained in L, such that the composition of ${\displaystyle \varphi }$ with ${\displaystyle d:L\{\tau \}\to L,a_0+a_1\tau +\cdots \mapsto a_0}$ coincides with ${\displaystyle \iota :A\to L}$.

The condition that the image of A is not in L is a non-degeneracy condition, put in to eliminate trivial cases, while the condition that ${\displaystyle d\circ \varphi =\iota }$ gives the impression that a Drinfeld module is simply a deformation of the map ${\displaystyle \iota }$.

As L{τ} can be thought of as endomorphisms of the additive group of L, a Drinfeld A-module can be regarded as an action of A on the additive group of L, or in other words as an A-module whose underlying additive group is the additive group of L.

Examples of Drinfeld modules

• Define A to be F_p[T], the usual (commutative!) ring of polynomials over the finite field of order p. In other words, A is the coordinate ring of an affine genus 0 curve. Then a Drinfeld module ψ is determined by the image ψ(T) of T, which can be any non-constant element of L{τ}. So Drinfeld modules can be identified with non-constant elements of L{τ}. (In the higher genus case the description of Drinfeld modules is more complicated.)
• The Carlitz module is the Drinfeld module ψ given by ψ(T) = T+τ, where A is F_p[T] and L is a suitable complete algebraically closed field containing A. It was described by L. Carlitz in 1935, many years before the general definition of Drinfeld module. See chapter 3 of Goss (1996) for more information about the Carlitz module. See also Carlitz exponential.