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arXiv · 1905.08002

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

Andrew Graham·Daniel R. Gulotta·Yujie Xu
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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 $L_p$.

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Andrew Graham, Daniel R. Gulotta, Yujie Xu. 2019-05-20. Bounding Selmer groups for the Rankin--Selberg convolution of Coleman families. https://doi.org/10.4153/s0008414x2000019x

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