Discretisation for odd quadratic twists

Non-vanishing of quadratic twists of modular $L$-functions of prime-related moduli

In this paper, we study central values of the family of quadratic twists of modular $L$-functions of moduli $8p$, with $p$ ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish a positive proportion non-vanishing result for the corresponding $L$-values.

Mean squares of quadratic twists of the Möbius function

In this paper, we evaluate asymptotically the sum \[ \sum_{d \leq X} \left( \sum_{n \leq Y} μ(n)\leg {8d}{n} \right)^2, \] where $\leg {8d}{n}$ is the Kronecker symbol and $d$ runs over positive, odd, square-free integers.

Plane quartic twists of X(5,3)

Given an odd representation of the absolute Galois group of Q onto PGL(2,3) and a positive integer N, there exists a twisted modular curve defined over Q whose rational points classify the quadratic Q-curves of degree N realizing the representation. The paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case N=5.

阿贝尔簇的挠率与扭形

Torsion and twists of abelian varieties

该研究探讨具好约化阿贝尔簇扭形的挠子群，证明相关二次扩张分歧素数定理，并指出满足Heegner假设的特定虚二次域上有理椭圆曲线无奇阶挠率增长。

Regulators of rank one quadratic twists

We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of an odd quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions.

Mordell-Lang 定理与某些超椭圆曲线的奇数扭的Selmer秩差异

Mordell--Lang and disparate Selmer ranks of odd twists of some superelliptic curves over global function fields

研究了某些超椭圆曲线的奇数扭的Selmer秩差异，证明了其有理点数量的上界及密度问题。