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3 - p-adic L-functions and Euler systems: a tale in two trilogies

Published online by Cambridge University Press:  05 October 2014

Massimo Bertolini
Affiliation:
Università di Milano
Francesc Castella
Affiliation:
University of California
Henri Darmon
Affiliation:
McGill University
Samit Dasgupta
Affiliation:
Univeristy of California
Kartik Prasanna
Affiliation:
University of Michigan
Victor Rotger
Affiliation:
Universitat Politècnica de Catalunya
Fred Diamond
Affiliation:
King's College London
Payman L. Kassaei
Affiliation:
King's College London
Minhyong Kim
Affiliation:
University of Oxford
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Publisher: Cambridge University Press
Print publication year: 2014

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References

[BD1] M., Bertolini, H., Darmon, Kato' Euler system and rational points on elliptic curves I: A p-adic Beilinson formula, Israel J. Math., to appear.
[BD2] M., Bertolini, H., Darmon, Kato' Euler system and rational points on elliptic curves II: The explicit reciprocity law, in preparation.
[BDP] M., Bertolini, H., Darmon, K., Prasanna, Generalised Heegner cycles and p-adic Rankin L-series, Duke Math. J. 162, (2013) no. 6, 1033–1148.
[BDR1] M., Bertolini, H., Darmon, V., Rotger, Beilinson-Flach elements and Euler systems I: syntomic regulators and p-adic Rankin L-series, submitted for publication.
[BDR2] M., Bertolini, H., Darmon, V., Rotger, Beilinson-Flach elements and Euler systems II: p-adic families and the Birch and Swinnerton-Dyer conjecture, in preparation.
[Bei1] A.A., Beilinson, Higher regulators and values of L-functions, in Current problems in mathematics 24, 181–238, Akad. Nauk SSSR, Vsesoyuz, Inst. Nauchn. Tekhn. Inform., Moscow, 1984.
[Bei2] A.A., Beilinson, 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.
[Bes1] A., Besser, A generalization of Coleman' p-adic integration theory, Invent. Math. 142 (2000), 397–434.Google Scholar
[Bes2] A., Besser, Syntomic regulators and p-adic integration I: rigid syntomic regulators, Israel J. Math. 120 (2000), 291–334.Google Scholar
[Bes3] A., Besser, Syntomic regulators and p-adic integration, II. K2 of curves. Proceedings of the Conference on p-adic Aspects of the Theory of Auto-morphic Representations (Jerusalem, 1998). Israel J. Math. 120 (2000), part B, 335–359.Google Scholar
[Bes4] A., Besser, On the syntomic regulator for K1 of a surface, Israel J. Math. 190 (2012) 29–66.Google Scholar
[BK] Bannai, Kenichi; Kings, Guido. p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure, Amer. J. Math. 132 (2010), no. 6, 1609–1654.Google Scholar
[Br] F., Brunault, Valeur en 2 de fonctions L de formes modulaires de poids 2: théorème de Beilinson expliref, Bull. Soc. Math. France 135 (2007), no. 2, 215–246.Google Scholar
[Cas1] F., Castella, Heegner cycles and higher weight specializations of big Heegner points, Math. Annalen 356 (2013), 1247–1282.Google Scholar
[Cas2] F., Castella, p-adic L-functions and the p-adic variation of Heegner points, submitted.
[Cit] C., Citro, ℒ -invariants of adjoint square Galois representations coming from modular forms, Int. Math. Res. Not. (2008), no. 14.Google Scholar
[Col] R.F., Coleman, A p-adic Shimura isomorphism and p-adic periods of modular forms, in p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), 21–51, Contemp. Math. 165, Amer. Math. Soc., Providence, RI, 1994.
[CW] J., Coates and A., Wiles. On the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 39 (3) (1977) 223–251.Google Scholar
[Cz1] P., Colmez. Fonctions L p-adiques, Séminaire Bourbaki, Vol. 1998/99. Astérisque 266 (2000), Exp. No. 851, 3, 21–58.Google Scholar
[Cz2] P., Colmez. La conjecture de Birch et Swinnerton-Dyer p-adique, Astérisque 294 (2004), ix, 251–319.Google Scholar
[DR1] H., Darmon, V., Rotger, Diagonal cycles and Euler systems I: A p-adic Gross-Zagier formula, Ann. Sci. Ec. Norm. Supé., to appear.
[DR2] H., Darmon, V., Rotger, Diagonal cycles and Euler systems II: the Birch– Swinnerton-Dyer conjecture for Hasse–Weil–Artin L-series, submitted.
