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The Eisenstein ideal at prime-square level has constant rank
1Department of Mathematics, Temple University, Philadelphia, PA 19122.
Abstract:
Let N and p be prime numbers with [Formula: see text] such that [Formula: see text]. In a previous paper, we showed that there is a cuspform f of weight 2 and level [Formula: see text] whose ℓ-th Fourier coefficient is congruent to [Formula: see text] modulo a prime above p for all primes ℓ. In this paper, we prove that this form f is unique up to Galois conjugacy, and the extension of [Formula: see text] generated by the coefficients of f is exactly [Formula: see text]. We also prove similar results when a higher power of p divides [Formula: see text].
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