×

A Dirichlet unit theorem for Drinfeld modules. (English) Zbl 1217.11062

Let \(k\) be a finite field of \(q\) elements, \(K\) a finite extension of the rational function field \(k(t)\), \(R\) the integral closure of \(k[t]\) in \(K\), \(E\) the additive group \({\mathbf G}_{a,R}\) equipped with an action \(\varphi\) of \(k[t]\) given by a \(k\)-algebra homomorphism \[ \varphi:k[t]\to \text{End}({\mathbf G}_{a,R}):t\mapsto t+a_1\tau+\dots+a_n\tau^n, \] \(\tau\) the \(q\)-th power Frobenius endomorphism, \(a_i\in R\), \(a_n\neq 0\). This is a model over \(R\) of a Drinfeld module of rank \(n\) over \(K\). Let \(\text{Lie}_E\) be the tangent space at zero of \(E\). Let \(K_\infty:=K\otimes_{k(t)}k((t^{-1}))\). There is a short exact sequence \[ \Lambda_E\to \text{Lie}_E(K_\infty^{\text{sep}})\twoheadrightarrow E(K_\infty^{\text{sep}}), \] where the \(k[t]\)-module \(\Lambda_E\) is discrete in \(\text{Lie}_E(K_\infty^{\text{sep}})\) and free of rank \(n\) times the separable degree of \(K\) over \(k(t)\). This sequence is \(G=\text{Gal}(k((t^{-1}))^{\text{sep}}/k((t^{-1})))\)-equivariant, and taking invariants gives an exact sequence \[ \Lambda_E^G\to \text{Lie}_E(K_\infty)\to E(K_\infty)\twoheadrightarrow H^1(G,\Lambda_E). \] The author proves that the cokernel of \(E(R)\to H^1(G,\Lambda_E)\) is finite. The inverse image of \(E(R)\) is a discrete and cocompact sub-\(k[t]\)-module of \(\text{Lie}_E(K_\infty)\). As a corollary, the kernel of \(E(R)\to H^1(G,\Lambda_E)\) is finitely generated. This should be seen as an analogue of Dirichlet’s unit theorem in number fields.

MSC:

11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R58 Arithmetic theory of algebraic function fields
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Anderson, G.W.: Log-algebraicity of twisted A-harmonic series and special values of L-series in characteristic p. J. Number Theory 60(1), 165–209 (1996). http://www.ams.org/mathscinet-getitem?mr=1405732 · Zbl 0868.11031
[2] Carlitz, L.: On certain functions connected with polynomials in a Galois field. Duke Math. J. 1(2), 137–168 (1935). http://www.ams.org/mathscinet-getitem?mr=1545872 · JFM 61.0127.01
[3] Drinfeld, V.G.: Elliptic modules. Mat. Sb. (N.S.) 94(136), 594–627, 656 (1974). http://www.ams.org/mathscinet-getitem?mr=0384707
[4] Goss, D.: L-series of t-motives and Drinfeld modules. In: The Arithmetic of Function Fields, vol. 2, pp. 313–402. Ohio State University Mathematical Research Institute Publications, Columbus (1991), de Gruyter, Berlin (1992). http://www.ams.org/mathscinet-getitem?mr=1196527 · Zbl 0806.11028
[5] Igusa, J.-I.: An Introduction to the Theory of Local Zeta Functions. AMS/IP Studies in Advanced Mathematics, vol. 14. American Mathematical Society, Providence (2000). http://www.ams.org/mathscinet-getitem?mr=1743467 · Zbl 0959.11047
[6] Lafforgue, V.: Valeurs spéciales des fonctions L en caractéristique p. J. Number Theory 129(10), 2600–2634 (2009). http://www.ams.org/mathscinet-getitem?mr=2541033 · Zbl 1194.11089
[7] Mazza, C., Voevodsky, V., Weibel, C.: Lecture Notes on Motivic Cohomology. Clay Mathematics Monographs, vol. 2. American Mathematical Society, Providence (2006). http://www.ams.org/mathscinet-getitem?mr=2242284 · Zbl 1115.14010
[8] Poonen, B.: Local height functions and the Mordell–Weil theorem for Drinfeld modules. Compos. Math. 97(3), 349–368 (1995). http://www.ams.org/mathscinet-getitem?mr=1353279 · Zbl 0839.11024
[9] Serre, J.-P.: Corps locaux. Publications de l’Institut de Mathématique de l’Université de Nancago, VIII. Actualités Sci. Indust., No. 1296. Hermann, Paris (1962). http://www.ams.org/mathscinet-getitem?mr=0150130
[10] Taelman, L.: Special L-values of t-motives: a conjecture. Int. Math. Res. Not. IMRN 16, 2957–2977 (2009). http://www.ams.org/mathscinet-getitem?mr=2533793 · Zbl 1236.11082
[11] Tate, J.: Les conjectures de Stark sur les fonctions L d’Artin en s = 0. Progress in Mathematics, vol. 47. Birkhäuser Boston Inc., Boston (1984). http://www.ams.org/mathscinet-getitem?mr=782485 · Zbl 0545.12009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.