Duke Mathematical Journal

A p-adic Waldspurger formula

Yifeng Liu, Shouwu Zhang, and Wei Zhang

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In this article, we study p-adic torus periods for certain p-adic-valued functions on Shimura curves of classical origin. We prove a p-adic Waldspurger formula for these periods as a generalization of recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic p-adic L-function of Rankin–Selberg type. At a character of positive weight, the p-adic L-function interpolates the central critical value of the complex Rankin–Selberg L-function. Its value at a finite-order character, which is outside the range of interpolation, essentially computes the corresponding p-adic torus period.

Article information

Duke Math. J., Volume 167, Number 4 (2018), 743-833.

Received: 19 October 2014
Revised: 31 August 2017
First available in Project Euclid: 9 February 2018

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Zentralblatt MATH identifier

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35] 11J95: Results involving abelian varieties

p-adic L-function p-adic Waldspurger formula


Liu, Yifeng; Zhang, Shouwu; Zhang, Wei. A $p$ -adic Waldspurger formula. Duke Math. J. 167 (2018), no. 4, 743--833. doi:10.1215/00127094-2017-0045. https://projecteuclid.org/euclid.dmj/1518166812

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  • [1] M. Bertolini, H. Darmon, and K. Prasanna, Generalized Heegner cycles and $p$-adic Rankin $L$-series, with an appendix by B. Conrad, Duke Math. J. 162 (2013), 1033–1148.
  • [2] S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean Analysis, Grundlehren Math. Wiss. 261, Springer, Berlin, 1984.
  • [3] N. Bourbaki, Lie Groups and Lie Algebras, Chapters 1–3, reprint of the 1975 edition, Elem. Math. (Berlin), Springer, Berlin, 1989.
  • [4] E. H. Brooks, Generalized Heegner cycles, Shimura curves, and special values of $p$-adic $L$-functions, Ph.D. dissertation, University of Michigan, Ann Arbor, Michigan, 2013.
  • [5] D. Bump, Automorphic Forms and Representations, Cambridge Stud. Adv. Math. 55, Cambridge Univ. Press, Cambridge, 1997.
  • [6] L. Cai, J. Shu, and Y. Tian, Explicit Gross-Zagier and Waldspurger formulae, Algebra Number Theory 8 (2014), 2523–2572.
  • [7] H. Carayol, Sur la mauvaise réduction des courbes de Shimura, Compos. Math. 59 (1986), 151–230.
  • [8] R. F. Coleman, Torsion points on curves and p-adic abelian integrals, Ann. of Math. (2) 121 (1985), 111–168.
  • [9] F. Diamond and R. Taylor, Nonoptimal levels of mod $l$ modular representations, Invent. Math. 115 (1994), 435–462.
  • [10] M. Emerton, Locally analytic vectors in representations of locally p-adic analytic groups, Mem. Amer. Math. Soc. 248 (2017), no. 1175.
  • [11] G. Faltings, Group schemes with strict $\mathcal{O}$-action, Mosc. Math. J. 2 (2002), 249–279.
  • [12] B. H. Gross and D. B. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), 225–320.
  • [13] P. L. Kassaei, $\mathcal{P}$-adic modular forms over Shimura curves over totally real fields, Compos. Math. 140 (2004), 359–395.
  • [14] N. M. Katz, “Travaux de Dwork” in Séminaire Bourbaki 1971/1972 nos. 400–417, Lecture Notes in Math. 317, Springer, Berlin, 1973, 167–200.
  • [15] N. M. Katz, Higher congruences between modular forms, Ann. of Math. (2) 101 (1975), 332–367.
  • [16] N. M. Katz, $p$-adic $L$-functions for CM fields, Invent. Math. 49 (1978), 199–297.
  • [17] N. M. Katz, “Serre–Tate local moduli” in Algebraic Surfaces (Orsay, 1976–78), Lecture Notes in Math. 868, Springer, Berlin, 1981, 138–202.
  • [18] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture, I, Invent. Math. 178 (2009), 485–504; II, 505–586.
  • [19] J. S. Milne, “The points on a Shimura variety modulo a prime of good reduction” in The Zeta Functions of Picard Modular Surfaces, Univ. Montréal, Montreal, 1992, 151–253.
  • [20] D. Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compos. Math. 24 (1972), 239–272.
  • [21] A. Murase and H.-a. Narita, Fourier expansion of Arakawa lifting, II: Relation with central $L$-values, Internat. J. Math. 27 (2016), 1650001.
  • [22] K. A. Ribet, “Abelian varieties over Q and modular forms” in Algebra and Topology 1992 (Taej\uon), Korea Adv. Inst. Sci. Tech., Taej\uon, 1992, 53–79.
  • [23] K. Rubin, $p$-adic $L$-functions and rational points on elliptic curves with complex multiplication, Invent. Math. 107 (1992), 323–350.
  • [24] H. Saito, On Tunnell’s formula for characters of ${\mathrm{GL}}(2)$, Compos. Math. 85 (1993), 99–108.
  • [25] P. Schneider, Nonarchimedean Functional Analysis, Springer Monogr. Math., Springer, Berlin, 2002.
  • [26] P. Schneider and J. Teitelbaum, $p$-adic Fourier theory, Doc. Math. 6 (2001), 447–481.
  • [27] J. L. Taylor, Notes on locally convex topological vector spaces, preprint, https://www.math.utah.edu/~taylor/LCS.pdf, 1995.
  • [28] J. B. Tunnell, Local $\epsilon$-factors and characters of $\operatorname{GL}(2)$, Amer. J. Math. 105 (1983), 1277–1307.
  • [29] J.-L. Waldspurger, Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie, Compos. Math. 54 (1985), 173–242.
  • [30] X. Yuan, S.-W. Zhang, and W. Zhang, The Gross–Zagier Formula on Shimura Curves, Ann. of Math. Stud. 184, Princeton Univ. Press, Princeton, 2013.