p-adic interpolation of Gauss--Manin connections on nearly overconvergent modular forms and p-adic L-functions

TL;DR

This work develops a geometric framework for nearly overconvergent modular forms and a robust $p$-adic interpolation of the Gauss--Manin connection. It constructs an LB-space $\mathscr{N}^{\dagger}$ carrying $U_p$, $\varphi$, $S_p$, a $T(\mathbb{Z}_p)$-action, and a locally analytic action of $C^{la}(\mathbb{Z}_p,\mathbb{Q}_p)$, with a slope filtration and a natural map to the classical space of $p$-adic modular forms, enabling interpolation of powers of the Gauss--Manin connection through $\nabla^{\chi}$. The main advancement is removing the analyticity restriction present in previous work (AI_LLL) by realizing a geometric construction via Igusa towers and unit-root splittings, yielding a full interpolation theory of the differential operator in the finite-slope setting. This framework underpins new constructions of triple-product and Rankin--Selberg $p$-adic L-functions in families, connecting finite-slope modular forms to central $L$-values through explicit Euler factors and period integrals. The results thus significantly extend the reach of $p$-adic L-functions to non-ordinary families and set the stage for higher-dimensional Shimura varieties, with broad potential for arithmetic applications in the study of special values and Iwasawa theory.

Abstract

In this paper, we give a new geometric definition of nearly overconvergent modular forms and $p$-adically interpolate the Gauss-Manin connection on this space. This can be seen as an ``overconvergent'' version of the unipotent circle action on the space of $p$-adic modular forms, as constructed by Gouvêa and Howe. This improves on results of Andreatta--Iovita and has applications to the construction of Rankin--Selberg and triple product $p$-adic $L$-functions.

Theorems & Definitions (154)

• Theorem 1
• Remark 1.1.2
• Remark 1.1.3
• Definition 2.1.1
• Proposition 2.1.2
• proof
• Remark 2.1.3
• Definition 2.1.4
• Theorem 2.1.5
• proof