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

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 $L_p$ .

Heegner points in Coleman families

“…In the language of Selmer complexes, this conjecture relates the index of a Heegner class in of the anticyclotomic Selmer complex to the torsion submodule of of the same complex. It would be interesting to formulate a generalisation of Perrin‐Riou's conjecture in the present setting; it may even be possible to prove some results in this direction using the ‘Euler system machine’ to bound Selmer groups, analogous to [18] in the Rankin–Selberg case. We shall not pursue this matter in the present paper, but it would be a very interesting direction for further work.…”