1. CJM 2011 (vol 64 pp. 588)
|Level Raising and Anticyclotomic Selmer Groups for Hilbert Modular Forms of Weight Two|
In this article we refine the method of Bertolini and Darmon and prove several finiteness results for anticyclotomic Selmer groups of Hilbert modular forms of parallel weight two.
Keywords:Hilbert modular forms, Selmer groups, Shimura curves
Categories:11G40, 11F41, 11G18
2. CJM 2009 (vol 62 pp. 456)
|The ChowlaâSelberg Formula and The Colmez Conjecture|
In this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.
Categories:11G15, 11F41, 14K22
3. CJM 2009 (vol 61 pp. 395)
|$L$-Functions for $\GSp(2)\times \GL(2)$: Archimedean Theory and Applications |
Let $\Pi$ be a generic cuspidal automorphic representation of $\GSp(2)$ defined over a totally real algebraic number field $\gfk$ whose archimedean type is either a (limit of) large discrete series representation or a certain principal series representation. Through explicit computation of archimedean local zeta integrals, we prove the functional equation of tensor product $L$-functions $L(s,\Pi \times \sigma)$ for an arbitrary cuspidal automorphic representation $\sigma$ of $\GL(2)$. We also give an application to the spinor $L$-function of $\Pi$.
Categories:11F70, 11F41, 11F46