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An Elementary Proof of a Weak Exceptional Zero Conjecture

Published online by Cambridge University Press:  20 November 2018

Louisa Orton
Louisa Orton*
Affiliation:
NWF I-Mathematik, Universität Regensburg, 93040 Regensburg, Germany e-mail: louisa.orton@mathematik.uni-regensburg.de
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Abstract

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In this paper we extend Darmon's theory of “integration on ${{\text{H}}_{p}}\times \text{H}$” to cusp forms $f$ of higher even weight. This enables us to prove a “weak exceptional zero conjecture”: that when the $p$-adic $L$-function of $f$ has an exceptional zero at the central point, the $\mathcal{L}$-invariant arising is independent of a twist by certain Dirichlet characters.

Keywords

11F1111F67
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 2 , 01 April 2004 , pp. 373 - 405
Copyright
Copyright © Canadian Mathematical Society 2004

References

[Col] Colmez, P., Théorie d'Iwasawa des Répresentations de de Rham d'un corps local. Ann. of Math. 148(1998), 485–571.Google Scholar
[Cr] Cremona, J. E., Algorithms for modular elliptic curves. CUP, (1992).Google Scholar
[Dar] Darmon, H., Integration on ℋp × ℋ and arithmetic applications. Ann. of Math. 154(2001), 589–639.Google Scholar
[GS] Greenberg, R. and Stevens, G., p-adic L-functions and p-adic periods of modular forms. Invent. Math. 111(1993), 401–447.Google Scholar
[Hi] Hida, H., Elementary theory of L-functions and Eisenstein series. London Mathematical Society Student Texts 26(1993).Google Scholar
[Kit] Kitagawa, K., On Standard p-adic L-functions of families of elliptic cusp forms. In: p-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture, Amer.Math. Soc., (1991).Google Scholar
[Mil] Mildenhall, S., Cycles in a product of elliptic curves and a group analogous to the class group. Duke Math J. (2) 67(1992), 387–406.Google Scholar
[Miy] Miyake, T., Modular Forms. Springer-Verlag, 1976.Google Scholar
[MTT] Mazur, B., Tate, J., Teitelbaum, J., On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84(1986), 1–48.Google Scholar
[Ro] Rohrlich, D. E., Nonvanishing of L-functions and structure of Mordell-Weil groups. J. Reine Angew. Math 417(1991), 1–26.Google Scholar
[Ser] Serre, J-P., Trees. Springer-Verlag, 1980.Google Scholar
[Tei] Teitelbaum, J., Values of p-adic L-functions and a p-adic Poisson Kernel. Invent. Math. 101(1990), 395–410.Google Scholar