## $$L$$-functions associated to overconvergent $$F$$-isocrystals. (Fonctions $$L$$ associées aux $$F$$-isocristaux surconvergents.)(French)Zbl 0789.14015

We prove the formulas that give the cohomological expressions of the $$L$$- function attached to an overconvergent $$F$$-isocristal. We deduce that such an $$L$$-function is always meromorphic. This applies in particular to the $$L$$-functions that arise in the study of classical exponential sums. We also show that Frobenius is always bijective on rigid cohomology with or without support and that the trace of Frobenius on the highest rigid cohomology space with compact support of a scheme of pure dimension $$n$$ is multiplication by $$q^ n$$.

### MSC:

 14F30 $$p$$-adic cohomology, crystalline cohomology 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14F20 Étale and other Grothendieck topologies and (co)homologies 11L03 Trigonometric and exponential sums (general theory)
Full Text:

### References:

 [1] Berthelot, P.: Cohomologie rigide et th?orie de Dwork: le cas des sommes exponentielles. Ast?risque119-120, 17-49 (1984) [2] Berthelot, P.: G?om?trie rigide et cohomologie des vari?t?s alg?briques de caract?ristiquep. Dans: Introduction aux cohomologiesp-adiques. Bull. Soc. Math. Fr.23 (1986) [3] Berthelot, P.: Cohomologie rigide et cohomologie rigide ? support propre. Ast?risque (a para?tre) [4] A rigid analytic version of M. Artin’s theorem on analytic equations. Math. Ann.255, 395-404 (1981) · Zbl 0462.14002 [5] Dwork, B.: A deformation theory for the zeta function of a hypersurface. In: Proc. Int. Cong. Math., pp. 247-259. Stockholm 1962. Djursholm: Institut Mittag-Leffler 1963 · Zbl 0173.48601 [6] Elkik, R.: Solution d’?quations ? coefficients dans un anneau hens?lien. Ann. Sci. Ec. Norm. Sup?r.6 (no 4), 553-604 (1973) · Zbl 0327.14001 [7] Etesse, J.-Y.: Rationalit? et valeurs de fonctionL en cohomologie cristalline. Ann. Inst. Fourrier38 (no 4), 33-92 (1988) · Zbl 0624.14016 [8] Etesse, J.-Y.: M?romorphie et rationalit? des fonctionsL en cohomologie rigide, version pr?liminaire (non publi?e) [9] Etesse, J.-Y., Le Stum, B.: FonctionsL associ?es auxF-isocristaux surconvergents. II. (en pr?paration) · Zbl 0911.14011 [10] Katz, N.: Nilpotent connections and the monodromy theorem. Publ. Math. Inst. Hautes ?tud. Sci. 175-232 (1970) · Zbl 0221.14007 [11] Katz, N.: Travaux de Dwork. S?minaire Bourbaki no409. (Lect. Notes Math., vol. 383) Berlin Heidelberg New York: Springer 1972 [12] Le Stum, B.: Lettre ? Stephen Sperber. Minneapolis (1986) [13] Le Stum, B.: Applications of rigid cohomology to arithmetic geometry. Th?se de Ph.D. Minneapolis (1988) [14] Monsky, P., Washnitzer, G.: Formal cohomology. I. Ann. Math.88 (no 2), 181-217 (1986) · Zbl 0162.52504 [15] Monsky, P.: Formal Cohomology. II. Ann. Math.88 (no2), 218-238 (1986) · Zbl 0162.52601 [16] Monsky, P.: Formal cohomology. III. Ann. Math.93, (no 2), 315-343 (1971) · Zbl 0213.47501 [17] Van der Put, M.: The cohomology of Monsky and Washnitzer. Dans: Introduction aux cohomologiesp-adiques. Bull. Soc. Math. Fr.23 (1986) · Zbl 0606.14018 [18] Reich, D.: Ap-adic fixed point formula. Am. J. Math.91, 835-850 (1969) · Zbl 0213.47502 [19] Robba, P.: Index ofp-adic differential operators. III. Applications to twisted exponential sums. Ast?risque119-120, 191-266 (1984)
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.