Regulators, L-Functions and Rational Points

Massimo Bertolini

Bollettino dell'Unione Matematica Italiana (2013)

  • Volume: 6, Issue: 1, page 191-204
  • ISSN: 0392-4041

Abstract

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This article is a revised version of the text of the plenary conference I gave at the XIX Congress of ``Unione Matematica Italiana'', held in Bologna in September 2011. It discusses the arithmetic significance of the values at integers of the complex and p-adic L-functions associated to Dirichlet characters and to elliptic curves.

How to cite

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Bertolini, Massimo. "Regulators, L-Functions and Rational Points." Bollettino dell'Unione Matematica Italiana 6.1 (2013): 191-204. <http://eudml.org/doc/294029>.

@article{Bertolini2013,
abstract = {This article is a revised version of the text of the plenary conference I gave at the XIX Congress of ``Unione Matematica Italiana'', held in Bologna in September 2011. It discusses the arithmetic significance of the values at integers of the complex and p-adic L-functions associated to Dirichlet characters and to elliptic curves.},
author = {Bertolini, Massimo},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {191-204},
publisher = {Unione Matematica Italiana},
title = {Regulators, L-Functions and Rational Points},
url = {http://eudml.org/doc/294029},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Bertolini, Massimo
TI - Regulators, L-Functions and Rational Points
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/2//
PB - Unione Matematica Italiana
VL - 6
IS - 1
SP - 191
EP - 204
AB - This article is a revised version of the text of the plenary conference I gave at the XIX Congress of ``Unione Matematica Italiana'', held in Bologna in September 2011. It discusses the arithmetic significance of the values at integers of the complex and p-adic L-functions associated to Dirichlet characters and to elliptic curves.
LA - eng
UR - http://eudml.org/doc/294029
ER -

