Quaternion algebras, Heegner points and the arithmetic of Hida families
Tóm tắt
Given a newform f, we extend Howard’s results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families of Heegner points on towers of Shimura curves. The novelty of our approach, which systematically exploits the theory of optimal embeddings, consists in treating both the case of definite quaternion algebras and the case of indefinite quaternion algebras in a uniform way. We prove results on the size of Nekovář’s extended Selmer groups attached to suitable big Galois representations and we formulate two-variable Iwasawa main conjectures both in the definite case and in the indefinite case. Moreover, in the definite case we propose refined conjectures à la Greenberg on the vanishing at the critical points of (twists of) the L-functions of the modular forms in the Hida family of f living on the same branch as f.
Tài liệu tham khảo
Bertolini M., Darmon H.: Heegner points on Mumford–Tate curves. Invent. Math. 126(3), 413–453 (1996)
Bertolini M., Darmon H.: A rigid analytic Gross–Zagier formula and arithmetic applications. Ann. Math. 2 146(1), 111–147 (1997)
Bertolini M., Darmon H.: Iwasawa’s Main Conjecture for elliptic curves over anticyclotomic \({\mathbb{Z}_p}\)-extensions. Ann. Math. 2 162(1), 1–64 (2005)
Bertolini M., Darmon H.: Hida families and rational points on elliptic curves. Invent. Math. 168(2), 371–431 (2007)
Bloch, S., Kato, K.: L-functions and Tamagawa numbers of motives. In: The Grothendieck Festschrift. Progress in Mathematics 86, vol. 1, pp. 333–400. Birkhõuser, Boston, MA (1990)
Carayol H.: Sur la mauvaise réduction des courbes de Shimura. Compos. Math. 59(2), 151–230 (1986)
Chida, M.: Selmer groups and central values of L-functions for modular forms (in preparation)
Chida, M.: On the anti-cyclotomic main conjecture for modular forms (in preparation)
Cornut C., Vatsal V.: CM points and quaternion algebras. Doc. Math. 10, 263–309 (2005)
Cornut, C., Vatsal, V.: Nontriviality of Rankin–Selberg L-functions and CM points. In: Burns, D., Buzzard, K., Nekovář, J. (eds.) L-functions and Galois representations. London Mathematical Society Lecture Note Series, vol. 320, pp. 121–186. Cambridge University Press, Cambridge (2007)
Delbourgo, D.: Elliptic curves and big Galois representations. In: London Mathematical Society Lecture Note Series, vol. 356. Cambridge University Press, Cambridge (2008)
Fouquet, O.: Tour de courbes de Shimura, systèmes de Kolyvagin et théorie d’Iwasawa des formes modulaires ordinaires. Ph. D. thesis, Université Paris VI (2007)
Fouquet, O.: Dihedral Iwasawa theory of ordinary quaternionic automorphic forms. submitted (2009)
Ghate E.: Ordinary forms and their local Galois representations. In: Tandon, R. (ed.) Algebra and Number Theory., pp. 226–242. Hindustan Book Agency, Delhi (2005)
Greenberg, R.: Elliptic curves and p-adic deformations. In: Kisilevsky, H., Ram Murty, M. (eds.) Elliptic Curves and Related Topics. CRM Proceedings and Lecture Notes, vol. 4, pp. 101–110. American Mathematical Society, Providence, RI (1994)
Greenberg R., Stevens G.: p-adic L-functions and p-adic periods of modular forms. Invent. Math. 111(2), 407–447 (1993)
Gross, B.H.: Heights and the special values of L-series. In: Kisilevsky, H., Labute, J. (eds.) Number Theory. CMS Conference Proceedings, vol. 7, pp. 115–187. American Mathematical Society, Providence, RI (1987)
Gross, B.H.: Kolyvagin’s work on modular elliptic curves. In: Coates, J., Taylor, M.J. (eds.) L-functions and Arithmetic. London Mathematical Society Lecture Note Series, vol. 153, pp. 235–256. Cambridge University Press, Cambridge (1991)
