Ranks of Elliptic Curves Twisted by Quadratic Forms

2026/07/28 by Mohammad H. Hamdar, Cihan Sabuncu
#math.NT

paper · pdf

Abstract

Let E be an elliptic curve over ℚ and let E^d be its twist by the quadratic character χ_d. We prove there are infinitely many twists d which are sums of two squares such that E^d has rank 1. This result is achieved using moments of derivatives of modular L-functions, and particularly captures the lower derivatives which were left out in the work of Munshi. Such a result, in particular, also gives us information on the elliptic fibration (1+t²)y²=f(x), where f(x) is a cubic polynomial.

Citations

• Moment of Derivatives of Quadratic Twists of Modular L-Functions
• Moments of derivatives of modular L-functions
• The first moment of quadratic twists of modular L-functions
• Quadratic Twists of Elliptic Curves

Related

• Rank frequencies for quadratic twists of elliptic curves
• A Family of Elliptic Curves with a Lower Bound on 2-Selmer Ranks of\n Quadratic Twists
• Parallelopipeds of Positive Rank Twists of Elliptic Curves
• Ranks of elliptic curves in families of quadratic twists
• On the rank of quadratic twists of elliptic curvers over function fields