Geometry of Fermat adeles

Buium, Alexandru

doi:10.1090/S0002-9947-04-03715-8

Geometry of Fermat adeles
HTML articles powered by AMS MathViewer

by Alexandru Buium PDF

Trans. Amer. Math. Soc. 357 (2005), 901-964 Request permission

Abstract:

If $L(a,s):=\sum _n c(n,a)n^{-s}$ is a family of “geometric” $L-$functions depending on a parameter $a$, then the function $(p,a)\mapsto c(p,a)$, where $p$ runs through the set of prime integers, is not a rational function and hence is not a function belonging to algebraic geometry. The aim of the paper is to show that if one enlarges algebraic geometry by “adjoining a Fermat quotient operation”, then the functions $c(p,a)$ become functions in the enlarged geometry at least for $L-$functions of curves and Abelian varieties.

References

Mugurel A. Barcau, Isogeny covariant differential modular forms and the space of elliptic curves up to isogeny, Compositio Math. 137 (2003), no. 3, 237–273. MR 1988499, DOI 10.1023/A:1024123915158
Mugurel Barcau and Alexandru Buium, Siegel differential modular forms, Int. Math. Res. Not. 28 (2002), 1457–1503. MR 1908022, DOI 10.1155/S1073792802110063
P. Berthelot and A. Ogus, $F$-isocrystals and de Rham cohomology. I, Invent. Math. 72 (1983), no. 2, 159–199. MR 700767, DOI 10.1007/BF01389319
Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822, DOI 10.1007/978-3-642-51438-8
Alexandru Buium, Differential characters of abelian varieties over $p$-adic fields, Invent. Math. 122 (1995), no. 2, 309–340. MR 1358979, DOI 10.1007/BF01231447
Alexandru Buium, Geometry of $p$-jets, Duke Math. J. 82 (1996), no. 2, 349–367. MR 1387233, DOI 10.1215/S0012-7094-96-08216-2
Alexandru Buium, Differential characters and characteristic polynomial of Frobenius, J. Reine Angew. Math. 485 (1997), 209–219. MR 1442195, DOI 10.1515/crll.1997.485.209
Alexandru Buium, Differential modular forms, J. Reine Angew. Math. 520 (2000), 95–167. MR 1748272, DOI 10.1515/crll.2000.024
Alexandru Buium, Infinitesimal Mordell-Lang, J. Number Theory 90 (2001), no. 2, 185–206. MR 1858073, DOI 10.1006/jnth.2001.2663
Bernard Dwork, A deformation theory for the zeta function of a hypersurface, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 247–259. MR 0175895
B. Dwork and A. Ogus, Canonical liftings of Jacobians, Compositio Math. 58 (1986), no. 1, 111–131. MR 834049
Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendix by David Mumford. MR 1083353, DOI 10.1007/978-3-662-02632-8
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523, DOI 10.1002/9781118032527
Michiel Hazewinkel, Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506881
Chris Hurlburt, Isogeny covariant differential modular forms modulo $p$, Compositio Math. 128 (2001), no. 1, 17–34. MR 1847663, DOI 10.1023/A:1017536003747
Yasutaka Ihara, On Fermat quotients and “the differentials of numbers”, Sūrikaisekikenkyūsho K\B{o}kyūroku 810 (1992), 324–341 (Japanese). Algebraic analysis and number theory (Kyoto, 1992). MR 1248209
Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723, DOI 10.1007/978-1-4612-0853-2
Nicholas M. Katz and William Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77. MR 332791, DOI 10.1007/BF01405203
William Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, Vol. 264, Springer-Verlag, Berlin-New York, 1972. MR 0347836, DOI 10.1007/BFb0058301
Leonhard Miller, Curves with invertible Hasse-Witt-matrix, Math. Ann. 197 (1972), 123–127. MR 314849, DOI 10.1007/BF01419588
P. Monsky and G. Washnitzer, The construction of formal cohomology sheaves, Proc. Nat. Acad. Sci. U.S.A. 52 (1964), 1511–1514. MR 171787, DOI 10.1073/pnas.52.6.1511
Frans Oort, A stratification of a moduli space of polarized abelian varieties in positive characteristic, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 47–64. MR 1722538
Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564, DOI 10.1007/978-1-4612-1035-1
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8
José Felipe Voloch, On a question of Buium, Canad. Math. Bull. 43 (2000), no. 2, 236–238. MR 1754028, DOI 10.4153/CMB-2000-031-6