# Bounding Selmer groups for the Rankin--Selberg convolution of Coleman families

## TL;DR

This paper constructs a sheaf interpolating Bloch-Kato Selmer groups for Rankin--Selberg convolutions of Coleman families, linking its support to the vanishing of the associated $p$-adic $L$-function.

## Contribution

It introduces a new coherent sheaf over weight space that interpolates Selmer groups in the critical range for Rankin--Selberg convolutions of Coleman families.

## Key findings

- Support of the sheaf is contained in the vanishing locus of the $p$-adic $L$-function.
- Provides a geometric framework connecting Selmer groups and $p$-adic $L$-functions.
- Extends previous work by interpolating Selmer groups across families of modular forms.

## Abstract

Let $f$ and $g$ be two cuspidal modular forms and let $\mathcal{F}$ be a Coleman family passing through $f$, defined over an open affinoid subdomain $V$ of weight space $\mathcal{W}$. Using ideas of Pottharst, under certain hypotheses on $f$ and $g$ we construct a coherent sheaf over $V \times \mathcal{W}$ which interpolates the Bloch-Kato Selmer group of the Rankin-Selberg convolution of two modular forms in the critical range (i.e. the range where the $p$-adic $L$-function $L_p$ interpolates critical values of the global $L$-function). We show that the support of this sheaf is contained in the vanishing locus of $L_p$.