-
UNVERDICTED
Local constancy of reduction type and related invariants for curves in p-adic families
The paper claims that reduction type, Tamagawa number, the BSD fudge factor, and the Galois representation of a curve are locally constant under p-adically small perturbations of the curve's defining equations.
arxiv:2508.12329 · math.NT · 2025-08
-
CONDITIONAL
On a Birch and Swinnerton-Dyer type conjecture for the Hasse-Weil-Artin L-functions in characteristic p>0
Under finiteness of the Tate-Shafarevich group and local Assumption 7.2, the normalized leading term at s=1 of the Hasse-Weil-Artin L-function of (A, chi) equals a product of a volume term, a ramification correction, a twisted regulator, and a characteristic ideal.
arxiv:2411.12404 · math.NT · 2024-11
-
ACCEPT
Secondary terms in the first moment of |{rm Sel}₂(E)|
For elliptic curves ordered by height, the first moment of |Sel2(E)| contains a secondary term of order X^{3/4}, with a power-saving error of size O(X^{3/4-1/3804+epsilon}).
arxiv:2412.00995 · math.NT · 2024-12
-
UNVERDICTED
A proof of the 4,7 cases of Sylvester's conjecture on cube sums
Proves that every prime p ≡ 4,7 mod 9 is a sum of two rational cubes, resolving 2/3 of Sylvester's conjecture via BSD progress and the solved Unbounded Denominators Conjecture.
arxiv:2605.25917 · math.NT · 2026-05
-
UNVERDICTED
Waldspurger's period integral for newforms
Authors compute Waldspurger's local period integral for newforms in new cases using minimal vectors and a representation theoretic trick, with an example tied to a setting used for the 3-part BSD conjecture.
arxiv:1907.11428 · math.NT · 2019-07
-
UNVERDICTED
Nontrivial torsion in the Tate--Shafarevich group of elliptic curves via visibility and twists
Visibility theorems imply nontrivial ℓ-torsion in Sha of quadratic twists of elliptic curves with additive reduction at ℓ; for ℓ=3 this yields pairs of curves with identical BSD data and Kodaira symbols but isomorphic Sha groups containing 3-torsion.
arxiv:2602.19861 · math.NT · 2026-02