## 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).

### MSC:

 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
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### References:

 [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
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