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

Let F be a number field and with rational Fourier coefficients. Under certain additional conditions, Guitart and colleagues [Guitart etal. 16] constructed a p-adic lattice which is conjectured to be the Tate lattice of an elliptic curve E-f 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 A(f) of dimension d. We also provide numerical evidence supporting the conjectural construction in the case of abelian surfaces.