Deformations of reducible Galois representations with large Selmer p-rank

2025/04/22 by Eknath Ghate, Anwesh Ray, Ghate, Eknath +1
Mathematics · #11F80 #11R23 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2504.16287

openalex publication_date 2025/04/22 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

Abstract

Let p≥ 5 be a prime number. In this paper, we construct Galois representations associated with modular forms for which the dimension of the p-torsion in the Bloch-Kato Selmer group can be made arbitrarily large. Our result extends similar results known for small primes, such as Matsuno's work on Tate-Shafarevich groups of elliptic curves. Extending the technique of Hamblen and Ramakrishna, we lift residually reducible Galois representations to modular representations for which the associated Greenberg Selmer groups are minimally generated by a large number of elements over the Iwasawa algebra. We deduce that there is an isogenous lattice for which the Bloch-Kato Selmer group has large p-rank.

Related

• Ranks of Selmer groups in an analytic family
• On Bloch-Kato Selmer groups and Iwasawa theory of p-adic Galois representations
• Selmer stability in families of congruent Galois representations
• Computing Selmer groups associated to mod p Galois representations
• The refined Tamagawa number conjectures for GL₂