Upper tail distributions of central $L$-values of quadratic twists of elliptic curves at the variance scale
N. Creighton
TL;DR
The paper develops a barrier-method framework to obtain upper bounds for large deviations of the central values $L\left(\tfrac{1}{2},E_d\right)$ in a degree-2 family of quadratic twists. By partitioning primes into disjoint intervals and modeling the logarithm of $L$-values as a random walk on primes, the authors bound the tail probability at the variance scale, using carefully crafted mollifiers and a twisted mollifier framework to control moments. A key technical advance is the adaptation of Rankin-trick lifting and diagonal-dominance analysis to a degree-2 setting, enabling precise upper bounds that improve on previous density estimates by Radziwiłł–Soundararajan. The results extend to restricted discriminant families via well-factorable twists and yield fractional-moment bounds that align with conjectured Gaussian tails, thereby contributing to a sharper understanding of the distribution of central $L$-values in orthogonal families and informing broader questions related to Lindelöf-type behavior for elliptic curves.
Abstract
We consider the large deviations at the order of the variance for the central value of a family of $L$-functions among the members with bounded discriminant. When there is an upper bound on an integer moment of the central value twisted by a short Dirichlet polynomial, we can establish upper bounds on the density of members exhibiting a large central value. We adapt the techniques from Arguin and Bailey for large deviations of the Riemann zeta function to prove results on the degree two family of quadratic twists of an elliptic curve. This upper bound improves on density results previously obtained by Radziwiłł and Soundararajan.