A 𝑝-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties

Balakrishnan, Jennifer; Müller, J.; Stein, William

doi:10.1090/mcom/3029

A $p$-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties
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by Jennifer S. Balakrishnan, J. Steffen Müller and William A. Stein PDF

Math. Comp. 85 (2016), 983-1016 Request permission

Abstract:

Mazur, Tate, and Teitelbaum gave a $p$-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the $p$-adic $L$-function of a modular abelian variety and showing that it satisfies the appropriate interpolation property. This relies on a careful normalization of the $p$-adic $L$-function, which we achieve by a comparison of periods. Our generalization agrees with the conjecture of Mazur, Tate, and Teitelbaum in dimension 1 and the classical Birch and Swinnerton-Dyer conjecture formulated by Tate in rank 0. We describe the theoretical techniques used to formulate the conjecture and give numerical evidence supporting the conjecture in the case when the modular abelian variety is of dimension 2.

References

Amod Agashe, Kenneth Ribet, and William A. Stein, The Manin constant, Pure Appl. Math. Q. 2 (2006), no. 2, Special Issue: In honor of John H. Coates., 617–636. MR 2251484, DOI 10.4310/PAMQ.2006.v2.n2.a11
Amod Agashe and William Stein, Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero, Math. Comp. 74 (2005), no. 249, 455–484. With an appendix by J. Cremona and B. Mazur. MR 2085902, DOI 10.1090/S0025-5718-04-01644-8
Jennifer S. Balakrishnan and Amnon Besser, Computing local $p$-adic height pairings on hyperelliptic curves, Int. Math. Res. Not. IMRN 11 (2012), 2405–2444. MR 2926986, DOI 10.1093/imrn/rnr111
Jennifer S. Balakrishnan, Robert W. Bradshaw, and Kiran S. Kedlaya, Explicit Coleman integration for hyperelliptic curves, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 6197, Springer, Berlin, 2010, pp. 16–31. MR 2721410, DOI 10.1007/978-3-642-14518-6_{6}
Dominique Bernardi and Bernadette Perrin-Riou, Variante $p$-adique de la conjecture de Birch et Swinnerton-Dyer (le cas supersingulier), C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 3, 227–232 (French, with English and French summaries). MR 1233417
Amnon Besser, The $p$-adic height pairings of Coleman-Gross and of Nekovář, Number theory, CRM Proc. Lecture Notes, vol. 36, Amer. Math. Soc., Providence, RI, 2004, pp. 13–25. MR 2076563, DOI 10.1090/crmp/036/02
Siegfried Bosch and Qing Liu, Rational points of the group of components of a Néron model, Manuscripta Math. 98 (1999), no. 3, 275–293. MR 1717533, DOI 10.1007/s002290050140
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein, Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic $L$-functions and their derivatives, Ann. of Math. (2) 131 (1990), no. 1, 53–127. MR 1038358, DOI 10.2307/1971508
John Coates, On $p$-adic $L$-functions, Astérisque 177-178 (1989), Exp. No. 701, 33–59. Séminaire Bourbaki, Vol. 1988/89. MR 1040567
Robert F. Coleman, The universal vectorial bi-extension and $p$-adic heights, Invent. Math. 103 (1991), no. 3, 631–650. MR 1091621, DOI 10.1007/BF01239529
Robert F. Coleman and Benedict H. Gross, $p$-adic heights on curves, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 73–81. MR 1097610, DOI 10.2969/aspm/01710073
E. Victor Flynn, Franck Leprévost, Edward F. Schaefer, William A. Stein, Michael Stoll, and Joseph L. Wetherell, Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, Math. Comp. 70 (2001), no. 236, 1675–1697. MR 1836926, DOI 10.1090/S0025-5718-01-01320-5
S. D. Galbraith, Equations for modular curves, Oxford Ph.D. thesis, 1996.
Grigor Grigorov, Andrei Jorza, Stefan Patrikis, William A. Stein, and Corina Tarniţǎ, Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves, Math. Comp. 78 (2009), no. 268, 2397–2425. MR 2521294, DOI 10.1090/S0025-5718-09-02253-4
Marc Hindry and Joseph H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR 1745599, DOI 10.1007/978-1-4612-1210-2
Paul Hriljac, Heights and Arakelov’s intersection theory, Amer. J. Math. 107 (1985), no. 1, 23–38. MR 778087, DOI 10.2307/2374455
Kazuya Kato, $p$-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 (2004), ix, 117–290 (English, with English and French summaries). Cohomologies $p$-adiques et applications arithmétiques. III. MR 2104361
