Abstract
This article is devoted to the elliptic Stark conjecture formulated by Darmon (Forum Math Pi 3:e8, 2015), which proposes a formula for the transcendental part of a p-adic avatar of the leading term at \(s=1\) of the Hasse–Weil–Artin L-series \(L(E,\varrho _1\otimes \varrho _2,s)\) of an elliptic curve \(E/\mathbb {Q}\) twisted by the tensor product \(\varrho _1\otimes \varrho _2\) of two odd 2-dimensional Artin representations, when the order of vanishing is two. The main ingredient of this formula is a \(2\times 2\) p-adic regulator involving the p-adic formal group logarithm of suitable Stark points on E. This conjecture was proved by Darmon (Forum Math Pi 3:e8, 2015) in the setting where \(\varrho _1\) and \(\varrho _2\) are induced from characters of the same imaginary quadratic field K. In this note, we prove a refinement of this result that was discovered experimentally by Darmon (Forum Math Pi 3:e8, 2015, [Remark 3.4]) in a few examples. Namely, we are able to determine the algebraic constant up to which the main theorem of Darmon (Forum Math Pi 3:e8, 2015) holds in a particular setting where the Hida–Rankin p-adic L-function associated to a pair of Hida families can be exploited to provide an alternative proof of the same result. This constant encodes local and global invariants of both E and K.
Similar content being viewed by others
Notes
Indeed, when g is cuspidal this hypothesis not only asks that \(\alpha \ne \beta \) but also that there should exist no real quadratic field F in which p splits such that \(\varrho _g \simeq {{\mathrm{Ind}}}_F^\mathbb {Q}(\xi )\) for some character \(\xi \) of F. However, in our CM setting, the existence of a character \(\xi \) of a real quadratic field F such that \(\mathrm {Ind}_\mathbb {Q}^K(\psi )\simeq \mathrm {Ind}_\mathbb {Q}^F(\xi )\) implies that \(\mathrm {Gal\,}(H/K)\simeq C_4\). Then F is the single real quadratic field contained in the quadratic extension of K cut out by \(\psi ^2\), and the condition \(\alpha \ne \beta \) implies that p cannot split in F.
Indeed, when \(\theta (\psi )\) is Eisenstein, it is shown in [10, Sect. 1] that a necessary condition for hypotheses C–C\({^\prime }\) to hold is that \(\alpha = \beta \), but this is automatically satisfied because \(\psi ^2=1\). As explained in loc. cit., this is also expected to be a sufficient condition.
References
Bertolini, M., Darmon, H.: Kato’s Euler system and rational points on elliptic curves I: a p-adic Beilinson formula. Isr. J. Math. 199(1), 163–188 (2014)
Bellaïche, J., Dimitrov, M.: On the eigencurve at classical weight one points. Duke Math. J. http://math.univ-lille1.fr/mladen/
Bertolini, M., Darmon, H., Prasanna, K.: Generalised Heegner cycles and \(p\)-adic Rankin L-series. Duke Math J. 162(6), 1033–1148 (2013)
Bertolini, M., Darmon, H., Prasanna, K.: p-Adic Rankin L-series and rational points on CM elliptic curves. Pac. J. Math. 260(2), 261–303 (2012)
Bertolini, M., Darmon, H., Rotger, V.: Beilinson-Flach elements and Euler systems I: syntomic regulators and \(p\)-adic Rankin L-series. J. Algebraic Geom. 24, 355–378 (2015)
Bertolini, M., Darmon, H., Rotger, V.: Beilinson-Flach elements and Euler systems II: the Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-series. J. Algebraic Geom. 24, 569–604 (2015)
Cox, D.A.: Primes of the Form \(x^2+ny^2\): Fermat, Class Field Theory and Complex Multiplication. Wiley, New York (1989)
Darmon, H.: Rational points on modular elliptic curves. In: CBMS Regional Conference Series in Mathematics, 101. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence (2004)
Darmon, H., Dasgupta, S.: Elliptic units for real quadratic fields. Ann. Math. 163(1), 301–346 (2006)
