The Eisenstein ideal at prime-square level has constant rank.
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- Record sourced from PubMed, PMID 40643979.
- Also identified by DOI 10.1073/pnas.2500729122 and PMC identifier 12280971.
- Licence recorded as CC BY-NC-ND.
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Abstract
Let <i>N</i> and <i>p</i> be prime numbers with [Formula: see text] such that [Formula: see text]. In a previous paper, we showed that there is a cuspform <i>f</i> of weight 2 and level [Formula: see text] whose <i>ℓ</i>-th Fourier coefficient is congruent to [Formula: see text] modulo a prime above <i>p</i> for all primes <i>ℓ</i>. In this paper, we prove that this form <i>f</i> is unique up to Galois conjugacy, and the extension of [Formula: see text] generated by the coefficients of <i>f</i> is exactly [Formula: see text]. We also prove similar results when a higher power of <i>p</i> divides [Formula: see text].