Periods and Reciprocity I

Abstract

Let F be a number field with adele ring AF, q, l two coprime integral ideals with q squarefree and π1, π2 two fixed unitary automorphic representations of PGL2(AF) unramified at all finite places. In this paper, we use regularized integrals to obtain a formula that links the first moment of L(π ⊗ π1 ⊗ π2, 1 2 ) twisted by the Hecke eigenvalues λπ(l), where π runs through unitary automorphic representations of PGL2(AF) with conductor dividing q, with some spectral expansion of periods over representations of conductor dividing l. In the special case where π1 = π2 = σ, this formula becomes a reciprocity relation between moments of L-functions. As applications, we obtain a subconvex estimate in the level aspect for the central value of the triple product L(π ⊗ π1 ⊗ π2, 1 2 ) and a simultaneous non-vanishing result for L(Sym 2 (σ) ⊗ π, 1

Spectral reciprocity for $\mathrm{GL}(n)$ and simultaneous non-vanishing of central $L$-values

“…A reciprocity formula, in this case, is implicit in [34], and, to our knowledge, the first instance of such an equality is in [1]. Later, a period approach to the equality was developed by Zacharias in [50] and [51]. The latter also deals with regularization and contain a strong subconvex bound for Lp1{2, σq, where σ is a PGLp2q cuspidal representation of prime level tending to infinity.…”

Mixed moment of GL(2) and GL(3)L‐functions

“…This result was refined by Young [31] and Bettin [5] and extended to the case of rational function fields by Djankovic [18]. Much activity of this type is currently being pursued for different families of L-functions, including Rankin-Selberg L-functions [1], cusp form L-functions [9,10] and triple product L-functions over a number field [33].…”

Motohashi’s fourth moment identity for non-archimedean test functions and applications

“…A heuristic argument based on a different strategy that we sketch in Subsection 1.4 suggests that we should expect something like This indicates that the period integral approach will not be straightforward to extend because at the very least some non-trivial combinatorics in the Hecke algebra (cf. [Zac18] how this could look like in a slightly different situation) have to happen to generate the Hecke eigenvalues on the right-hand side.…”