Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Diagonal cycles and Euler systems II: The Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin $ L$-functions

Authors: Henri Darmon and Victor Rotger
Journal: J. Amer. Math. Soc. 30 (2017), 601-672
MSC (2010): Primary 11G05; Secondary 11G40
Published electronically: June 10, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This article establishes new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank 0, for elliptic curves over $ \mathbb{Q}$ viewed over the fields cut out by certain self-dual Artin representations of dimension at most $ 4$. When the associated $ L$-function vanishes (to even order $ \ge 2$) at its central point, two canonical classes in the corresponding Selmer group are constructed and shown to be linearly independent assuming the non-vanishing of a Garrett-Hida $ p$-adic $ L$-function at a point lying outside its range of classical interpolation. The key tool for both results is the study of certain $ p$-adic families of global Galois cohomology classes arising from Gross-Kudla-Schoen diagonal cycles in a tower of triple products of modular curves.

References [Enhancements On Off] (What's this?)

  • [AL78] A. O. L. Atkin and Wen Ch'ing Winnie Li, Twists of newforms and pseudo-eigenvalues of $ W$-operators, Invent. Math. 48 (1978), no. 3, 221-243. MR 508986,
  • [BeDi] Joël Bellaïche and Mladen Dimitrov, On the eigencurve at classical weight $ 1$ points, Duke Math. J. 165 (2016), no. 2, 245-266. MR 3457673,
  • [BCDDPR] M. Bertolini, F. Castella, H. Darmon, S. Dasgupta, K. Prasanna, and V. Rotger, $ p$-adic $ L$-functions and Euler systems: a tale in two trilogies, in Automorphic forms and Galois representations, vol. 1, LMS Lecture Notes 414, CUP (2014) 52-102.
  • [BD97] Massimo Bertolini and Henri Darmon, A rigid analytic Gross-Zagier formula and arithmetic applications, Ann. of Math. (2) 146 (1997), no. 1, 111-147. With an appendix by Bas Edixhoven. MR 1469318,
  • [BD99] M. Bertolini and H. Darmon, Euler systems and Jochnowitz congruences, Amer. J. Math. 121 (1999), no. 2, 259-281. MR 1680333
  • [BD05] M. Bertolini and H. Darmon, Iwasawa's main conjecture for elliptic curves over anticyclotomic $ \mathbb{Z}_p$-extensions, Ann. of Math. (2) 162 (2005), no. 1, 1-64. MR 2178960,
  • [BDD07] Massimo Bertolini, Henri Darmon, and Samit Dasgupta, Stark-Heegner points and special values of $ L$-series, $ L$-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 1-23. MR 2392351,
  • [BDR1] Massimo Bertolini, Henri Darmon, and Victor Rotger, Beilinson-Flach elements and Euler systems I: Syntomic regulators and $ p$-adic Rankin $ L$-series, J. Algebraic Geom. 24 (2015), no. 2, 355-378. MR 3311587,
  • [BDR2] Massimo Bertolini, Henri Darmon, and Victor Rotger, Beilinson-Flach elements and Euler systems II: the Birch-Swinnerton-Dyer conjecture for Hasse-Weil-Artin $ L$-series, J. Algebraic Geom. 24 (2015), no. 3, 569-604. MR 3344765,
  • [Be00] Amnon Besser, A generalization of Coleman's $ p$-adic integration theory, Invent. Math. 142 (2000), no. 2, 397-434. MR 1794067,
  • [BK93] Spencer Bloch and Kazuya Kato, $ L$-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333-400. MR 1086888
  • [BLZ] A. Besser, D. Loefffler, and S. Zerbes, Finite polynomial cohomology for general varieties, submitted.
  • [BrEm10] Christophe Breuil and Matthew Emerton, Représentations $ p$-adiques ordinaires de $ {\rm GL}_2(\mathbf {Q}_p)$ et compatibilité local-global, Astérisque 331 (2010), 255-315 (French, with English and French summaries). MR 2667890