Qing Liu, Conducteur et discriminant minimal de courbes de genre $2$, Compositio Math. 94 (1994), no. 1, 51–79 (French). MR 1302311
Qing Liu, Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète, Trans. Amer. Math. Soc. 348 (1996), no. 11, 4577–4610 (French, with English summary). MR 1363944, DOI 10.1090/S0002-9947-96-01684-4
Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
Barry Mazur, William Stein, and John Tate, Computation of $p$-adic heights and log convergence, Doc. Math. Extra Vol. (2006), 577–614. MR 2290599
B. Mazur and J. Tate, Canonical height pairings via biextensions, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 195–237. MR 717595
B. Mazur, J. Tate, and J. Teitelbaum, On $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), no. 1, 1–48. MR 830037, DOI 10.1007/BF01388731
Robert L. Miller, Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one, LMS J. Comput. Math. 14 (2011), 327–350. MR 2861691, DOI 10.1112/S1461157011000180
Jan Steffen Müller, Computing canonical heights using arithmetic intersection theory, Math. Comp. 83 (2014), no. 285, 311–336. MR 3120591, DOI 10.1090/S0025-5718-2013-02719-6
David Mumford, Tata lectures on theta. I, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007. With the collaboration of C. Musili, M. Nori, E. Previato and M. Stillman; Reprint of the 1983 edition. MR 2352717, DOI 10.1007/978-0-8176-4578-6
Yukihiko Namikawa and Kenji Ueno, The complete classification of fibres in pencils of curves of genus two, Manuscripta Math. 9 (1973), 143–186. MR 369362, DOI 10.1007/BF01297652
Jan Nekovář, On $p$-adic height pairings, Séminaire de Théorie des Nombres, Paris, 1990–91, Progr. Math., vol. 108, Birkhäuser Boston, Boston, MA, 1993, pp. 127–202. MR 1263527, DOI 10.1007/s10107-005-0696-y
Vivek Pal, Periods of quadratic twists of elliptic curves, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1513–1525. With an appendix by Amod Agashe. MR 2869136, DOI 10.1090/S0002-9939-2011-11014-1
Bernadette Perrin-Riou, Arithmétique des courbes elliptiques et théorie d’Iwasawa, Mém. Soc. Math. France (N.S.) 17 (1984), 130 (French, with English summary). MR 799673
Bernadette Perrin-Riou, Fonctions $L$ $p$-adiques, théorie d’Iwasawa et points de Heegner, Bull. Soc. Math. France 115 (1987), no. 4, 399–456 (French, with English summary). MR 928018
Robert Pollack, On the $p$-adic $L$-function of a modular form at a supersingular prime, Duke Math. J. 118 (2003), no. 3, 523–558. MR 1983040, DOI 10.1215/S0012-7094-03-11835-9
Robert Pollack and Glenn Stevens, Overconvergent modular symbols and $p$-adic $L$-functions, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 1, 1–42 (English, with English and French summaries). MR 2760194, DOI 10.24033/asens.2139
Karl Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), no. 1, 25–68. MR 1079839, DOI 10.1007/BF01239508
Peter Schneider, $p$-adic height pairings. I, Invent. Math. 69 (1982), no. 3, 401–409. MR 679765, DOI 10.1007/BF01389362
Peter Schneider, $p$-adic height pairings. II, Invent. Math. 79 (1985), no. 2, 329–374. MR 778132, DOI 10.1007/BF01388978
Goro Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), 523–544. MR 318162, DOI 10.2969/jmsj/02530523
Goro Shimura, On the periods of modular forms, Math. Ann. 229 (1977), no. 3, 211–221. MR 463119, DOI 10.1007/BF01391466
Christopher Skinner and Eric Urban, The Iwasawa main conjectures for $\rm GL_2$, Invent. Math. 195 (2014), no. 1, 1–277. MR 3148103, DOI 10.1007/s00222-013-0448-1
William Stein, Modular forms, a computational approach, Graduate Studies in Mathematics, vol. 79, American Mathematical Society, Providence, RI, 2007. With an appendix by Paul E. Gunnells. MR 2289048, DOI 10.1090/gsm/079
W. A. Stein et al, Sage Mathematics Software (Version 6.2). The Sage Development Team, 2014. http://www.sagemath.org.
William Stein and Christian Wuthrich, Algorithms for the arithmetic of elliptic curves using Iwasawa theory, Math. Comp. 82 (2013), no. 283, 1757–1792. MR 3042584, DOI 10.1090/S0025-5718-2012-02649-4
Michael Stoll, On the height constant for curves of genus two. II, Acta Arith. 104 (2002), no. 2, 165–182. MR 1914251, DOI 10.4064/aa104-2-6
John Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 306, 415–440. MR 1610977
V. Vatsal, Canonical periods and congruence formulae, Duke Math. J. 98 (1999), no. 2, 397–419. MR 1695203, DOI 10.1215/S0012-7094-99-09811-3