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Bounding Selmer groups for the Rankin-Selberg convolution of Coleman families. Justification of hypotheses. (English) Zbl 1477.11093

Summary: 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 \(V \times \mathcal{W}\) that 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 \(L_p\).

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F85 \(p\)-adic theory, local fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G35 Modular and Shimura varieties
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References:

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