  • [Co97] Robert F. Coleman, Classical and overconvergent modular forms of higher level, J. Théor. Nombres Bordeaux 9 (1997), no. 2, 395-403 (English, with English and French summaries). MR 1617406
  • [CI99] Robert Coleman and Adrian Iovita, The Frobenius and monodromy operators for curves and abelian varieties, Duke Math. J. 97 (1999), no. 1, 171-215. MR 1682268,
  • [CI10] Robert Coleman and Adrian Iovita, Hidden structures on semistable curves, Astérisque 331 (2010), 179-254 (English, with English and French summaries). MR 2667889
  • [Da] Henri Darmon, Integration on $ \mathcal {H}_p\times \mathcal {H}$ and arithmetic applications, Ann. of Math. (2) 154 (2001), no. 3, 589-639. MR 1884617,
  • [DDT] Henri Darmon, Fred Diamond, and Richard Taylor, Fermat's last theorem, Elliptic curves, modular forms & Fermat's last theorem (Hong Kong, 1993), Int. Press, Cambridge, MA, 1997, pp. 2-140. MR 1605752
  • [DLR] Henri Darmon, Alan Lauder, and Victor Rotger, Stark points and $ p$-adic iterated integrals attached to modular forms of weight one, Forum Math. Pi 3 (2015), e8, 95p.. MR 3456180,
  • [DR13] Henri Darmon and Victor Rotger, Diagonal cycles and Euler systems I: A $ p$-adic Gross-Zagier formula, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 4, 779-832 (English, with English and French summaries). MR 3250064
  • [DR15] H. Darmon and V. Rotger, Elliptic curves of rank two and generalised Kato classes, submitted.
  • [DS05] Fred Diamond and Jerry Shurman, A first course in modular forms, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005. MR 2112196
  • [Em99] Matthew Emerton, A new proof of a theorem of Hida, Internat. Math. Res. Not. IMRN 9 (1999), 453-472. MR 1692599,
  • [Fa89] Gerd Faltings, Crystalline cohomology and $ p$-adic Galois-representations, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25-80. MR 1463696
  • [Fa97] Gerd Faltings, Crystalline cohomology of semistable curve--the $ {\bf Q}_p$-theory, J. Algebraic Geom. 6 (1997), no. 1, 1-18. MR 1486990
  • [Fl90] Matthias Flach, A generalisation of the Cassels-Tate pairing, J. Reine Angew. Math. 412 (1990), 113-127. MR 1079004,
  • [Gr89] Ralph Greenberg, Iwasawa theory for $ p$-adic representations, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 97-137. MR 1097613
  • [GZ86] Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $ L$-series, Invent. Math. 84 (1986), no. 2, 225-320. MR 833192,
  • [Hi85] Haruzo Hida, A $ p$-adic measure attached to the zeta functions associated with two elliptic modular forms. I, Invent. Math. 79 (1985), no. 1, 159-195. MR 774534,
  • [Hi86] Haruzo Hida, Galois representations into $ {\rm GL}_2({\bf Z}_p[[X]])$ attached to ordinary cusp forms, Invent. Math. 85 (1986), no. 3, 545-613. MR 848685,
  • [Hi93] Haruzo Hida, Elementary theory of $ L$-functions and Eisenstein series, London Mathematical Society Student Texts, vol. 26, Cambridge University Press, Cambridge, 1993. MR 1216135
  • [Ja] Uwe Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988), no. 2, 207-245. MR 929536,
  • [Ka98] 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
  • [KM85] Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569
  • [Ko89] V. A. Kolyvagin, Finiteness of $ E({\bf Q})$ and SH $ (E,{\bf Q})$ for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 670-671 (Russian); English transl., Math. USSR-Izv. 32 (1989), no. 3, 523-541. MR 954295
