Periods of Modular GL2-type Abelian Varieties and p-adic Integration

Let F be a number field and an integral ideal. Let f be a modular newform over F of level with rational Fourier coefficients. Under certain additional conditions, Guitart and colleagues [Guitart et al. 16[Guitart et al. 16] X. Guitart, M. Masdeu, and M. Haluk Şengün. "Uniformization of Modular Elliptic Curves via p-adic Periods." J. Algebra 445 (2016), 458-502. MR 3418066 [Crossref], [Web of Science ®] , [Google Scholar] ] constructed a p-adic lattice which is conjectured to be the Tate lattice of an elliptic curve Ef whose L-function equals that of f. The aim of this note is to generalize this construction when the Hecke eigenvalues of f generate a number field of degree d ⩾ 1, in which case the geometric object associated with f is expected to be, in general, an abelian variety Af of dimension d. We also provide numerical evidence supporting the conjectural construction in the case of abelian surfaces

An automorphic approach to Darmon points

We give archimedean and non-archimedean constructions of Darmon points on modular abelian varieties attached to automorphic forms over arbitrary number fields and possibly non-trivial central character. An effort is made to present a unifying point of view, emphasizing the automorphic nature of the construction.Peer ReviewedPostprint (author's final draft

Periods of Modular GL2-type Abelian Varieties and p-adic Integration

Let F be a number field and an integral ideal. Let f be a modular newform over F of level with rational Fourier coefficients. Under certain additional conditions, Guitart and colleagues [Guitart et al. 16[Guitart et al. 16] X. Guitart, M. Masdeu, and M. Haluk Şengün. "Uniformization of Modular Elliptic Curves via p-adic Periods." J. Algebra 445 (2016), 458-502. MR 3418066 [Crossref], [Web of Science ®] , [Google Scholar] ] constructed a p-adic lattice which is conjectured to be the Tate lattice of an elliptic curve Ef whose L-function equals that of f. The aim of this note is to generalize this construction when the Hecke eigenvalues of f generate a number field of degree d ⩾ 1, in which case the geometric object associated with f is expected to be, in general, an abelian variety Af of dimension d. We also provide numerical evidence supporting the conjectural construction in the case of abelian surfaces