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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A new fibration theorem implies solvable descent, solving the Grunwald problem for solvable groups up to the Brauer-Manin obstruction.
Establishes first-order definability of Campana and Darmon points in algebraic function fields over number fields by extending quadratic Pfister form methods from prior number field results.
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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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Solvable Descent and the Grunwald Problem for Solvable Groups
A new fibration theorem implies solvable descent, solving the Grunwald problem for solvable groups up to the Brauer-Manin obstruction.
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First-order definability of Campana Points and Darmon Points in algebraic function fields in one variable over number fields
Establishes first-order definability of Campana and Darmon points in algebraic function fields over number fields by extending quadratic Pfister form methods from prior number field results.