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Bounding Selmer Groups for the Rankin–Selberg Convolution of Coleman Families

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  17 July 2020

Andrew Graham ,
Daniel R. Gulotta  and
Yujie Xu
Andrew Graham*
Affiliation:
Department of Mathematics, South Kensington Campus, Imperial College London, LondonSW7 2AZ, UK
Daniel R. Gulotta
Affiliation:
Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK e-mail: Daniel.Gulotta@maths.ox.ac.uk
Yujie Xu
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA02138 e-mail: yujiex@math.harvard.edu
*
e-mail: a.graham17@imperial.ac.uk
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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 $V \times \mathcal {W}$ 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 $L_p$ .

Keywords

Euler systemsSelmer complexesColeman familiesp-adic L-functionsGalois representations

MSC classification

Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Secondary: 11F85: $p$-adic theory, local fields 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G35: Modular and Shimura varieties
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 3 , June 2021 , pp. 805 - 853
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The Winter School was supported by NSF grant DMS-1504537 and by the Clay Mathematics Institute. AG was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London and Imperial College London. DG was supported in part by Royal Society grant RP∖EA∖180020. YX was supported by Harvard University scholarships.

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