Journal of the Ramanujan Mathematical Society
Volume 41, Issue 3, September 2026 pp. 253–261.
Reducibility and rational torsion in modular abelian varieties
Authors:
Amod Agashe and Matthew Winters
Author institution:Department of Mathematics, Florida State University, 208 Love Building, 1017 Academic Way Tallahassee, FL 32306, U.S.A.
Summary:
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.
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