Year: 2025
Type: article
Abstract: In the early 2000s, Ramakrishna asked the question: for the elliptic curve \[E\;:\; y^2 = x^3 - x,\] what is the density of primes p for which the Fourier coefficient $a_p(E)$ is a cube modulo p? As a generalisation of this question, Weston–Zaurova formulated conjectures concerning the distribution of power residues of degree m of the Fourier coefficients of elliptic curves $E/\mathbb{Q}$ with complex multiplication. In this paper, we prove the conjecture of Weston–Zaurova for cubic residues using the analytic theory of spin. Our proof works for all elliptic curves E with complex multiplication.
Authors: PETER KOYMANS, PETER VANG UTTENTHAL
Institutions: Utrecht University, Aarhus University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W4402853510