On the Bloch-Kato conjecture for the Asai L-function

Abstract

Following Ribet's seminal 1976 paper there have been many results employing congruences between stable cuspforms and lifted forms to construct non-split extensions of Galois representations. We show how this strategy can be extended to construct elements in the Bloch-Kato Selmer groups of +/--Asai (or tensor induction) representations associated to Bianchi modular forms. We prove, in particular, how the Galois representation associated to a suitable low weight Siegel modular form produces elements in the Selmer group for exactly the Asai representation (+ or -) that is critical in the sense of Deligne. We further outline a strategy using an orthogonal-symplectic theta correspondence to prove the existence of such a Siegel modular form and explain why we expect this to be governed by the divisibility of the near-central critical value of the Asai L-function, in accordance with the Bloch-Kato conjecture.

18 pages, v2: article revised to focus on Bianchi modular form case, higher-dimensional results now form part of article "Oddness of residually reducible Galois representations"

References in corpus (1)

• p-adic families of Galois representations and higher rank Selmer groups

Cited by in corpus (3)

• On the images of Galois representations attached to low weight Siegel modular forms
• Explicit inner product formulas and Bessel period formulas for HST lifts
• Special L-values and Selmer groups of Siegel modular forms of genus 2