References

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  1. BEILINSON, A. A.Higher regulators of modular curves, Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), 1-34, Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 1986. MR862627DOI10.1090/conm/055.1/862627
  2. BREUIL, C. - CONRAD, B. - DIAMOND, F. - TAYLOR, R., On the modularity of elliptic curves over : wild 3-adic exercises, J. Amer. Math. Soc., 14, no. 4 (2001), 843-939. Zbl0982.11033MR1839918DOI10.1090/S0894-0347-01-00370-8
  3. BERTOLINI, M. - DARMON, H., Kato's Euler system and rational points on elliptic curves I: A p-adic Beilinson formula, Israel Journal of Math., to appear. Zbl1317.11071MR3219532DOI10.1007/s11856-013-0047-2
  4. BERTOLINI, M. - DARMON, H., Kato's Euler system and rational points on elliptic curves II: The explicit reciprocity law, in preparation. Zbl1317.11071
  5. BERTOLINI, M. - DARMON, H., Kato's Euler system and rational points on elliptic curves III: The conjecture of Perrin-Riou, in preparation. 
  6. BERTOLINI, M. - DARMON, H. - PRASANNA, K., Generalised Heegner cycles and p-adic Rankin L-series, Duke Math. J., to appear. MR3053566DOI10.1215/00127094-2142056
  7. BERTOLINI, M., Report on the Birch and Swinnerton-Dyer conjecture, Milan J. Math., 78, no. 1 (2010), 153-178. MR2684777DOI10.1007/s00032-010-0123-6
  8. BESSER, A., Syntomic regulators and p-adic integration. I. Rigid syntomic regulators, Proceedings of the Conference on p-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998) Israel J. Math., 120, part B (2000), 291-334. Zbl1001.19003MR1809626DOI10.1007/BF02834843
  9. BESSER, A., Syntomic regulators and p-adic integration. II. K 2 of curves, Proceedings of the Conference on p-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998). Israel J. Math., 120, part B (2000), 335-359. Zbl1001.19004MR1809627DOI10.1007/BF02834844
  10. BLOCH, S. J., Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, CRM Monograph Series, 11. American Mathematical Society, Providence, RI, 2000. x+97 pp. MR1760901
  11. BRUNAULT, F., Régulateurs p-adiques explicites pour le K 2 des courbes elliptiques, Actes de la Conférence ``Fonctions L et Arithmétique'', 29-57, Publ. Math. Besançon Algèbre Théorie Nr., Lab. Math. Besançon, Besançon, 2010. MR2744770
  12. COLEMAN, R. F., Dilogarithms, regulators and p-adic L-functions, Invent. Math., 69, no. 2 (1982), 171-208. Zbl0516.12017MR674400DOI10.1007/BF01399500
  13. COLEMAN, R. F. - DE SHALIT, E., p-adic regulators on curves and special values of p-adic L-functions, Inventiones Math., 93 (1988), 239-266. Zbl0655.14010MR948100DOI10.1007/BF01394332
  14. COLMEZ, P., La conjecture de Birch et Swinnerton-Dyer p-adique, (French) Astérisque No. 294 (2004), ix, 251-319. MR2111647
  15. DELIGNE, P., Valeurs de fonctions L et périodes d'intégrales, With an appendix by N. Koblitz and A. Ogus. Proc. Sympos. Pure Math., XXXIII, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 313-346, Amer. Math. Soc., Providence, R.I., 1979. MR546622
  16. DARMON, H. - ROTGER, V., Diagonal cycles and Euler systems I: A p-adic Gross- Zagier formula, submitted. Zbl1356.11039MR3250064DOI10.24033/asens.2227
  17. GEALY, M., On the Tamagawa number conjecture for motives attached to modular forms, PhD Thesis, California Institute of Technology, 2006. MR2709207
  18. GROSS, B. H. - ZAGIER, D. B., Heegner points and derivatives of L-series, Invent. Math., 84, no. 2 (1986), 225-320. Zbl0608.14019MR833192DOI10.1007/BF01388809
  19. KATO, K., p-adic Hodge theory and values of zeta functions of modular forms. Cohomologies p-adiques et applications arithmétiques. III, Astérisque No. 295 (2004), ix, 117-290. MR2104361
  20. KIM, M., Classical motives and motivic L-functions, Autour des motifs-École d'été Franco-Asiatique de Géométrie Algébrique et de Théorie des Nombres/Asian-French Summer School on Algebraic Geometry and Number Theory. Volume I, 1-25, Panor. Synthèses, 29, Soc. Math. France, Paris, 2009. MR2730655
  21. KITAGAWA, K., On standard p-adic L-functions of families of elliptic cusp forms, in p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), 81-110, Contemp. Math., 165, Amer. Math. Soc., Providence, RI, 1994. MR1279604DOI10.1090/conm/165/01611
  22. KOLYVAGIN, V. A., Euler systems, The Grothendieck Festschrift, Vol. II, 435-483, Progr. Math., 87, Birkhäuser Boston, Boston, MA, 1990. MR1106906
  23. LANG, S., Cyclotomic fields I and II. Combined second edition. With an appendix by Karl Rubin. Graduate Texts in Mathematics, 121. Springer-Verlag, New York, 1990, xviii+433. Zbl0704.11038MR1029028DOI10.1007/978-1-4612-0987-4
  24. MANIN, JU. I., Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 19-66. Zbl0243.14008MR314846
  25. MAZUR, B. - SWINNERTON-DYER, P., Arithmetic of Weil curves, Invent. Math., 25 (1974), 1-61. Zbl0281.14016MR354674DOI10.1007/BF01389997
  26. MAZUR, B. - TATE, J. - TEITELBAUM, J., On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math., 84, no. 1 (1986), 1-48. Zbl0699.14028MR830037DOI10.1007/BF01388731
  27. PERRIN-RIOU, B., Points de Heegner et dérivées de fonctions L p-adiques, Invent. Math., 89, no. 3 (1987), 455-510. Zbl0645.14010MR903381DOI10.1007/BF01388982
  28. PERRIN-RIOU, B., Fonctions L p-adiques d'une courbe elliptique et points rationnels, Ann. Inst. Fourier (Grenoble), 43, no. 4 (1993), 945-995. Zbl0840.11024MR1252935
  29. RUBIN, K. C., The main conjecture, Appendix to [La]. 
  30. SILVERMAN, J. H., The arithmetic of elliptic curves. Second edition. Graduate Texts in Mathematics, 106. Springer, Dordrecht, 2009. xx+513 pp. Zbl1194.11005MR2514094DOI10.1007/978-0-387-09494-6
  31. SOULÉ, C. , Éléments cyclotomiques en K-théorie, Journées arithmétiques de Besançon, Astérisque No. 147-148 (1987), 225-257. MR891430
  32. SKINNER, C. - URBAN, E., The Iwasawa Main Conjecture for G L 2 , preprint. MR3148103DOI10.1007/s00222-013-0448-1
  33. TAYLOR, R. - WILES, A., Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2), 141, no. 3 (1995), 553-572. Zbl0823.11030MR1333036DOI10.2307/2118560
  34. WASHINGTON, L. C., Introduction to cyclotomic fields. Second edition. Graduate Texts in Mathematics, 83. Springer-Verlag, New York, 1997. xiv+487 pp. Zbl0966.11047MR1421575DOI10.1007/978-1-4612-1934-7
  35. WASHINGTON, L. C., Euler factors for p-adic L-functions, Mathematika, 25, no. 1 (1978), 68-75. MR506178DOI10.1112/S002557930000927X
  36. WILES, A., Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2), 141, no. 3 (1995), 443-551. Zbl0823.11029MR1333035DOI10.2307/2118559
  37. WILES, A., The Birch and Swinnerton-Dyer conjecture, on the Clay Mathematics Institute web site: http://www.claymath.org/millennium/ Zbl1194.11006

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