$p$-adic $L$-functions for $\mathrm{GSp}(4)\times \mathrm{GL}(2)$, II

Zheng Liu

doi:10.5802/jep.342

$p$-adic $L$-functions for $\mathrm{GSp}(4)\times \mathrm{GL}(2)$, II
[Fonctions $L$ $p$-adiques pour $\mathrm{GSp}(4)\times \mathrm{GL}(2)$, II]

Zheng Liu ¹

¹Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA 93106-3080, USA

Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1029-1060

Abstract (VO)
Résumé

We construct a four-variable $p$-adic $L$-function for cuspidal Hida families on $\mathrm{GSp}(4)\times \mathrm{GL}(2)$ and prove a complete interpolation formula for it. The Archimedean zeta integrals are computed by using a partial interpolation formula for the four-variable $p$-adic $L$-function, combined with Yoshida lifts and some previously constructed $p$-adic $L$-functions—specifically Kubota–Leopoldt $p$-adic $L$-functions, Rankin–Selberg $p$-adic $L$-functions, and $p$-adic (standard) $L$-functions for $\mathrm{Sp}(4)$.

Nous construisons une fonction $L$ $p$-adique à quatre variables pour les familles de Hida cuspidales sur $\mathrm{GSp}(4)\times \mathrm{GL}(2)$ et en démontrons une formule d’interpolation complète. Les intégrales zêta archimédiennes sont calculées à l’aide d’une formule d’interpolation partielle pour la fonction $L$ $p$-adique à quatre variables, combinée à des relèvements de Yoshida et à certaines fonctions $L$ $p$-adiques déjà construites auparavant — notamment les fonctions $L$ $p$-adiques de Kubota-Leopoldt, les fonctions $L$ $p$-adiques de Rankin-Selberg et les fonctions $L$ $p$-adiques (standard) pour $\mathrm{Sp}(4)$.

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Reçu le : 2025-04-02
Accepté le : 2026-04-10
Publié le : 2026-05-22

DOI : 10.5802/jep.342

Classification : 11F67, 11F33, 11F55, 11F27, 11R23
Keywords: $p$-adic L-functions, $p$-adic families of automorphic forms, Archimedean zeta integrals
Mots-clés : Fonctions $L$ $p$-adiques, familles $p$-adiques de formes automorphes, intégrales zêta archimédiennes

Affiliations des auteurs :

Zheng Liu ¹

¹ Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA 93106-3080, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

Zheng Liu. $p$-adic $L$-functions for $\mathrm{GSp}(4)\times \mathrm{GL}(2)$, II. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1029-1060. doi: 10.5802/jep.342

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     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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