Abstract
Let E be a modular elliptic curve over a totally real number field F. We prove the weak exceptional zero conjecture which links a (higher) derivative of the p-adic L-function attached to E to certain p-adic periods attached to the corresponding Hilbert modular form at the places above p where E has split multiplicative reduction. Under some mild restrictions on p and the conductor of E we deduce the exceptional zero conjecture in the strong form (i.e. where the automorphic p-adic periods are replaced by the \(\mathcal {L}\)-invariants of E defined in terms of Tate periods) from a special case proved earlier by Mok. Crucial for our method is a new construction of the p-adic L-function of E in terms of local data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In principle our construction is related to Manin’s [19]. However in our set-up the measure μ π is build in a simple manner from local distributions \(\mu_{\pi_{v}}\) at each place v of F.
Due to V. Paskunas.
References
Barthel, L., Livne, R.: Modular representations of GL 2 of a local field: the ordinary, unramified case. J. Number Theory 55, 1–27 (1995)
Bertolini, M., Darmon, H., Iovita, A.: Families of automorphic forms on definite quaternion algebras and Teitelbaum’s conjecture. Astérisque 331, 29–64 (2010)
Breuil, C.: Invariant \(\mathcal {L}\) et série spéciale p-adique. Ann. Sci. Éc. Norm. Super. 37, 559–610 (2004)
Brown, K.S.: Cohomology of Groups. Graduate Texts in Mathematics, vol. 87. Springer, Berlin (1982)
Bump, D.: Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics, vol. 55. Cambridge University Press, Cambridge (1996)
Bushnell, C., Henniart, G.: The Local Langlands Conjecture for \(\operatorname{GL}(2)\). Springer, Berlin (2006)
Carayol, H.: Sur les représentations ℓ-adiques associées aux formes modulaires de Hilbert. Ann. Sci. Éc. Norm. Super. 19, 409–468 (1986)
Casselman, W.: On some results of Atkin and Lehner. Math. Ann. 201, 301–314 (1973)
Charollois, P., Dasgupta, S.: Integral Eisenstein cocycles on \(\operatorname{GL}_{n}\), I: Szech’s cocycle and p-adic L-functions of totally real fields. Preprint, available at arXiv:1206.3050
Dabrowski, A.: P-adic L-functions of Hilbert modular forms. Ann. Inst. Fourier 44, 1025–1041 (1994)
Darmon, H.: Integration on \(\mathcal {H}_{p} \times \mathcal {H}\) and arithmetic applications. Ann. Math. 154, 589–639 (2001)
Friedberg, S., Hoffstein, J.: Nonvanishing theorems for automorphic L-functions on \(\operatorname{GL}(2)\). Ann. Math. 142, 385–423 (1995)
Greenberg, M.: Stark-Heegner points and the cohomology of quaternionic Shimura varieties. Duke Math. J. 147, 541–575 (2009)
Greenberg, M., Iovita, A., Spieß, M.: (in preparation)
Greenberg, R., Stevens, G.: p-adic L-functions and p-adic periods of modular forms. Invent. Math. 111, 407–447 (1993)
Harder, G.: Eisenstein cohomology of arithmetic groups. The case \(\operatorname{GL}_{2}\). Invent. Math. 89, 37–118 (1987)
Hida, H.: \(\mathcal {L}\)-invariants of Tate curves. Pure Appl. Math. Q. 5, 1343–1384 (2009)
Mac Lane, S.: Homology. Springer, Berlin (1963)
Manin, Y.: Non-archimedean integration and Jacquet-Langlands p-adic L-functions. Usp. Mat. Nauk 31, 5–54 (1976) (in Russian)
Mazur, B., Tate, J., Teitelbaum, J.: On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84, 1–48 (1986)
Mok, C.P.: The exceptional zero conjecture for Hilbert modular forms. Compos. Math. 145, 1–55 (2007)
Orton, L.: An elementary proof of a weak exceptional zero conjecture. Can. J. Math. 56, 373–405 (2004)
Panchishkin, A.A.: Non-Archimedean L-Functions Associated with Siegel and Hilbert Modular Forms. Lect. Notes in Math., vol. 1471. Springer, Berlin (1991)
Serre, J.-P.: Cohomologie des groupes discrets. Ann. Math. Stud. 70, 77–169 (1971)
Serre, J.-P.: Cours au Collège de France 1987–1988
Spieß, M.: Shintani cocycles and vanishing order of L p (χ,s) at s=0. Preprint, available at arXiv:1203.6689
Taylor, R.: On Galois representations associated to Hilbert modular forms. Invent. Math. 98, 265–280 (1989)
Teitelbaum, J.: Values of p-adic L-functions and a p-adic Poisson kernel. Invent. Math. 101, 395–410 (1990)
Waldspurger, J.: Correspondances de Shimura et quaternions. Forum Math. 3, 219–307 (1991)
Wintenberger, J.-P.: Potential modularity of elliptic curves over totally real fields. Appendix to: J. Nekovar, On the parity of ranks of Selmer groups IV. Compos. Math. 145, 1351–1359 (2009)
Acknowledgements
I thank Vytautas Paskunas for several helpful conversations and Kumar Murty for providing me with the reference [12]. I am grateful to H. Deppe, L. Gehrmann, S. Molina and M. Seveso for useful comments on an earlier draft. Also the referee suggested several useful improvements.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Spieß, M. On special zeros of p-adic L-functions of Hilbert modular forms. Invent. math. 196, 69–138 (2014). https://doi.org/10.1007/s00222-013-0465-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-013-0465-0