Reducibility and rational torsion in modular abelian varieties
Amod Agashe, Matthew Winters
TL;DR
The paper links the reducibility of mod-$r$ Galois representations attached to modular abelian varieties to explicit arithmetic constraints: if $N$ is square-free and $r mid 6N$, reducibility of $ ho_{ rak m}$ or $A[ rak m]$ forces a nontrivial intersection with the cuspidal subgroup, implying $rigm| |C|$, a nontrivial rational $r$-torsion point on $A$, and a rational point on $A[ rak m]$. The authors realize this by constructing Eisenstein congruences to the modular form $f$ modulo a prime over $r$, proving ordinarity, and invoking Tang’s results to force cuspidal-intersection, with the outcome extending to applications such as partial BSD statements. The results yield concrete consequences for level-lowering phenomena when Tamagawa factors contribute to the BSD formula and provide a framework for analyzing rational torsion and irreducibility of $A[ rak m]$ in terms of the cuspidal subgroup. Overall, the work connects Galois-representation reducibility, Eisenstein congruences, and the cuspidal subgroup to arithmetic invariants of modular abelian varieties and BSD-type conjectures.
Abstract
Let N be a square-free positive integer and let f be a newform of weight 2 on Γ_0(N). Let A denote the abelian subvariety of J_0(N) associated to f and let m be a maximal ideal of the Hecke algebra T that contains Ann_T(f) and has residue characteristic r such that r does not divide 6N. We show that if either A[m] or the canonical representation ρ_m over T/m associated to m is reducible, then r divides the order of the cuspidal subgroup of J_0(N) and A[m] has a nontrivial rational point. We mention some applications of this result, including an application to the second part of the Birch and Swinnerton-Dyer conjecture for A.