On Twists of A Family of Elliptic Curves and Their $ L-$Function
Derong Qiu

TL;DR
This paper investigates quadratic twists of specific elliptic curves, analyzing their $L$-functions and related arithmetic properties to provide evidence supporting the Birch and Swinnerton-Dyer conjecture.
Contribution
It offers explicit results on the vanishing of $L$-functions and arithmetic invariants for a family of elliptic curves, using classical and Iwasawa theory methods.
Findings
Criteria for $L(1)$ vanishing in quadratic twists
Explicit determination of norm indices and root numbers
Analysis of Selmer groups and Mordell-Weil ranks
Abstract
Let be an elliptic curve defined over a number field, the conjecture of Birch and Swinnerton-Dyer (BSD, for short) asserts a deep relation between the group of rational points and the function of at Very few explicit results about and are known, even no general method is known to determine vanishing or not for a given elliptic curve. In this paper, we study some quantities related to BSD of a special class of elliptic curves, more precisely, we study the arithmetic of quadratic twists of elliptic curves and their function. Based on some classical works, especially those of Greenberg, Kramer-Tunnell, Kato-Rohrlich, Manin and Mazur, under some conditions, we obtain results about the vanishing of the value at of the -function, and explicitly…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Analytic Number Theory Research · Cryptography and Residue Arithmetic
