Bounding Selmer Groups for the Rankin–Selberg Convolution of Coleman Families. (17th June 2021)

• Record Type: Journal Article
• Authors: Graham, Andrew; Gulotta, Daniel R.; Xu, Yujie
• Is Part Of: Canadian journal of mathematics. Volume 73:Number 3(2021)
• Page Start: 805
• Page End: 853
• Publication Date: 2021-06-17
• Subjects: 11F67 -- 11F85 -- 11G40 -- 14G35; Euler systems -- Selmer complexes -- Coleman families -- p-adic L-functions -- Galois representations; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/S0008414X2000019X
• Languages: English
• ISSNs: 0008-414X

Abstract

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 $L_p$ .