Abstract
We show that the Euler system associated with Rankin–Selberg convolutions of modular forms, introduced in our earlier works with Lei and Kings, varies analytically as the modular forms vary in p-adic Coleman families. We prove an explicit reciprocity law for these families and use this to prove cases of the Bloch–Kato conjecture for Rankin–Selberg convolutions.
Highlights
Theorem A Let f, g be eigenforms of weights 2 and levels Nf, Ng coprime to p whose Hecke polynomials at p have distinct roots, and let fα, gα be non-critical p-stabilisations of f, g
The proof of Theorem 5.4.2 reveals some new phenomena which may be of independent interest; the Galois modules in which these classes lie are, in a natural way, étale counterparts of the modules of “nearly overconvergent modular forms” introduced by Urban [32]
In order to define the Perrin-Riou logarithm in this context, one needs to work with triangulations of (φ, Γ )-modules over the Robba ring; we use here results of Liu [21], showing that the (φ, Γ )-modules of the Galois representations MV1 (F )∗ and MV2 (G)∗ admit canonical triangulations
Summary
For λ ∈ R 0, we define the Banach space Cλ(Zp, Qp) of order λ functions on Zp as in [11]. All the modules in this sequence are finite, since HI2w(K∞, V )Γ vanishes by assumption; this implies that there is a uniform power of p (independent of n) which annihilates HI2w(K∞, T )[νn] for all n 1 (compare the proof of [20, Proposition A.2.10], which is a similar argument with νn = (γ − 1)n replaced by γ pn − 1). With this in hand we may proceed as in [10]. The convention that if p is not invertible on Y , He∗t(Y, −) is a shorthand for He∗t(Y [1/p], −)
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