On coefficients of Poincaré series and single-valued periods of modular forms

We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level $$\varGamma _0(N)$$ Γ 0 ( N ) and integral weight $$k\ge 2$$ k ≥ 2 coincides with the field generated by the single-valued periods of a certain motive attached to $$\varGamma _0(N)$$ Γ 0 ( N ) . This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres–Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann–Ono’s construction of harmonic lifts of Poincaré series.

Periods of Modular Forms and Imaginary Quadratic Base Change

Let f be a classical newform of weight 2 on the upper half-plane , E the corresponding strong Weil curve, K a class number one imaginary quadratic field, and F the base change of f to K. Under a mild hypothesis on the pair (f, K), we prove that the period ratio is in ℚ. Here ΩF is the unique minimal positive period of F, and ΩE the area of E(ℂ). The claim is a specialization to base change forms of a conjecture proposed and numerically verified by Cremona and Whitley.

The exceptional zero conjecture for Hilbert modular forms

Using ap-adic analogue of the convolution method of Rankin–Selberg and Shimura, we construct the two-variablep-adicL-function of a Hida family of Hilbert modular eigenforms of parallel weight. It is shown that the conditions of Greenberg–Stevens [R. Greenberg and G. Stevens,p-adic L-functions and p-adic periods of modular forms, Invent. Math.111(1993), 407–447] are satisfied, from which we deduce special cases of the Mazur–Tate–Teitelbaum conjecture in the Hilbert modular setting.