2018/10/17 by Tobias Berger, Berger, Tobias, Krzysztof Klosin +1
Mathematics · #11F33 #11F80 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1810.07808
openalex publication_date 2018/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a totally real field F, a finite extension F of Fp and a Galois character χ: GF → F× unramified away from a finite set of places Σ⊃ \\mathfrakp | p\ consider the Bloch-Kato Selmer group H:=H1Σ(F, χ-1). In an earlier paper of the authors it was proved that the number d of isomorphism classes of (non-semisimple, reducible) residual representations ρ giving rise to lines in H which are modular by some ρf (also unramified outside Σ) satisfies d ≥ n:= dimF H. This was proved under the assumption that the order of a congruence module is greater than or equal to that of a divisible Selmer group. We show here that if in addition the relevant local Eisenstein ideal J is non-principal, then d >n. When F=Q we prove the desired bounds on the congruence module and the Selmer group. We also formulate a congruence condition implying the non-principality of J that can be checked in practice, allowing us to furnish an example where d>n.