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, 11F67 show english descriptions Holomorphic modular forms of integral weight
Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 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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