On $p$-adic $L$-functions for $\mathrm{GSp}_4 \times \mathrm{GL}_2$

David Loeffler and Óscar Rivero

Vol. 335 (2025), No. 2, 373–400

DOI: 10.2140/pjm.2025.335.373

Abstract

We use higher Coleman theory to construct a new $p$-adic $L$-function for ${\mathrm{GSp}<!--FUNCTION\; APPLICATION-->}_4\times {\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_2$. While Loeffler et al. (2021) had considered the $p$-adic variation of classes in the $H^2$ of Shimura varieties for ${\mathrm{GSp}<!--FUNCTION\; APPLICATION-->}_4$, here we explore the interpolation of classes in the $H^1$, which detect critical values for a different range of weights, disjoint from the range covered by this earlier construction. Using the algebraicity result established in our earlier work (Loeffler and Rivero 2024) we further show an interpolation property in terms of complex $L$-values.

Keywords

$p$-adic $L$-functions, higher Coleman theory, Siegel Shimura varieties

Mathematical Subject Classification

Primary: 11F46

Secondary: 11F67, 11R23

Milestones

Received: 13 April 2024

Revised: 29 January 2025

Accepted: 3 April 2025

Published: 12 May 2025