Euler systems obtained from congruences between Hilbert modular forms
Rendiconti del Seminario Matematico della Università di Padova, Tome 118 (2007), pp. 1-34.
@article{RSMUP_2007__118__1_0,
     author = {Longo, Matteo},
     title = {Euler systems obtained from congruences between {Hilbert} modular forms},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {1--34},
     publisher = {Seminario Matematico of the University of Padua},
     volume = {118},
     year = {2007},
     mrnumber = {2378387},
     language = {en},
     url = {http://www.numdam.org/item/RSMUP_2007__118__1_0/}
}
TY  - JOUR
AU  - Longo, Matteo
TI  - Euler systems obtained from congruences between Hilbert modular forms
JO  - Rendiconti del Seminario Matematico della Università di Padova
PY  - 2007
SP  - 1
EP  - 34
VL  - 118
PB  - Seminario Matematico of the University of Padua
UR  - http://www.numdam.org/item/RSMUP_2007__118__1_0/
LA  - en
ID  - RSMUP_2007__118__1_0
ER  - 
%0 Journal Article
%A Longo, Matteo
%T Euler systems obtained from congruences between Hilbert modular forms
%J Rendiconti del Seminario Matematico della Università di Padova
%D 2007
%P 1-34
%V 118
%I Seminario Matematico of the University of Padua
%U http://www.numdam.org/item/RSMUP_2007__118__1_0/
%G en
%F RSMUP_2007__118__1_0
Longo, Matteo. Euler systems obtained from congruences between Hilbert modular forms. Rendiconti del Seminario Matematico della Università di Padova, Tome 118 (2007), pp. 1-34. http://www.numdam.org/item/RSMUP_2007__118__1_0/

[BD1] M. Bertolini - H. Darmon, A rigid analytic Gross-Zagier formula and arithmetic applications. With an appendix by Bas Edixhoven. Ann. of Math. (2) 146, no. 1 (1997), pp. 111-147. | MR | Zbl

[BD2] M. Bertolini - H. Darmon, Iwasawa's main conjecture for elliptic curves over anticyclotomic Zp-extensions. Ann. of Math. (2) 162, no. 1 (2005), pp. 1-64. | MR | Zbl

[BLR] N. Boston - H. W. Lenstra - K. A. Ribet, Quotients of group rings arising from two-dimensional representations. (English. French summary) C. R. Acad. Sci. Paris Sér. I Math., 312, no. 4 (1991), pp. 323-328 | MR | Zbl

[BC] J.-F. Boutot - H. Carayol, Uniformisation p-adique des courbes de Shimura: les théoròmes de CÏerednik et de Drinfeld. (French) [p-adic uniformization of Shimura curves: the theorems of Cherednik and Drinfeld] Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). Astérisque No. 7 (1991), pp. 196-197 (1992), pp. 45-158 . | MR | Zbl

[Ca] H. Carayol. Sur les représentations galoisiennes modulo l attachées aux formes modulaires. (French) [On Galois representations modulo l associated with modular forms] Duke Math. J., 59, no. 3 (1989), pp. 785-801. | MR | Zbl

[Ce] I. V. Cïerednik, Uniformization of algebraic curves by discrete arithmetic subgroups of PGL2(kw) with compact quotient spaces. (Russian) Mat. Sb. (N.S.) 100(142), 165, no. 1 (1976), pp. 59-88. | MR | Zbl

[CG] J. Coates - R. Greenberg, Kummer theory for abelian varieties over local fields. Invent. Math., 124, no. 1-3 (1996), pp. 129-174. | MR | Zbl

[DT] F. Diamond - R. Taylor, Nonoptimal levels of mod l modular representations. Invent. Math., 115, no. 3 (1994), pp. 435-462. | MR | Zbl

[Dr] V. G. Drinfeld, Coverings of p-adic symmetric domains. (Russian) Funkcional. Anal. i PrilozÏen., 10, no. 2 (1976), pp. 29-40. | MR | Zbl

[GP] L. Gerritzen, M. Van Der Put, Schottky groups and Mumford curves. Lecture Notes in Mathematics, 817. Springer, Berlin, 1980. viii+317 pp. | MR | Zbl

[Go] E. Goren, Lectures on Hilbert modular varieties and modular forms. With the assistance of Marc-Hubert Nicole. CRM Monograph Series, 14. American Mathematical Society, Providence, RI, 2002. | MR | Zbl

[Gr] B. H. Gross, Heights and the special values of L-series. Number theory (Montreal, Que., 1985), pp. 115-187, CMS Conf. Proc., 7, Amer. Math. Soc., Providence, RI, 1987. | MR | Zbl

[GZ] B. H. Gross - D. B. Zagier, Heegner points and derivatives of L-series. Invent. Math., 84, no. 2 (1986), pp. 225-320. | MR | Zbl

[Gt] A. Grothendieck, Groupes de monodromie en géométrie algébrique. I. (French) Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (SGA 7 I). Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim. Lecture Notes in Mathematics, Vol. 288. Springer-Verlag, Berlin-New York, 1972. viii+523 pp. | MR | Zbl

