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# An Elementary Proof of a Weak Exceptional Zero Conjecture

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Published:2004-04-01
Printed: Apr 2004
• Louisa Orton
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## Abstract

In this paper we extend Darmon's theory of integration on $\uh_p\times \uh$'' 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.
 MSC Classifications: 11F11 - Holomorphic modular forms of integral weight 11F67 - Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols

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