Riv. Mat. Univ. Parma, Vol. 7, No. 1, 2016

Matteo Longo[1] and Stefano Vigni[2]

Quaternionic Darmon points on abelian varieties

Pages: 39-70
Received: 29 December 2015
Accepted in revised form: 18 February 2016
Mathematics Subject Classification (2010): 14G05, 11G10.
Keywords: Darmon points, modular abelian varieties, \(p\)-adic logarithm, genus fields.
Author address:
[1] : Dipartimento di Matematica , UniversitÓ degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy
[2] : Dipartimento di Matematica, UniversitÓ degli Studi di Genova , Via Dodecaneso 35 , 16146 Genova, Italy

The authors are partially supported by PRIN 2010-11 "Arithmetic Algebraic Geometry and Number Theory".

Abstract: In the first part of the paper we prove formulas for the \(p\)-adic logarithm of quaternionic Darmon points on modular abelian varieties over \(\mathbb{Q}\) with toric reduction at \(p\). These formulas are amenable to explicit computations and are the first to treat Stark-Heegner type points on higher-dimensional abelian varieties. In the second part of the paper we explain how these formulas, together with a mild generalization of results of Bertolini and Darmon on Hida families of modular forms and rational points, can be used to obtain rationality results over genus fields of real quadratic fields for Darmon points on abelian varieties.