Darmon, H., Lauder, A., Rotger, V.: Stark points and p-adic iterated integrals attached to modular forms of weight one. Forum Math. Pi 3, e8–95 (2015)
Darmon, H., Pollack, R.: The efficient calculation of Stark-Heegner points via overconvergent modular symbols. Isr. J. Math. 153, 319–354 (2006)
Darmon, H., Rotger, V.: Diagonal cycles and Euler systems I: a p-adic Gross-Zagier formula. Ann. Scie. de l’Ecol. Norm. Supér. 47(4), 779–832 (2014)
Darmon, H., Rotger, V.: Diagonal cycles and Euler systems II: \(p\)-adic families and the Birch and Swinnerton-Dyer conjecture. J. Am. Math. Soc. (to appear)
De Shalit, E.: Iwasawa Theory of Elliptic Curves with Complex Multiplication. \(p\)-Adic \(L\)-Functions. Academic Press, Boston (1987)
Dokchitser, V.: \(L\)-functions of non-abelian twists of elliptic curves. Ph.D. dissertation, Cambridge (2005)
Gross, B.: Heegner points on \(X_0(N)\), Modular forms (Durham,1983): Ellis Horwood Ser. Math. Appli.: Statist. Oper. Res. Horwood, Chichester, pp. 87–105 (1984)
Gross, B.: On the factorization of \(p\)-adic L-series. Invent. Math. 57(1), 83–95 (1980)
Gross, B., Zagier, D.: Heegner points and derivatives of \(L\)-series. Invent. Math. 84, 225–320 (1986)
Hida, H.: Congruences of cusp forms and special values of their zeta functions. Invent. Math. 63, 225–261 (1981)
Hida, H.: Elementary Theory of \(L\)-Functions and Eisenstein Series, vol. 26. London Mathematical Society Student Texts, Cambridge (1993)
Kani, E.: The space of binary theta series. Ann. Sci. Math. Quebec 36, 501–534 (2012)
Kani, E.: Binary theta series and modular forms with complex multiplication. Int. J. Number Theory 10, 1025–1042 (2014)
Katz, N.M.: \(p\)-Adic interpolation of real analytic Eisenstein series. Ann. Math. 104(3), 459–571 (1976)
Katz, N.M.: \(p\)-Adic \(L\)-functions for CM fields. Invent. Math. 49, 199–297 (1978)
Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture (I). Invent. Math. 178, 485–504 (2009)
Lauder, A.: Computations with classical and p-adic modular forms. LMS J. Comput. Math. 14, 214–231 (2011)
Lauder, A.: Efficient computation of Rankin \(p\)-adic \(L\)-functions. In: Boeckle G., Wiese G. (eds.) Computations with Modular Forms, Proceedings of a Summer School and Conference, Heidelberg, August/September 2011, Springer, Berlin, pp. 181–200 (2014)
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)
Petersson, H.: Uber die Berechnung der skalarprodukte ganzer modulformen. Commun. Math. Helv. 22, 168–199 (1949)
Prasad, D.: Trilinear forms for representations of GL(2) and local epsilon factors. Compos. Math. 75, 1–46 (1990)
Roberts, G.: Unites elliptiques et formules pour le nombre de classes des extensions abeliennes d’un corps quadratique imaginaire. Bull. Soc. Math. France, Mém. 36, 77 (1973)
Rohrlich, D.: Root Numbers, Notes from PCMI/IAS: Arithmetic of L-Functions at Park City Mathematics Institute, Park City (2009)
Rubin, K.: \(p\)-Adic \(L\)-functions and rational points on elliptic curves with complex multiplication. Invent. Math. 107(2), 323–350 (1992)
Rubin, K.: \(p\)-Adic variants of the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication. In: \(p\)-Adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), pp. 71–80, Contemp. Math., vol. 165, Am. Math. Soc., Providence (1994)
Shimura, G.: The special values of the zeta functions associated with cusp forms. Commun. Pure. Appl. Math. 29, 783–804 (1976)
Silverman, J.H.: Advanced topics in the arithmetic of elliptic curves. Grad. Texts Math., vol. 151 (1994)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Casazza, D., Rotger, V. Stark points and the Hida–Rankin p-adic L-function. Ramanujan J 45, 451–473 (2018). https://doi.org/10.1007/s11139-016-9824-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-016-9824-y