[DRS1] H., Darmon, V., Rotger, I., Sols, Iterated integrals, diagonal cycles, and rational points on elliptic curves, Publications Mathématiques de Besançon 2 (2012), 19–46.Google Scholar
[Das] S., Dasgupta, Factorization of p-adic Rankin L-series, in preparation.
[Gar] P., Garrett, Decomposition of Eisenstein series: Rankin triple products, Ann. Math. 125 (1987), 209–235.Google Scholar
[Gr] B., Gross. On the factorization of p-adic L-series. Invent. Math. 57 (1980), no. 1, 83–95.Google Scholar
[GrKu] B., Gross, S., Kudla, Heights and the central critical values of triple product L-functions, Compositio Math. 81 (1992), no. 2, 143–209.Google Scholar
[GrSc] B., Gross, C., Schoen, The modified diagonal cycle on the triple product of a pointed curve, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 3, 649–679.Google Scholar
[GZ] B.H., Gross and D.B., Zagier. Heegner points and derivatives of L-series. Invent. Math. 84 (1986), no. 2, 225–320.Google Scholar
[HaKu] M., Harris, S., Kudla, The central critical value of a triple product L-function, Ann. Math. (2) 133 (1991), 605–672.Google Scholar
[HaTi] M., Harris and J., Tilouine, p-adic measures and square roots of special values of triple product L-functions, Math. Annalen 320 (2001), 127–147.Google Scholar
[Hi1] H., Hida, A p-adic measure attached to the zeta functions associated with two elliptic modular forms II, Ann. Inst. Fourier (Grenoble) 38 (1988) 1–83.Google Scholar
[Hi2] H., Hida, Elementary theory of L-functions and Eisenstein series, London Mathematical Society Student Texts 26, 1993.
[How] B., Howard, Variation of Heegner points in Hida families, Invent. Math. 167 (2007), 91–128.Google Scholar
[Kato] K., Kato, 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.Google Scholar
[Katz] N.M., Katz, p-adic interpolation of real analytic Eisenstein series, Ann. Math. (2) 104 (1976), no. 3, 459–571.Google Scholar
[Ko] V.A., Kolyvagin, Euler systems, The Grothendieck Festschrift, Vol. II, 435–483, Progr. Math. 87, Birkhäuser Boston, Boston, MA, 1990.
[Lau] A., Lauder, Efficient computation of Rankin p-adic L-functions, to appear in the proceedings of the Heidelberg conference “Computations with Modular Forms 2011”.
[LLZ] A., Lei, D., Loeffler, S. L., Zerbes, Euler systems for Rankin-Selberg convolutions of modular forms, to appear in Ann. Math.
[MR] B., Mazur, K., Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168, no. 799, 2004.
[MTT] B., Mazur, J., Tate, J., Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), no. 1, 1–48.Google Scholar
[Nik] M., Niklas, Rigid syntomic regulators and the p-adic L-function of a modular form, Regensburg PhD Thesis, 2010, available at http://epub.uniregensburg.de/19847/
[PR1] B., Perrin-Riou, Point de Heegner et dérivées de fonctions L p-adiques, Invent. Math. 89 (1987), no. 3, 455–510.Google Scholar
[PR2] B., Perrin-Riou, Fonctions L p-adiques d'une courbe elliptique et points rationnels, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 4, 945–995.Google Scholar
[PR3] B., Perrin-Riou, Théorie d'Iwasawa des représentations p-adiques sur un corps local, with an appendix by J.-M. Fontaine, Invent. Math. 115 (1994), no. 1, 81–161.Google Scholar
[PR4] B., Perrin-Riou, La fonction L p-adique de Kubota-Leopoldt, Arithmetic geometry (Tempe, AZ, 1993), 65–93, Contemp. Math. 174, Amer. Math. Soc., Providence, RI, 1994.
[Ru] K., Rubin, Euler systems. Annals of Mathematics Studies 147, Hermann Weyl Lectures, The Institute for Advanced Study, Princeton University Press, Princeton, NJ, 2000.
[Se] J.-P., Serre, Formes modulaires et fonctions zêta p-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), pp. 191–268. Lecture Notes in Math. 350, Springer, Berlin, 1973.
[Sh1] G., Shimura, The special values of the zeta functions associated with cusp forms, Commun. Pure and Appl. Math. 29, (1976) 783–795.Google Scholar
[Sh2] G., Shimura, On a class of nearly holomorphic automorphic forms, Ann. Math. 123 (1986), 347–406.Google Scholar
[YZZ] X., Yuan, S., Zhang, W., Zhang, Triple product L-series and Gross–Schoen cycles I: split case, preprint.
[Zh] S., Zhang, Arithmetic of Shimura curves, Sci. China Math. 53 (2010), 573–592.Google Scholar
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