D. Wachs

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math.NT2026

Murmurations, Periods, and Local Factors

Dane Wachs

This note replaces arXiv:2603.22807v1 and withdraws its three theorems, which do not stand as stated. Theorem A, that the order of the Tate-Shafarevich group of $y^{2} = x^{3} + x + D (t)$ ov…

math.NT2026

Frobenius in $W (E_{8})$ for $y^{2} = x^{3} + x + D (t)$ over $F_{q} (t)$ : ranks, Tate--Shafarevich groups and trace statistics

Dane Wachs

Let $q$ be a power of a prime $p \geq 5$ and let $F_{5} (q)$ be the set of curves $E_{D} : y^{2} = x^{3} + x + D (t)$ over $F_{q} (t)$ with $D$ monic of degree $5$ and $4 + 27 D^{2}$ squ…

math.NT2026

BSD Invariants and Murmurations of Elliptic Curves

Dane Wachs

We study how the murmurations of elliptic curves over $Q$ depend on the invariants in the Birch-Swinnerton-Dyer formula, using the Frobenius traces at all primes up to $10…

Murmurations, Periods, and Local Factors

Frobenius in W(E8​) for y2=x3+x+D(t) over Fq​(t): ranks, Tate--Shafarevich groups and trace statistics

BSD Invariants and Murmurations of Elliptic Curves

Frobenius in $W (E_{8})$ for $y^{2} = x^{3} + x + D (t)$ over $F_{q} (t)$ : ranks, Tate--Shafarevich groups and trace statistics