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

ABSTRACT

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] 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.

KEYWORDS:

• Modular Abelian varieties
• p-adic L-invariants
• p-adic uniformization

2000 AMS Subject Classification:

• 11G40
• 11F41
• 11Y99

Acknowledgments

We wish to thank Lassina Dembele, Ariel Pacetti, Haluk Sengun, John Voight, and Xavier Xarles for feedback and helpful conversations during this project. Marc Masdeu thanks the Number Theory group of the University of Warwick for providing an outstanding working environment, and Xavier Guitart is thankful to the Essen Seminar for Algebraic Geometry and Arithmetic for their hospitality during his stay.

Funding

Xavier Guitart was supported by MTM2015-66716-P and MTM2015-63829, and Marc Masdeu was supported by MSC–IF–H2020–ExplicitDarmonProg. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement no. 682152).

Notes

2 In the context of L-simple algebras, by a subfield of D′ one understands a field contained in D′ and that contains L. But this is the case for M. Indeed, the compositum LM is a subfield in this sense, and therefore its dimension over is ⩽ 2d; then it has to equal 2d and LM = M.

3 We remark that the construction of the lattice works for any choice of signs at infinity, and we expect Conjecture 3.3 to hold for any choice. However, our numerical experiments have been done for eigenclasses with eigenvalues + 1 at infinity, so we prefer to consider this case.