p-adic L-functions and the Rationality of Darmon Cycles
Canadian journal of mathematics, Tome 64 (2012) no. 5, pp. 1122-1181
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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 ${{\Gamma }_{0}}\left( N \right)$ of even weight ${{k}_{0}}\,\ge \,2$ . 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 $\mathbb{Q}$ .
Seveso, Marco Adamo. p-adic L-functions and the Rationality of Darmon Cycles. Canadian journal of mathematics, Tome 64 (2012) no. 5, pp. 1122-1181. doi: 10.4153/CJM-2011-076-8
@article{10_4153_CJM_2011_076_8,
author = {Seveso, Marco Adamo},
title = {p-adic {L-functions} and the {Rationality} of {Darmon} {Cycles}},
journal = {Canadian journal of mathematics},
pages = {1122--1181},
year = {2012},
volume = {64},
number = {5},
doi = {10.4153/CJM-2011-076-8},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2011-076-8/}
}
TY - JOUR AU - Seveso, Marco Adamo TI - p-adic L-functions and the Rationality of Darmon Cycles JO - Canadian journal of mathematics PY - 2012 SP - 1122 EP - 1181 VL - 64 IS - 5 UR - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2011-076-8/ DO - 10.4153/CJM-2011-076-8 ID - 10_4153_CJM_2011_076_8 ER -
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