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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Lang-Trotter conjecture for elliptic curve pairs implies new Zilber-Pink cases for curves in A_3 via André's G-functions method, without boundary intersection assumptions.
Two Zilber-Pink-type statements are proved in Y(1)^n assuming a weak Lang-Trotter conjecture for pairs of elliptic curves.
citing papers explorer
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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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Lang-Trotter phenomena and unlikely intersections
Lang-Trotter conjecture for elliptic curve pairs implies new Zilber-Pink cases for curves in A_3 via André's G-functions method, without boundary intersection assumptions.
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A note on Zilber-Pink in $Y(1)^n$
Two Zilber-Pink-type statements are proved in Y(1)^n assuming a weak Lang-Trotter conjecture for pairs of elliptic curves.