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

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

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 ${\displaystyle \infty }$ of F. Define A to be the ring of elements in F that are regular at every place except possibly ${\displaystyle \infty }$. In particular, A is a Dedekind domain and it is discrete in F (with the topology induced by ${\displaystyle \infty }$). 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 ${\displaystyle \phi :A\to L\{\tau \}}$ whose image is not contained in L, such that the composition of ${\displaystyle \phi }$ 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 ${\displaystyle d\circ \phi =\iota }$ 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.

Drinfeld modules

• Drinfeld, Vladimir (1974), "Elliptic modules", Matematicheskii Sbornik (in Russian), 94, MR 0384707. English translation in Math. USSR Sbornik 23 (1974) 561–592.
• Goss, David (1996), Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-61480-4, ISBN 978-3-540-61087-8, MR 1423131
• Gekeler, E.-U. (2001) [1994], "Drinfel'd module", Encyclopedia of Mathematics, EMS Press.
• Laumon, Gérard (1996), Cohomology of Drinfeld Modular Varieties, Part 1, Geometry, Counting of Points and Local Harmonic Analysis, Cambridge Studies in Advanced Mathematics, vol. 41, Cambridge University Press, ISBN 978-0-521-47060-5
• Laumon, Gérard; Waldspurger, Jean-Loup (1996), Cohomology of Drinfeld Modular Varieties, Part 2, Automorphic Forms, Trace Formulas and Langlands Correspondence, Cambridge Studies in Advanced Mathematics, vol. 56, Cambridge University Press, ISBN 978-0-521-47061-2
• Rosen, Michael (2002), "13. Drinfeld modules: an introduction", Number theory in function fields, Graduate Texts in Mathematics, vol. 210, New York, NY: Springer-Verlag, ISBN 0-387-95335-3, Zbl 1043.11079.

Shtukas

• Drinfeld, V. G. Cohomology of compactified moduli varieties of F-sheaves of rank 2. (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 162 (1987), Avtomorfn. Funkts. i Teor. Chisel. III, 107–158, 189; translation in J. Soviet Math. 46 (1989), no. 2, 1789–1821
• Drinfeld, V. G. (1987), "Moduli varieties of F-sheaves", Funktsional. Anal. I Prilozhen. (in Russian), 21 (2): 23–41. English translation: Functional Anal. Appl. 21 (1987), no. 2, 107–122.
• Goss, D. (2003), "What is a shtuka?" (PDF), Notices of the American Mathematical Society, 50 (1)
• Kazhdan, David A. (1979), "An introduction to Drinfeld's Shtuka", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 347–356, ISBN 978-0-8218-1437-6, MR 0546623