Special \(L\)-values of Drinfeld modules. (English) Zbl 1323.11039

From the text: We state and prove a formula for a certain value of the Goss \(L\)-function of a Drinfeld module. This gives characteristic-\(p\)-valued function field analogues of the class number formula and of the Birch and Swinnerton-Dyer conjecture. The formula and its proof are presented in an entirely self-contained fashion.
In this paper we prove the following result.
Theorem 1. For every Drinfeld module \(E\) over \(R\) we have
\[ L(E/R) = \left[R: \exp_E^{-1} E(R)\right]\cdot | H(E/R)|. \]
Here \(L(E/R) :=\prod_{\mathfrak m}\frac{| R/\mathfrak m|}{| E(R/\mathfrak m)|}\in 1+T^{-1} k[[T^{-1}]]\), \(H(E/R)\)is defined to be the \(k[t]\)-module \(H(E/R) :=\frac{E(K_\infty)}{E(R)+\exp_E K_\infty}\) and is shown to be finite (Proposition 5).


11G09 Drinfel’d modules; higher-dimensional motives, etc.
11M38 Zeta and \(L\)-functions in characteristic \(p\)
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
Full Text: DOI arXiv


[1] G. W. Anderson, ”An elementary approach to \(L\)-functions mod \(p\),” J. Number Theory, vol. 80, iss. 2, pp. 291-303, 2000. · Zbl 0977.11036
[2] G. W. Anderson, ”\(t\)-motives,” Duke Math. J., vol. 53, iss. 2, pp. 457-502, 1986. · Zbl 0679.14001
[3] G. W. Anderson, ”Log-algebraicity of twisted \(A\)-harmonic series and special values of \(L\)-series in characteristic \(p\),” J. Number Theory, vol. 60, iss. 1, pp. 165-209, 1996. · Zbl 0868.11031
[4] G. W. Anderson and D. S. Thakur, ”Tensor powers of the Carlitz module and zeta values,” Ann. of Math., vol. 132, iss. 1, pp. 159-191, 1990. · Zbl 0713.11082
[5] G. Böckle and R. Pink, Cohomological Theory of Crystals over Function Fields, European Mathematical Society (EMS), Zürich, 2009, vol. 9. · Zbl 1186.14002
[6] L. Carlitz, ”On certain functions connected with polynomials in a Galois field,” Duke Math. J., vol. 1, iss. 2, pp. 137-168, 1935. · Zbl 0012.04904
[7] C. Chang and M. A. Papanikolas, Algebraic independence of periods and logarithms of Drinfeld modules, 2010. · Zbl 1271.11079
[8] C. Chang and M. A. Papanikolas, ”Algebraic relations among periods and logarithms of rank 2 Drinfeld modules,” Amer. J. Math., vol. 133, iss. 2, pp. 359-391, 2011. · Zbl 1216.11065
[9] V. G. Drinfelcprimed, ”Elliptic modules,” Mat. Sb., vol. 94(136), pp. 594-627, 656, 1974. · Zbl 0321.14014
[10] F. Gardeyn, ”A Galois criterion for good reduction of \(\tau\)-sheaves,” J. Number Theory, vol. 97, iss. 2, pp. 447-471, 2002. · Zbl 1053.11054
[11] D. Goss, ”\(L\)-series of \(t\)-motives and Drinfel\cprime d modules,” in The Arithmetic of Function Fields, Berlin: de Gruyter, 1992, vol. 2, pp. 313-402. · Zbl 0806.11028
[12] J. Igusa, An Introduction to the Theory of Local Zeta Functions, Providence, RI: Amer. Math. Soc., 2000, vol. 14. · Zbl 0959.11047
[13] V. Lafforgue, ”Valeurs spéciales des fonctions \(L\) en caractéristique \(p\),” J. Number Theory, vol. 129, iss. 10, pp. 2600-2634, 2009. · Zbl 1194.11089
[14] M. A. Papanikolas, ”Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms,” Invent. Math., vol. 171, iss. 1, pp. 123-174, 2008. · Zbl 1235.11074
[15] B. Poonen, ”Local height functions and the Mordell-Weil theorem for Drinfel\cprime d modules,” Compositio Math., vol. 97, iss. 3, pp. 349-368, 1995. · Zbl 0839.11024
[16] H. P. F. Swinnerton-Dyer, ”An application of computing to class field theory,” in Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 280-291. · Zbl 0153.07403
[17] L. Taelman, ”A Dirichlet unit theorem for Drinfeld modules,” Math. Ann., vol. 348, iss. 4, pp. 899-907, 2010. · Zbl 1217.11062
[18] L. Taelman, ”The Carlitz shtuka,” J. Number Theory, vol. 131, iss. 3, pp. 410-418, 2011. · Zbl 1221.11137
[19] Y. Taguchi and D. Wan, ”Entireness of \(L\)-functions of \(\phi\)-sheaves on affine complete intersections,” J. Number Theory, vol. 63, iss. 1, pp. 170-179, 1997. · Zbl 0908.11026
[20] J. Tate, ”Residues of differentials on curves,” Ann. Sci. École Norm. Sup., vol. 1, pp. 149-159, 1968. · Zbl 0159.22702
[21] J. Tate, Les Conjectures de Stark sur les Fonctions \(L\) d’Artin en \(s=0\), Boston, MA: Birkhäuser, 1984, vol. 47. · 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.