arXiv subjects

Dane Wachs

Publications and source records attributed to Dane Wachs.

4 recordsLinked to original sources

arXiv2026

Murmurations, Periods, and Local Factors

We prove that over function fields F_q(t), the Tate-Shafarevich group |Sha| is an invariant of the cyclotomic type of the L-polynomial, so that |Sha|-stratified murmuration densities reduce to type-weighted densities with no within-type zero displacement (Theorem A). Over Q, the obstruction vanishes because Satake parameters are continuous: conditioning on L(f,1) = c biases each theta_p through the Euler product constraint, creating covariance between the Frobenius trace a_p and the real period Omega_f that the L-value regression does not absorb. A new inequality for modified Bessel functions (Theorem B) establishes the positivity of this single-prime covariance under a linearized tilt; the full Euler-factor positivity follows by perturbation for large p (Theorem 10) and is verified numerically for small primes (Theorem 11). We establish that the conditional covariance Cov(a_p, Omega_f | L(f,1) ~ c, N) converges to an explicit function C(c)/sqrt(p) as N -> infinity (Theorem C). The function C(c) changes sign -- positive at small c, negative at large c -- a prediction confirmed empirically using 657,000 curves from the Cremona database. Empirically, the covariance is concentrated entirely in the Tamagawa product prod c_v: at fine L(1)-conditioning, the Tamagawa channel accounts for 100% of the signal, and cross-validated regression confirms that the nonlinear adjoint-Euler-factor weighting carries independent Tamagawa information beyond a full trace basis but adds nothing for |Sha|. The |Sha|-modulation of murmurations discovered in [Wac26a] is a consequence of the BSD identity linking |Sha| to local factors -- the same local-factor mechanism operates discretely over function fields and continuously over Q.

math.NT↗

arXiv2026

Murmurations of Elliptic Curves over Function Fields

We compute the first murmurations for elliptic curves over function fields F_q(t): oscillatory patterns in average Frobenius traces that separate rank-0 from rank-1 curves, with z-scores up to 256. For the family E_D: y^2 = x^3 + x + D(t) with D monic squarefree of degree 5, we enumerate 534,745 curves across q = 7, 11, 13 with exact BSD invariants. All L-polynomials factor into cyclotomic polynomials -- a weight-2 consequence of the Weil conjectures and Kronecker's theorem, independent of CM. Since |Sha| = L(1/q) in this family (a consequence of BSD with trivial torsion and Tamagawa numbers), the |Sha| modulation of murmurations is entirely a composition effect: different |Sha| strata have different mixtures of L-polynomial types, and hence different mean traces. This yields an exact reweighting identity for the |Sha|-stratified murmuration density: M_s(d,q) = -sum_lambda f_{lambda,s} p_d(lambda), where lambda ranges over cyclotomic types, f_{lambda,s} is the type composition of the |Sha| = s stratum, and p_d(lambda) is the degree-d power sum of the unitarized roots. Within each |Sha| stratum, joint cells -- distinct L-polynomial types sharing the same |Sha| -- show that the murmuration profile carries arithmetic information strictly finer than |Sha| alone.

math.NT↗

arXiv2026

BSD Invariants and Murmurations of Elliptic Curves

We investigate the interaction between Birch and Swinnerton-Dyer (BSD) invariants and the murmuration phenomenon for elliptic curves over the rational numbers. Our study, based on a dataset of 3,064,705 curves from the Cremona database with conductor up to 499,998, yields three results. First, the BSD invariants themselves - real period, Tamagawa product, analytic order of the Tate-Shafarevich group, regulator, and torsion order - do not exhibit murmuration-type oscillations when averaged in sliding conductor windows. Second, these invariants modulate the shape of the standard Frobenius trace murmurations: within a fixed rank, curves stratified by Tamagawa product, analytic order of the Tate-Shafarevich group, or real period display significantly different murmuration profiles, with p-values less than 0.001 against permutation null models, and these differences are scale-invariant across conductor ranges. Third, the Tate-Shafarevich group modulation survives controlling simultaneously for the L-value at 1, the real period, and the conductor, establishing that the order of the Tate-Shafarevich group encodes information about the distribution of Frobenius traces at good primes that is not captured by any other standard BSD invariant. We further show that this modulation is a pure mean shift in the Frobenius trace distribution - variance, skewness, and kurtosis are identical between Tate-Shafarevich group strata - and that it concentrates at small primes. Computing low-lying L-function zeros for 2,000 curves at fixed L-value, we find that curves with Tate-Shafarevich group order at least four have systematically different low-lying zero distributions, with the first zero displaced higher and subsequent zeros more tightly packed. The explicit formula connects this zero displacement to the observed murmuration modulation, consistent with the zero distribution acting as a mediating mechanism.

math.NT↗

arXiv2025

The Derived Adelic Cohomology Conjecture for Elliptic Curves

We propose a novel derived cohomological framework for the Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves. In our approach, local arithmetic data are encoded in derived sheaves which, when glued via a mapping cone construction, yield an adelic complex. A natural Postnikov filtration on this complex gives rise to a spectral sequence whose first nonzero differential detects the analytic and algebraic rank of the curve. Moreover, the determinant of this differential equals the combination of classical invariants appearing in the BSD formula. We present rigorous constructions of the derived sheaves involved and establish their key properties, including explicit connections to L-functions through cohomological interpretations. Extensive numerical evidence across curves of various ranks, including those with non-trivial Tate-Shafarevich groups, supports these structural predictions. Our framework unifies several existing approaches to the BSD conjecture, providing a cohomological interpretation that explains both the rank equality and the precise formula where previous methods addressed only partial aspects of the conjecture.

math.GM↗