  • [Liu1] Y. Liu, Gross-Kudla-Schoen cycles and twisted triple product Selmer groups. Preprint.
  • [Liu2] Y. Liu, Bounding cubic-triple product Selmer groups of elliptic curves. Preprint.
  • [LLZ1] Antonio Lei, David Loeffler, and Sarah Livia Zerbes, Euler systems for Rankin-Selberg convolutions of modular forms, Ann. of Math. (2) 180 (2014), no. 2, 653-771. MR 3224721,
  • [LZ12] David Loeffler and Sarah Livia Zerbes, Iwasawa theory and $ p$-adic $ L$-functions over $ \mathbb{Z}_p^2$-extensions, Int. J. Number Theory 10 (2014), no. 8, 2045-2095. MR 3273476,
  • [LRV13] Matteo Longo, Victor Rotger, and Stefano Vigni, Special values of $ L$-functions and the arithmetic of Darmon points, J. Reine Angew. Math. 684 (2013), 199-244. MR 3181561
  • [MW84] B. Mazur and A. Wiles, Class fields of abelian extensions of $ {\bf Q}$, Invent. Math. 76 (1984), no. 2, 179-330. MR 742853,
  • [Mi] Dzh. Miln, Etalnye kogomologii, ``Mir,'' Moscow, 1983 (Russian). Translated from the English by O. N. Vvedenskiĭ, Yu. G. Zarkhin, and V. V. Shokurov; Translation edited and with a preface by I. R. Shafarevich. MR 733256
  • [Ne1] J. Nekovář, Syntomic cohomology and $ p$-adic regulators, preprint, 1998.
  • [Ne2] Jan Nekovář, $ p$-adic Abel-Jacobi maps and $ p$-adic heights, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) CRM Proc. Lecture Notes, vol. 24, Amer. Math. Soc., Providence, RI, 2000, pp. 367-379. MR 1738867
  • [Ne3] Jan Nekovář, Level raising and anticyclotomic Selmer groups for Hilbert modular forms of weight two, Canad. J. Math. 64 (2012), no. 3, 588-668. MR 2962318,
  • [NeNi] J. Nekovář, W. Niziol, Syntomic cohomology and $ p$-adic regulators for varieties over $ p$-adic fields, preprint.
  • [Oh95] Masami Ohta, On the $ p$-adic Eichler-Shimura isomorphism for $ \Lambda $-adic cusp forms, J. Reine Angew. Math. 463 (1995), 49-98. MR 1332907,
  • [Pr] Dipendra Prasad, Trilinear forms for representations of $ {\rm GL}(2)$ and local $ \epsilon $-factors, Compos. Math. 75 (1990), no. 1, 1-46. MR 1059954
  • [Sa97] Takeshi Saito, Modular forms and $ p$-adic Hodge theory, Invent. Math. 129 (1997), no. 3, 607-620. MR 1465337,
  • [SU] Christopher Skinner and Eric Urban, Vanishing of $ L$-functions and ranks of Selmer groups, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 473-500. MR 2275606
  • [Wa] P. Wake, The $ \Lambda $-adic Eichler-Shimura isomorphism and $ p$-adic étale cohomology, submitted.
  • [W88] A. Wiles, On ordinary $ \lambda $-adic representations associated to modular forms, Invent. Math. 94 (1988), no. 3, 529-573. MR 969243,

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 11G05, 11G40

Retrieve articles in all journals with MSC (2010): 11G05, 11G40

Additional Information

Henri Darmon
Affiliation: Department of Mathematics, McGill University, Montréal H3A-0B9, Canada

Victor Rotger
Affiliation: Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona 08034, Spain

Keywords: Elliptic curves, Artin representations, equivariant Birch and Swinnerton-Dyer conjecture, Gross-Kudla-Schoen diagonal cycles, $p$-adic families of modular forms, Euler Systems
Received by editor(s): September 21, 2014
Received by editor(s) in revised form: October 16, 2015, and April 27, 2016
Published electronically: June 10, 2016
Additional Notes: The first author was supported by an NSERC Discovery grant.
The second author was supported by Grants MTM2012-34611 and MTM2015-63829-P
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society