For primes N and p with N ≡ 1 mod p, the rank r of Mazur's Eisenstein Hecke algebra equals one plus the vanishing order of a mod-p zeta element interpolating L-values at -1 when r is 2 or 3, with a uniform extension to level N² and partial results for higher ranks.
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Proves H^{binom(n,2)}(Γ_{0,n}^+(p); ℚ) vanishes for p in {2,3,5,7,13} (n≥3) and p≤6n-14, and is nonzero for n=2 (all p) and n=3 (p not in that set).
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A new perspective on the rank of Mazur's Eisenstein Hecke algebra
For primes N and p with N ≡ 1 mod p, the rank r of Mazur's Eisenstein Hecke algebra equals one plus the vanishing order of a mod-p zeta element interpolating L-values at -1 when r is 2 or 3, with a uniform extension to level N² and partial results for higher ranks.
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Top-dimensional rational cohomology of the congruence subgroup $\Gamma_{0,n}^+(p)$
Proves H^{binom(n,2)}(Γ_{0,n}^+(p); ℚ) vanishes for p in {2,3,5,7,13} (n≥3) and p≤6n-14, and is nonzero for n=2 (all p) and n=3 (p not in that set).