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

2015/07/02 by Tobias Berger, Berger, Tobias
Mathematics · #11F46 #11F55 #11F80 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1507.00684

openalex publication_date 2015/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Galois representations associated to Siegel modular forms of low weight
• On Certain L-Functions

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