Triple product p-adic L-functions for balanced weights

Abstract

We construct p-adic triple product L-functions that interpolate (square roots of) central critical L-values in the balanced region. Thus, our construction complements that of M. Harris and J. Tilouine.There are four central critical regions for the triple product L-functions and two opposite settings, according to the sign of the functional equation. In the first case, three of these regions are of interpolation, having positive sign; they are called the unbalanced regions and one gets three p-adic L-functions, one for each region of interpolation (this is the Harris-Tilouine setting). In the other setting there is only one region of interpolation, called the balanced region. An especially interesting feature of our construction is that we get three different p-adic triple product L-functions with the same (balanced) region of interpolation. To the best of the authors' knowledge, this is the first case where an interpolation problem is solved on a single critical region by different p-adic L-functions at the same time. This is possible due to the structure of the Eulerlike factors at p arising in the interpolation formulas, the vanishing of which are related to the dimensions of certain Nekovar period spaces. Our triple product p-adic L-functions arise as specializations of p-adic period integrals interpolating normalizations of the local archimedean period integrals. The latter encode information about classical representation theoretic branching laws. The main step in our construction of p-adic period integrals is showing that these branching laws vary in a p-adic analytic fashion. This relies crucially on the Ash-Stevens theory of highest weight representations over affinoid algebras.2010 Mathematics Subject Classification. 11F67.

Parabolic eigenvarieties via overconvergent cohomology

“…-Triple product p-adic L-functions. In [GS20], the authors use overconvergent methods to construct triple product p-adic L-functions in the balanced region of weight space, interpolating the central critical value of the triple product L-function of a trio of Coleman families over this region.…”

Arithmetic applications of the Euler systems of Beilinson-Flach elements and diagonal cycles

“…[BD14, Section 3]. If only h is Eisenstein but both f and g are cuspidal, we obtain the so-called three-variable Hida-Rankin p-adic L-function L p (f , g) attached to two Hida families, already constructed in [Hi85] and [Hi88]; and when all three Hida families are cuspidal, this is the triple product p-adic L-function constructed in different instances by Ichino, Harris-Tilouine, Darmon-Rotger or more recently by Hsieh [Hs20], whose construction also works for interpolating along the balanced region, where k ≤ + m; ≤ m + k; and m ≤ k + (this generalizes earlier work of Greenberg and Seveso [GS20]). We extensively discuss these different instances and the main properties of each p-adic L-function along the text.…”

Overconvergent cohomology, p-adic L-functions and families for GL(2) over CM fields

“…• Triple product p-adic L-functions. In [21], the authors use overconvergent methods to construct triple product p-adic L-functions in the balanced region of weight space, interpolating the central critical value of the triple product L-function of a trio of Coleman families over this region.…”