[JL] H. Jacquet - R. P. Langlands, Automorphic forms on GL(2). Lecture Notes in Mathematics, Vol. 114. Springer-Verlag, Berlin-New York, 1970. vii+548 pp. | MR | Zbl

[J1] F. Jarvis, Correspondences on Shimura curves and Mazur's principle at p. Pacific J. Math., 213, no. 2 (2004), pp. 267-280. | MR | Zbl

[JL] Jordan - Bruce W. - Livné - Ron A., Local Diophantine properties of Shimura curves. Math. Ann., 270, no. 2 (1985), pp. 235-248. | MR | Zbl

[J2] F. Jarvis, Mazur's principle for totally real fields of odd degree. Compositio Math., 116, no. 1 (1999), pp. 39-79. | MR | Zbl

[J3] F. Jarvis, Level lowering for modular mod l representations over totally real fields. Math. Ann., 313, no. 1 (1999), pp. 141-160. | MR | Zbl

[K1] V. A. Kolyvagin, Finiteness of E(Q) and I-I-I(E; Q) for a subclass of Weil curves. (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 52, no. 3 (1988), pp. 522-540, 670-671; translation in Math. USSR-Izv., 32, no. 3 (1989), pp. 523-541. | MR | Zbl

[K2] V. A. Kolyvagin, The Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves. (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 52, no. 6 (1988), pp. 1154-1180, 1327; translation in Math. USSR-Izv., 33, no. 3 (1989), pp. 473-499. | MR | Zbl

[K3] V. A. Kolyvagin, Euler systems. The Grothendieck Festschrift, Vol. II, 435-483, Progr. Math., 87, Birkäuser Boston, Boston, MA, 1990. | MR | Zbl

[KL1] V. A. Kolyvagin - D. Y. Logachëv, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties. (Russian) Algebra i Analiz, 1, no. 5 (1989), pp. 171-196; translation in Leningrad Math. J., 1, no. 5 (1990), pp. 1229-1253. | MR | Zbl

[KL2] V. A. Kolyvagin - D. Y. Logachëv, Finiteness of I-I-I over totally real fields. (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 55, no. 4 (1991), pp. 851-876; translation in Math. USSR-Izv., 39, no. 1 (1992), pp. 829-853. | MR | Zbl

[L1] M. Longo, On the Birch and Swinnerton-Dyer conjecture over totally real fields. PhD thesis, Dipartimento di Matematica P. e A., Padova, 2004.

[L2] M. Longo, On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields. Ann. Inst. Fourier (Grenoble) 56, no. 3 (2006), pp. 689-733. | Numdam | MR | Zbl

[L3] M. Longo, On the anticyclotomic Iwasawa's main conjecture for Hilbert modular forms. Submitted.

[Mi] J. S. Milne, Arithmetic duality theorems. Perspectives in Mathematics, 1. Academic Press, Inc., Boston, MA, 1986. x+421 pp. | MR | Zbl

[Ra] A. Rajaei, On the levels of mod l Hilbert modular forms. J. Reine Angew. Math. 537 (2001), pp. 33-65. | MR | Zbl

[Ri] K. A. Ribet. On modular representations of Gal(Q=Q) arising from modular forms. Invent. Math., 100, no. 2 (1990), pp. 431-476. | MR | Zbl

[Ru] K. Rubin, Euler systems. Annals of Mathematics Studies, 147. Hermann Weyl Lectures. The Institute for Advanced Study. Princeton University Press, Princeton, NJ, 2000. | MR | Zbl

[Se] J.-P. Serre. Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Invent. Math., 15, no. 4 (1972), pp. 259-331. | MR | Zbl

[S1] G. Shimura, The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J., 45, no. 3 (1978), pp. 637-679. | MR | Zbl

[S2] G. Shimura, Introduction to the arithmetic theory of automorphic functions. KanÎ Memorial Lectures, No. 1. Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. xiv+267 pp. | MR | Zbl

[Ta] R. Taylor, On Galois representations associated to Hilbert modular forms. Invent. Math., 98, no. 2 (1989), pp. 265-280. | MR | Zbl

[Va] V. Vatsal, Special value formulae for Rankin L-functions. Math. Sci. Res. Inst. Publ., 49, Cambridge Univ. Press, Cambridge, 2004. | MR | Zbl

[Vi] M.-F. Vignéras. Arithmétique des algèbres de quaternions. (French) | Zbl

[Arithmetic of quaternion algebras] Lecture Notes in Mathematics, 800. Springer, Berlin, 1980. vii+169 pp. | MR

[Wa] J.-L. Waldspurger, Correspondances de Shimura et quaternions. (French) [Shimura correspondences and quaternions] Forum Math., 3, no. 3 (1991), pp. 219-307. | MR | Zbl

[Wi] A. Wiles, On ordinary l-adic representations associated to modular forms. Invent. Math., 94, no. 3 (1988), pp. 529-573. | MR | Zbl

[Z1] S. Zhang, Heights of Heegner points on Shimura curves. Ann. of Math. (2) 153, no. 1 (2001), pp. 27-147. | MR | Zbl

[Z2] S. Zhang, Gross-Zagier formula for GL2. Asian J. Math., 5, no. 2 (2001), pp. 183-290. | MR | Zbl