We introduce a “geometric” method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the...

This paper draws connections between the double shuffle equations and structure of associators; Hain and Matsumoto’s universal mixed elliptic motives; and the Rankin–Selberg method for modular forms for SL2(Z). We write down explicit formulae for zeta elements σ2n−1 (generators of the Tannaka Lie algebra of the category of mixed Tate motives over Z) in depths up to four, give applications to th...

In this paper, we propose a conjectural identity between the Fourier–Jacobi periods on symplectic groups and the central value of certain Rankin–Selberg L-functions. This identity can be viewed as a refinement to the global Gan–Gross–Prasad conjecture for Sp(2n)×Mp(2m). To support this conjectural identity, we show that when n = m and n = m±1, it can be deduced from the Ichino–Ikeda’s conjectur...

All known ways to analytically continue automorphic L-functions involve integral representations using the corresponding automorphic forms. The simplest cases, extending Hecke’s treatment of GL2, need no further analytic devices and very little manipulation beyond Fourier-Whittaker expansions. [1] Poisson summation is a sufficient device for several accessible classes of examples, as in Riemann...

We prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points semi-stable elliptic curves E over a quadratic imaginary field K satisfying certain generalized hypothesis, at ordinary prime p. It states that the square of index family in equals characteristic ideal torsion part its Bloch-Kato Selmer group (see Theorem 1.3 precise statement). As byproduct we also eq...

Let K/F be a quadratic extension of p-adic fields, and n a positive integer. A smooth irreducible representation of the group GL(n, K) is said to be distinguished, if it admits on its space a nonzero GL(n, F )-invariant linear form. In the present work, we classify distinguished generic representations of the group GL(n, K) in terms of inducing quasi-square-integrable representations. This has ...

Let [Formula: see text] be a newform of even weight for text], where is possibly split indefinite quaternion algebra over text]. quadratic imaginary field splitting and an odd prime in We extend our theory text]-adic measures attached to the power series expansions around Galois orbit CM point corresponding embedding forms with any nebentypus dividing level For latter we restrict considerations...