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# $p$-adic $L$-functions and the Rationality of Darmon Cycles

Published:2011-10-05
Printed: Oct 2012
• Marco Adamo Seveso,
Dipartimento di Matematica Federigo Enriques, Università degli studi di Milano
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## Abstract

Darmon cycles are a higher weight analogue of Stark--Heegner points. They yield local cohomology classes in the Deligne representation associated with a cuspidal form on $\Gamma _{0}( N)$ of even weight $k_{0}\geq 2$. They are conjectured to be the restriction of global cohomology classes in the Bloch--Kato Selmer group defined over narrow ring class fields attached to a real quadratic field. We show that suitable linear combinations of them obtained by genus characters satisfy these conjectures. We also prove $p$-adic Gross--Zagier type formulas, relating the derivatives of $p$-adic $L$-functions of the weight variable attached to imaginary (resp. real) quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express the second derivative of the Mazur--Kitagawa $p$-adic $L$-function of the weight variable in terms of a global cycle defined over a quadratic extension of $\mathbb{Q}$.
 MSC Classifications: 11F67 - Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 14G05 - Rational points

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