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Transactions of the American Mathematical Society

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Leading terms of anticyclotomic Stickelberger elements and $ p$-adic periods


Authors: Felix Bergunde and Lennart Gehrmann
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11F67; Secondary 11F75, 11G18, 11G40
DOI: https://doi.org/10.1090/tran/7120
Published electronically: February 21, 2018
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Abstract: Let $ E$ be a quadratic extension of a totally real number field. We construct Stickelberger elements for Hilbert modular forms of parallel weight 2 in anticyclotomic extensions of $ E$. Extending methods developed by Dasgupta and Spieß  from the multiplicative group to an arbitrary one-dimensional torus we bound the order of vanishing of these Stickelberger elements from below and, in the analytic rank zero situation, we give a description of their leading terms via automorphic $ \mathcal {L}$-invariants. If the field $ E$ is totally imaginary, we use the $ p$-adic uniformization of Shimura curves to show the equality between automorphic and arithmetic $ \mathcal {L}$-invariants. This generalizes a result of Bertolini and Darmon from the case that the ground field is the field of rationals to arbitrary totally real number fields.


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  • [1] Felix Bergunde and Lennart Gehrmann, On the order of vanishing of Stickelberger elements of Hilbert modular forms, Proc. Lond. Math. Soc. (3) 114 (2017), no. 1, 103-132. MR 3653078, https://doi.org/10.1112/plms.12004
  • [2] Massimo Bertolini and Henri Darmon, Heegner points, $ p$-adic $ L$-functions, and the Cerednik-Drinfeld uniformization, Invent. Math. 131 (1998), no. 3, 453-491. MR 1614543, https://doi.org/10.1007/s002220050211
  • [3] Massimo Bertolini and Henri Darmon, $ p$-adic periods, $ p$-adic $ L$-functions, and the $ p$-adic uniformization of Shimura curves, Duke Math. J. 98 (1999), no. 2, 305-334. MR 1695201, https://doi.org/10.1215/S0012-7094-99-09809-5
  • [4] A. Borel and J.-P. Serre, Corners and arithmetic groups, avec un appendice: Arrondissement des variétés à coins, par A. Douady et L. Hérault, Comment. Math. Helv. 48 (1973), 436-491. MR 0387495, https://doi.org/10.1007/BF02566134
  • [5] A. Borel and J.-P. Serre, Cohomologie d'immeubles et de groupes $ S$-arithmétiques, Topology 15 (1976), no. 3, 211-232 (French). MR 0447474, https://doi.org/10.1016/0040-9383(76)90037-9
  • [6] J. Boutot and T. Zink, The p-adic uniformisation of Shimura curves, Universität Bielefeld, Sonderforschungsbereich 343:95-107, 1995.
  • [7] Christophe Breuil, Invariant $ \mathcal{L}$ et série spéciale $ p$-adique, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 4, 559-610 (French, with English and French summaries). MR 2097893, https://doi.org/10.1016/j.ansens.2004.02.001
  • [8] William Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301-314. MR 0337789, https://doi.org/10.1007/BF01428197
  • [9] Laurent Clozel, Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic forms, Shimura varieties, and $ L$-functions, Vol.I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 77-159 (French). MR 1044819
  • [10] Christophe Cornut and Vinayak Vatsal, Nontriviality of Rankin-Selberg $ L$-functions and CM points, $ L$-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 121-186. MR 2392354, https://doi.org/10.1017/CBO9780511721267.005
  • [11] Samit Dasgupta, Stark-Heegner points on modular Jacobians, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 3, 427-469 (English, with English and French summaries). MR 2166341, https://doi.org/10.1016/j.ansens.2005.03.002
  • [12] S. Dasgupta and M. Spieß, The Eisenstein cocycle, partial zeta values and Gross-Stark units, Journal of the European Mathematical Society, to appear.
  • [13] Daniel File, Kimball Martin, and Ameya Pitale, Test vectors and central $ L$-values for $ {\rm GL}(2)$, Algebra Number Theory 11 (2017), no. 2, 253-318. MR 3641876, https://doi.org/10.2140/ant.2017.11.253
  • [14] Matthew Greenberg, Stark-Heegner points and the cohomology of quaternionic Shimura varieties, Duke Math. J. 147 (2009), no. 3, 541-575. MR 2510743, https://doi.org/10.1215/00127094-2009-017
  • [15] Ralph Greenberg and Glenn Stevens, $ p$-adic $ L$-functions and $ p$-adic periods of modular forms, Invent. Math. 111 (1993), no. 2, 407-447. MR 1198816, https://doi.org/10.1007/BF01231294
  • [16] Xavier Guitart, Marc Masdeu, and Mehmet Haluk Şengün, Darmon points on elliptic curves over number fields of arbitrary signature, Proc. Lond. Math. Soc. (3) 111 (2015), no. 2, 484-518. MR 3384519, https://doi.org/10.1112/plms/pdv033
  • [17] H. Jacquet and R. P. Langlands, Automorphic forms on $ {\rm GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654
  • [18] B. Mazur, J. Tate, and J. Teitelbaum, On $ p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), no. 1, 1-48. MR 830037, https://doi.org/10.1007/BF01388731
  • [19] S. Molina Blanco, Anticyclotomic p-adic L-functions and the exceptional zero phenomenon, arXiv e-prints, September 2015.
  • [20] Hiroshi Saito, On Tunnell's formula for characters of $ {\rm GL}(2)$, Compositio Math. 85 (1993), no. 1, 99-108. MR 1199206
  • [21] Jean-Pierre Serre, Cohomologie des groupes discrets, Séminaire Bourbaki, 23ème année (1970/1971), Exp. No. 399, Lecture Notes in Math., Vol. 244, pp. 337-350, Springer, Berlin, 1971 (French). MR 0422504
  • [22] Jean-Pierre Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell; Corrected 2nd printing of the 1980 English translation. MR 1954121
  • [23] Michael Spieß, On special zeros of $ p$-adic $ L$-functions of Hilbert modular forms, Invent. Math. 196 (2014), no. 1, 69-138. MR 3179573, https://doi.org/10.1007/s00222-013-0465-0
  • [24] Jerrold B. Tunnell, Local $ \epsilon$-factors and characters of $ {\rm GL}(2)$, Amer. J. Math. 105 (1983), no. 6, 1277-1307. MR 721997, https://doi.org/10.2307/2374441
  • [25] J. Van Order, p-adic interpolation of automorphic periods for GL(2), arXiv e-prints, September 2014.

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Additional Information

Felix Bergunde
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Universitätsstraße 25, 33615 Bielefeld, Germany
Email: fbergund@math.uni-bielefeld.de

Lennart Gehrmann
Affiliation: Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany
Email: lennart.gehrmann@uni-due.de

DOI: https://doi.org/10.1090/tran/7120
Received by editor(s): August 2, 2016
Received by editor(s) in revised form: November 7, 2016
Published electronically: February 21, 2018
Additional Notes: The first-named author was financially supported by the DFG within the CRC 701 ‘Spectral Structures and Topological Methods in Mathematics’.
Article copyright: © Copyright 2018 American Mathematical Society

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