Gross B.H., Zagier D.B.: Heegner points and derivatives of L-series. Invent. Math. 84(2), 225–320 (1986)
Hida H.: Iwasawa modules attached to congruences of cusp forms. Ann. Sci. École Norm. Sup. 4 19(2), 231–273 (1986)
Hida H.: Galois representations into \({{\rm GL} _2(\mathbb{Z}_p{\left[\left[{X}\right]\right]})}\) attached to ordinary cusp forms. Invent. Math. 85(3), 545–613 (1986)
Hida H.: On p-adic Hecke algebras for GL2 over totally real fields. Ann. Math. 2 128(2), 295–384 (1988)
Hida H.: p-ordinary cohomology groups for SL(2) over number fields. Duke Math. J. 69(2), 259–314 (1993)
Hida H.: Hilbert Modular Forms and Iwasawa Theory, Oxford Mathematical Monographs. Oxford University Press, Oxford (2006)
Howard B.: Variation of Heegner points in Hida families. Invent. Math. 167(1), 91–128 (2007)
Howard B.: Central derivatives of L-functions in Hida families. Math. Ann. 339(4), 803–818 (2007)
Longo M., Vigni S.: On the vanishing of Selmer groups for elliptic curves over ring class fields. J. Number Theory 150(1), 128–163 (2010)
Matsumura H.: Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1989)
Mazur B.: An introduction to the deformation theory of Galois representations. In: Cornell, G., Silverman, J.H., Stevens, G. (eds) Modular Forms and Fermat’s Last Theorem, pp. 243–311. Springer, New York (1997)
Mazur, B., Ribet, K.: Two-dimensional representations in the arithmetic of modular curves. In: Courbes modulaire et courbes de Shimura (Orsay 1987/1988), vol. 196–197, pp. 215–255. Astérisque (1991)
Mazur B., Tilouine J.: Représentations galoisiennes, différentielles de Kähler et “Conjectures Principales”. Publ. Math. Inst. Hautes Études Sci. 71, 65–103 (1990)
Mazur B., Tate J., Teitelbaum J.: On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84(1), 1–48 (1986)
Nekovář J.: Kolyvagin’s method for Chow groups of Kuga–Sato varieties. Invent. Math. 107(1), 99–125 (1992)
Nekovář, J.: Selmer Complexes, vol. 310. Astérisque (2006)
Nekovář J., Plater A.: On the parity of ranks of Selmer groups. Asian J. Math. 4(2), 437–498 (2000)
Neukirch J.: Class Field Theory, Grundlehren der mathematischen Wissenschaften, vol. 280. Springer-Verlag, Berlin (1986)
Ochiai T.: On the two-variable Iwasawa main conjecture. Compos. Math. 142(5), 1157–1200 (2006)
Ogg A.: Real points on Shimura curves. In: Artin, M., Tate, J. (eds) Arithmetic and Geometry I. Progress in Mathematics, vol. 35, pp. 277–307. Birkhäuser, Boston (1983)
Perrin-Riou B.: Fonctions L p-adiques, théorie d’Iwasawa et points de Heegner. Bull. Soc. Math. France 115, 399–456 (1987)
Pizer A.: On the arithmetic of quaternion algebras. Acta Arith. 31(1), 61–89 (1976)
Pizer A.: Theta series and modular forms of level p 2 M. Compos. Math. 40(2), 177–241 (1980)
Ribet, K.: Galois representations attached to eigenforms with Nebentypus. In: Serre, J.-P., Zagier, D.B. (eds.) Modular functions of one variable V. Lecture Notes in Mathematics, vol. 601, pp. 17–52. Springer-Verlag, Berlin (1977)
Ribet, K.: Multiplicities of Galois representations in Jacobians of Shimura curves. In: Gelbart, S., Howe, R., Sarnak, P. (eds.) Festschrift in Honor of I.I. Piatetski- Shapiro on the Occasion of his Sixtieth Birthday, Part II. Israel Mathematical Conference Proceedings, vol. 3, pp. 221–236. Weizmann Science Press of Israel, Jerusalem (1990)
Shimura G.: Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, NJ (1994)
Swanson, I., Huneke, C.: Integral closure of ideals, rings, and modules. In: London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)
Vignéras, M.-F.: Arithmétique des algèbres de quaternions. In: Lecture Notes in Mathematics, vol. 800. Springer-Verlag, Berlin (1980)
Zhang S.-W.: Gross–Zagier formula for GL2Asian J. Math. 5(2), 183–290 (2001)