# zbMATH — the first resource for mathematics

Congruences and rationality of Stark-Heegner points. (English) Zbl 1308.11063
Summary: Let $$A/\mathbb Q$$ be a modular abelian variety attached to a weight $$2$$ new modular form of level $$N=pM$$, where $$p$$ is a prime and $$M$$ is an integer prime to $$p$$. When $$K/\mathbb Q$$ is an imaginary quadratic extension the Heegner points, that are defined over the ring class fields $$H/K$$, can contribute to the growth of the rank of the Selmer groups over $$H$$. When $$K/\mathbb Q$$ is a real quadratic field the theory of Stark-Heegner points provides a conjectural explanation of the growth of these ranks under suitable sign conditions on the $$L$$-function of $$f/K$$. The main result of the paper relates the growth of the Selmer groups to the conjectured rationality of the Stark-Heegner points over the expected field of definition.

##### MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 11F11 Holomorphic modular forms of integral weight 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11G05 Elliptic curves over global fields
Full Text:
##### References:
 [1] Bertolini, M.; Darmon, H., Derived heights and generalized Mazur-Tate regulators, Duke math. J., 76, 1, 75-111, (1994) · Zbl 0853.14013 [2] Bertolini, M.; Darmon, H., Derived p-adic heights, Amer. J. math., 117, 1-38, (1995) · Zbl 0882.11036 [3] Bertolini, M.; Darmon, H., The rationality of Stark-Heegner points over genus fields of real quadratic fields, Ann. of math. (2), 170, 343-369, (2009) · Zbl 1203.11045 [4] Bertolini, M.; Darmon, H.; Dasgupta, S., Stark-Heegner points and special values of L-series, (), 1-23 · Zbl 1170.11015 [5] Bloch, S.; Kato, K., L-functions and Tamagawa numbers of motives, (), 333-400 · Zbl 0768.14001 [6] Conrad, B.; Stein, W., Component groups of purely toric quotients, Math. res. lett., 8, 745-766, (2001) · Zbl 1081.11040 [7] Darmon, H., Integration on $$\mathcal{H}_p \times \mathcal{H}$$ and arithmetic applications, Ann. of math. (2), 154, 3, 589-639, (2001) · Zbl 1035.11027 [8] Dasgupta, S., Stark-Heegner points on modular Jacobians, Ann. sci. école norm. sup. (4), 38, 427-469, (2005) · Zbl 1173.11334 [9] Deligne, P.; Rapoport, M., LES schémas de modules de courbes elliptiques, (), 143-316 · Zbl 0281.14010 [10] B.H. Gross, J. Parson, On the local divisibility of Heegner points, preprint. · Zbl 1276.11091 [11] Grothendieck, A., Éléments de géométrique algébrique IV, Publ. math. inst. hautes études sci., 32, 2, (1967) · Zbl 0153.22301 [12] Grothendieck, A.; Grothendieck, A., Groupes de monodromie en géometrie algébrique, (), Lecture notes in math., vol. 340, (1973), Springer-Verlag Berlin · Zbl 0355.14005 [13] Mazur, B.; Rubin, K., Finding large Selmer rank via an arithmetic theory of local constants, Ann. of math. (2), 166, 2, 579-612, (2007) · Zbl 1219.11084 [14] Raynaud, M., Spécialisation du foncteur de Picard, Publ. math. inst. hautes études sci., 38, 27-76, (1970) · Zbl 0207.51602 [15] K. Ribet, Congruence relations between modular forms, in: Proceedings of the International Congress of Mathematicians, Warszawa, August 1983, pp. 16-24. [16] Ribet, K., On modular representation of $$\mathit{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$$ arising from modular forms, Invent. math., 100, 431-476, (1990) · Zbl 0773.11039 [17] Serre, Jean-Pierre, Local fields, Grad. texts in math., vol. 67, (1979), Springer, translated from the French · Zbl 0423.12016 [18] M.A. Seveso, Stark-Heegner points and Selmer groups of abelian varieties, PhD thesis, University of Milan, Federigo Enriques Department of Mathematics. · Zbl 1308.11063 [19] M.A. Seveso, The arithmetic theory of local constants for abelian varieties with real multiplication, Rend. Semin. Mat. Univ. Padova, in press. · Zbl 1257.14032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.