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

Darmon, Henri¹ ; Rotger, Victor²

¹Department of Mathematics, McGill University, Montréal H3A-0B9, Canada
²Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona 08034, Spain

Journal of the American Mathematical Society, Tome 30 (2017) no. 3, pp. 601-672

Résumé

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.

Détail

DOI : 10.1090/jams/861

Bibliographie

[1] Atkin, A. O. L., Li, Wen Ch’Ing Winnie Twists of newforms and pseudo-eigenvalues of 𝑊-operators Invent. Math. 1978 221 243

[2] Bellaïche, Joël, Dimitrov, Mladen On the eigencurve at classical weight 1 points Duke Math. J. 2016 245 266

[3] Bertolini, Massimo, Darmon, Henri A rigid analytic Gross-Zagier formula and arithmetic applications Ann. of Math. (2) 1997 111 147

[4] Bertolini, M., Darmon, H. Euler systems and Jochnowitz congruences Amer. J. Math. 1999 259 281

[5] Bertolini, M., Darmon, H. Iwasawa’s main conjecture for elliptic curves over anticyclotomic ℤ_{𝕡}-extensions Ann. of Math. (2) 2005 1 64

[6] Bertolini, Massimo, Darmon, Henri, Dasgupta, Samit Stark-Heegner points and special values of 𝐿-series 2007 1 23

[7] Bertolini, Massimo, Darmon, Henri, Rotger, Victor Beilinson-Flach elements and Euler systems I: Syntomic regulators and 𝑝-adic Rankin 𝐿-series J. Algebraic Geom. 2015 355 378

[8] Bertolini, Massimo, Darmon, Henri, Rotger, Victor Beilinson-Flach elements and Euler systems II: the Birch-Swinnerton-Dyer conjecture for Hasse-Weil-Artin 𝐿-series J. Algebraic Geom. 2015 569 604

[9] Besser, Amnon A generalization of Coleman’s 𝑝-adic integration theory Invent. Math. 2000 397 434

[10] Bloch, Spencer, Kato, Kazuya 𝐿-functions and Tamagawa numbers of motives 1990 333 400

[11] Breuil, Christophe, Emerton, Matthew Représentations 𝑝-adiques ordinaires de 𝐺𝐿₂(𝐐_{𝐩}) et compatibilité local-global Astérisque 2010 255 315

[12] Coleman, Robert F. Classical and overconvergent modular forms of higher level J. Théor. Nombres Bordeaux 1997 395 403

[13] Coleman, Robert, Iovita, Adrian The Frobenius and monodromy operators for curves and abelian varieties Duke Math. J. 1999 171 215

[14] Coleman, Robert, Iovita, Adrian Hidden structures on semistable curves Astérisque 2010 179 254

[15] Darmon, Henri Integration on ℋ_{𝓅}×ℋ and arithmetic applications Ann. of Math. (2) 2001 589 639

[16] Darmon, Henri, Diamond, Fred, Taylor, Richard Fermat’s last theorem 1997 2 140

[17] Darmon, Henri, Lauder, Alan, Rotger, Victor Stark points and 𝑝-adic iterated integrals attached to modular forms of weight one Forum Math. Pi 2015

[18] Darmon, Henri, Rotger, Victor Diagonal cycles and Euler systems I: A 𝑝-adic Gross-Zagier formula Ann. Sci. Éc. Norm. Supér. (4) 2014 779 832

[19] Diamond, Fred, Shurman, Jerry A first course in modular forms 2005

[20] Emerton, Matthew A new proof of a theorem of Hida Internat. Math. Res. Notices 1999 453 472

[21] Faltings, Gerd Crystalline cohomology and 𝑝-adic Galois-representations 1989 25 80

[22] Faltings, Gerd Crystalline cohomology of semistable curve—the 𝑄_{𝑝}-theory J. Algebraic Geom. 1997 1 18

[23] Flach, Matthias A generalisation of the Cassels-Tate pairing J. Reine Angew. Math. 1990 113 127

[24] Greenberg, Ralph Iwasawa theory for 𝑝-adic representations 1989 97 137

[25] Gross, Benedict H., Zagier, Don B. Heegner points and derivatives of 𝐿-series Invent. Math. 1986 225 320

[26] Hida, Haruzo A 𝑝-adic measure attached to the zeta functions associated with two elliptic modular forms. I Invent. Math. 1985 159 195

[27] Hida, Haruzo Galois representations into 𝐺𝐿₂(𝑍_{𝑝}[[𝑋]]) attached to ordinary cusp forms Invent. Math. 1986 545 613

[28] Hida, Haruzo Elementary theory of 𝐿-functions and Eisenstein series 1993

[29] Jannsen, Uwe Continuous étale cohomology Math. Ann. 1988 207 245

[30] Kato, Kazuya 𝑝-adic Hodge theory and values of zeta functions of modular forms Astérisque 2004

[31] Katz, Nicholas M., Mazur, Barry Arithmetic moduli of elliptic curves 1985

[32] Kolyvagin, V. A. Finiteness of 𝐸(𝑄) and SH(𝐸,𝑄) for a subclass of Weil curves Izv. Akad. Nauk SSSR Ser. Mat. 1988

[33] Lei, Antonio, Loeffler, David, Zerbes, Sarah Livia Euler systems for Rankin-Selberg convolutions of modular forms Ann. of Math. (2) 2014 653 771

[34] Loeffler, David, Zerbes, Sarah Livia Iwasawa theory and 𝑝-adic 𝐿-functions over ℤ_{𝕡}²-extensions Int. J. Number Theory 2014 2045 2095

[35] Longo, Matteo, Rotger, Victor, Vigni, Stefano Special values of 𝐿-functions and the arithmetic of Darmon points J. Reine Angew. Math. 2013 199 244

[36] Mazur, B., Wiles, A. Class fields of abelian extensions of 𝑄 Invent. Math. 1984 179 330

[37] Miln, Dzh. Ètal′nye kogomologii 1983 392

[38] Nekovář, Jan 𝑝-adic Abel-Jacobi maps and 𝑝-adic heights 2000 367 379

[39] Nekovář, Jan Level raising and anticyclotomic Selmer groups for Hilbert modular forms of weight two Canad. J. Math. 2012 588 668

[40] Ohta, Masami On the 𝑝-adic Eichler-Shimura isomorphism for Λ-adic cusp forms J. Reine Angew. Math. 1995 49 98

[41] Prasad, Dipendra Trilinear forms for representations of 𝐺𝐿(2) and local 𝜀-factors Compositio Math. 1990 1 46

[42] Saito, Takeshi Modular forms and 𝑝-adic Hodge theory Invent. Math. 1997 607 620

[43] Skinner, Christopher, Urban, Eric Vanishing of 𝐿-functions and ranks of Selmer groups 2006 473 500

[44] Wiles, A. On ordinary 𝜆-adic representations associated to modular forms Invent. Math. 1988 529 573