2022/06/13 by Shaunak V. Deo, Deo, Shaunak V.
Computer Science · Engineering · Mathematics · #11F33 #11F80 (Primary) #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2206.06209
openalex publication_date 2022/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p ≥ 5 be a prime, N be an integer not divisible by p, ρ0 be a reducible, odd and semi-simple representation of Gℚ,Np of dimension 2 and \ℓ1,⋯,ℓr\ be a set of primes not dividing Np. After assuming that a certain Selmer group has dimension at most 1, we find sufficient conditions for the existence of a cuspidal eigenform f of level N∏i=1rℓi and appropriate weight lifting ρ0 such that f is new at every ℓi. Moreover, suppose p | ℓi0+1 for some 1 ≤ i0 ≤ r. Then, after assuming that a certain Selmer group vanishes, we find sufficient conditions for the existence of a cuspidal eigenform of level Nℓi02 ∏j ≠ i0 ℓj and appropriate weight which is new at every ℓi and which lifts ρ0. As a consequence, we prove a conjecture of Billerey--Menares in many cases.