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 Γ_0(N) of even weight k≥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 Q.
p-adic L-functions and the rationality of Darmon cycles / M.A. Seveso. - In: CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES. - ISSN 0008-414X. - 64:5(2012), pp. 1122-1181. [10.4153/CJM-2011-076-8]
p-adic L-functions and the rationality of Darmon cycles
M.A. SevesoPrimo
2012
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 Γ_0(N) of even weight k≥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 Q.Pubblicazioni consigliate
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