[1] M. Bertolini and H. Darmon, Hida families and rational points on elliptic curves, Invent. Math. 168 (2007), no. 2, 371-431. MR2289868
[2] M. Bertolini and H. Darmon, The rationality of Stark-Heegner points over genus fields of real quadratic fields, Ann. of Math. (2) 170 (2009), no. 1, 343-370. MR2521118
[3] S. Bloch and K. Kato, \(L\)-functions and Tamagawa numbers of motives, in "The Grothendieck Festschrift", vol. I, P. Cartier, L. Illusie, N. M. Katz, G. Laumon, Yu. Manin and K. A. Ribet (eds.), Progr. Math., 86 , Birkhäuser, Boston, MA 1990, 333-400. MR1086888
[4] N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie - Chapitres 2 et 3, Reprint of the 1972 original, Springer, Berlin 2006. Zbl1102.22001
[5] H. Cohn, A classical invitation to algebraic numbers and class fields, Universitext, Springer-Verlag, New York-Heidelberg 1978. MR0506156
[6] R. F. Coleman, Reciprocity laws on curves, Compositio Math. 72 (1989), no. 2, 205-235. MR1030142
[7] H. Darmon, Integration on \(\mathcal H_p\times\mathcal H\) and arithmetic applications, Ann. of Math. (2) 154 (2001), no. 3, 589-639. MR1884617
[8] S. Dasgupta, Stark-Heegner points on modular Jacobians, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 3, 427-469. MR2166341
[9] M. Greenberg and M. A. Seveso, \(p\)-adic families of modular forms and \(p\)-adic Abel-Jacobi maps, Special volume in honor of Glenn Stevens' 60th birthday, Ann. Math. Qué 40 (2016), n. 2, 397-434. DOI: 10.1007/s40316-016-0060-z
[10] M. Greenberg , M. A. Seveso and S. Shahabi , Modular \(p\)-adic \(L\)-functions attached to real quadratic fields and arithmetic applications, J. Reine Angew. Math., to appear. DOI: 10.1515/crelle-2014-0088
[11] R. Greenberg and G. Stevens, \(p\)-adic \(L\)-functions and \(p\)-adic periods of modular forms, Invent. Math 111 (1993), no. 2, 407-447. MR1198816
[12] X. Guitart and M. Masdeu, Overconvergent cohomology and quaternionic Darmon points, J. Lond. Math. Soc. (2) 90 (2014), no. 2, 495-524. MR3263962
[13] H. Hida, Hilbert modular forms and Iwasawa theory, Oxford Mathematical Monographs, Oxford University Press, Oxford 2006. MR2243770
[14] V. A. Kolyvagin and D. Yu. Logachëv , Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties, Leningrad Math. J. 1 (1990), no. 5, 1229-1253. MR1036843
[15] V. A. Kolyvagin and D. Yu. Logachëv, Finiteness of Ш over totally real fields, Math. USSR-Izv. 39 (1992), no. 1, 829-853. MR1137589
[16] W.-C. W. Li, Newforms and functional equations, Math. Ann. 212 (1975), no. 4, 285-315. MR0369263
[17] M. Longo , V. Rotger and S. Vigni, On rigid analytic uniformizations of Jacobians of Shimura curves, Amer. J. Math. 134 (2012), no. 5, 1197-1246. MR2975234
[18] M. Longo , V. Rotger and S. Vigni, Special values of \(L\)-functions and the arithmetic of Darmon points, J. Reine Angew. Math. 684 (2013), 199-244. MR3181561
[19] M. Longo and S. Vigni, Quaternion algebras, Heegner points and the arithmetic of Hida families, Manuscripta Math. 135 (2011), no. 3-4, 273-328. MR2813438
[20] M. Longo and S. Vigni, A note on control theorems for quaternionic Hida families of modular forms, Int. J. Number Theory 8 (2012), no. 6, 1425-1462. MR2965758
[21] M. Longo and S. Vigni, The rationality of quaternionic Darmon points over genus fields of real quadratic fields, Int. Math. Res. Not. IMRN 2014 , no. 13, 3632-3691. MR3229764
[22] A. Mattuck, Abelian varieties over \(p\)-adic ground fields, Ann. of Math. (2) 62 (1955), no. 1, 92-119. MR0071116
[23] M. Papikian, Rigid-analytic geometry and the uniformization of abelian varieties, in "Snowbird lectures in algebraic geometry", R. Vakil (ed.), Contemp. Math., 388 , Amer. Math. Soc., Providence, RI 2005, 145-160. MR2182895
[24] A. A. Popa, Central values of Rankin \(L\)-series over real quadratic fields, Compos. Math. 142 (2006), no. 4, 811-866. MR2249532
[25] K. A. Ribet, Endomorphisms of semi-stable abelian varieties over number fields, Ann. Math. (2) 101 (1975), no. 3, 555-562. MR0371903
[26] K. A. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98 (1976), no. 3, 751-804. MR0457455
[27] K. A. Ribet, Twists of modular forms and endomorphisms of abelian varieties, Math. Ann. 253 (1980), no. 1, 43-62. MR0594532
[28] V. Rotger and M. A. Seveso, \(\mathcal L\)-invariants and Darmon cycles attached to modular forms, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 6, 1955-1999. MR2984593
[29] J.-P. Serre, Lie algebras and Lie groups, 2nd edition, Lecture Notes in Math., 1500 , Springer-Verlag, Berlin 1992. MR1176100
[30] J.-P. Serre, Abelian \(l\)-adic representations and elliptic curves, revised reprint of the 1968 original, Research Notes in Mathematics, 7 , A K Peters, Wellesley, MA 1998. MR1484415
[31] M. A. Seveso, \(p\)-adic \(L\)-functions and the rationality of Darmon cycles, Canad. J. Math. 64 (2012), no. 5, 1122-1181. MR2979580
[32] M. A. Seveso, The Teitelbaum conjecture in the indefinite setting, Amer. J. Math. 135 (2013), no. 6, 1525-1557. MR3145003
[33] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, Princeton, NJ 1971. MR0314766
[34] Yu. G. Zarhin, \(p\)-adic abelian integrals and commutative Lie groups, J. Math. Sci. 81 (1996), no. 3, 2744-2750. MR1420227
[35] S.-W. Zhang, Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), no. 1, 27-147. MR1826411
[36] S.-W. Zhang, Gross-Zagier formula for \(\mathrm{GL} _2\), Asian J. Math. 5 (2001), no. 2, 183-290. MR1868935

Home Riv.Mat.Univ.Parma