Stark–Heegner points and special values of L-series (original) (raw)

Abstract

sparkles

AI

This paper investigates Stark-Heegner points associated with elliptic curves over real quadratic fields and their relation to special values of L-series. It builds on conjectures regarding the incorporation of these points into class fields of the quadratic fields, proposing generalizations to incorporate twisted special values of the L-series and examining the implications on the Mordell-Weil and Shafarevich-Tate groups. The findings suggest a connection between the vanishing of L-values and the structure of related algebraic groups, highlighting the broader arithmetic applications of Stark-Heegner points.

Loading...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (18)

  1. M. Bertolini, H.Darmon. The p-adic L-functions of modular ellip- tic curves. in Mathematics unlimited-2001 and beyond, 109-170, Springer, Berlin, 2001.
  2. M. Bertolini, H. Darmon. The main conjecture of Iwasawa theory for elliptic curves over anticyclotomic Z p -extensions. Annals of Mathe- matics (2) 162 (2005), no. 1, 1-64.
  3. M. Bertolini, H. Darmon. The rationality of Stark-Heegner points over genus fields of real quadratic fields. Annals of Mathematics, to appear.
  4. M. Bertolini, H. Darmon, and P. Green. Periods and points attached to quadratic algebras. MSRI Publ. 49, Cambridge Univ. Press, 323- 367, 2004.
  5. H. Darmon, Integration on H p × H and arithmetic applications. Ann. of Math. (2) 154 (2001), no. 3, 589-639.
  6. S. Dasgupta, Stark-Heegner points on modular Jacobians. Ann. Sci- ent. Éc. Norm. Sup., 4e sér., 38 (2005), 427-469.
  7. H. Darmon, P. Green, Elliptic curves and class fields of real quadratic fields: algorithms and verifications. Experimental Mathematics, 11:1, 2002, 37-55.
  8. L. Gerritzen, M. van der Put, Shottky Groups and Mumford Curves. Lecture Notes in Mathematics, 817. Springer, Berlin, 1980.
  9. R. Greenberg, G. Stevens, p-adic L-functions and p-adic periods of modular forms. Invent. Math. 111 (1993), no. 2, 407-447.
  10. B. H. Gross, D. B. Zagier, Heegner points and derivatives of L-series. Invent. Math. 84 (1986), no. 2, 225-320.
  11. M. Longo, On the Birch and Swinnerton-Dyer for modular elliptic curves over totally real fields, Ann. Inst. Fourier, to appear.
  12. J.I. Manin, Parabolic Points and Zeta Functions of Modular Curves. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), no. 1, 19-66.
  13. B. Mazur, J. Tate, J. Teitelbaum, On p-adic analogues of the conjec- tures of Birch and Swinnerton-Dyer. Invent. Math. 84 (1986), no. 1, 1-48.
  14. A. Popa. Central values of Rankin L-series over real quadratic fields. Submitted.
  15. K. Ribet, Congruence Relations Between Modular Forms. Proceed- ings of the International Congress of Mathematicians, Warsaw, Au- gust 16-24, 1983.
  16. K. Ribet, On modular representation of Gal( Q/Q) arising from mod- ular forms, Invent. Math. 100 (1990) 431-476.
  17. J.P. Serre, Trees. Springer, Berlin, 1980.
  18. A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. Math. 141 (1995) 443-551.