Rankin--Eisenstein classes and explicit reciprocity laws

Abstract

We construct three-variable $p$-adic families of Galois cohomology classes attached to Rankin convolutions of modular forms, and prove an explicit reciprocity law relating these classes to critical values of L-functions. As a consequence, we prove finiteness results for the Selmer group of an elliptic curve twisted by a 2-dimensional odd irreducible Artin representation when the associated $L$-value does not vanish.

Updated Nov 2023 to add a correction (included separately at the end of the file)

References in corpus (3)

• Rankin--Eisenstein classes and explicit reciprocity laws
• Rankin--Eisenstein classes for modular forms
• Images of adelic Galois representations for modular forms

Cited by in corpus (48)

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• Rankin--Eisenstein classes for modular forms
• Euler systems for GSp(4)
• Images of adelic Galois representations for modular forms
• Iwasawa Main Conjecture for Supersingular Elliptic Curves and BSD conjecture
• Higher Hida theory and p-adic L-functions for GSp(4)
• Euler systems for Hilbert modular surfaces
• Topological polylogarithms and $p$-adic interpolation of $L$-values of totally real fields
• On the $p$-adic variation of Heegner points
• Iwasawa theory for Rankin--Selberg products of $p$-non-ordinary eigenforms
• Iwasawa theory for the symmetric square of a modular form
• An Euler system for GU(2, 1)
• On the Iwasawa main conjectures for modular forms at non-ordinary primes
• Heegner points in Coleman families
• Iwasawa Main Conjecture for Non-Ordinary Modular Forms
• Syntomic regulators of Asai--Flach classes
• Euler systems with local conditions
• Anticyclotomic Euler systems for unitary groups
• P-adic Asai L-functions of Bianchi modular forms
• Interpolation of Generalized Heegner Cycles in Coleman Families
• Rank--two Euler systems for symmetric squares
• P-adic L-functions and diagonal cycles for GSp(4) x GL(2) x GL(2)
• Anticyclotomic p-ordinary Iwasawa Theory of Elliptic Modular Forms
• On refined conjectures of Birch and Swinnerton-Dyer type for Hasse-Weil-Artin L-series
• Interpolation of Beilinson-Kato elements and $p$-adic $L$-functions
• $p$-adic heights of Heegner points and Beilinson-Flach elements
• Main conjectures for higher rank nearly ordinary families -- I
• On the Bloch--Kato conjecture for GSp(4) x GL(2)
• Triple product p-adic L-functions for Shimura curves over totally real number fields
• A class number formula for Picard modular surfaces
• Iwasawa theory for quadratic Hilbert modular forms
• Eisenstein degeneration of Euler systems
• Codimension two cycles in Iwasawa theory and tensor product of Hida families
• On $p$-adic $L$-functions for ${\text{GSp}}_4\times {\text{GL}}_2$
• Generalized Kato classes and exceptional zero conjectures
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• Beilinson-Kato and Beilinson-Flach elements, Coleman-Rubin-Stark classes, Heegner points and the Perrin-Riou Conjecture
• An introduction to $p$-adic $L$-functions
• Control Theorems for Hilbert Modular Varieties
• P-adic Rankin-Selberg L-functions in universal deformation families and functional equations
• Hirzebruch-Zagier classes and rational elliptic curves over quintic fields
• P-adic L-functions for GL(3)
• On the Birch-Swinnerton-Dyer conjecture for modular abelian surfaces
• On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case
• Finite slope Triple Product p-adic L-functions over totally real fields
• Euler Systems and Selmer Bounds for GU(2,1)
• A canonical generator for congruence ideals of Hida families
• Bounding Selmer groups for the Rankin--Selberg convolution of Coleman families