Konstantin Miagkov | Searcharxiv

arXiv2024

Potential automorphy of certain non self-dual 3-dimensional Galois representations

In a series of papers, van Geemen and Top have defined a family of surfaces $S_z$ indexed by a nonzero integer parameter $z$, and a compatible family of 3-dimensional Galois representations over $\Q(i)$ attached to each surface. In this note we use recent advancements in potential automorphy and automorphy lifting to show that these compatible families are potentially automophic for all values of $z$, and hence that their L-functions have analytic continuation and a functional equation.

math.NT↗

arXiv2023

Mean Estimation Under Heterogeneous Privacy Demands

Differential Privacy (DP) is a well-established framework to quantify privacy loss incurred by any algorithm. Traditional formulations impose a uniform privacy requirement for all users, which is often inconsistent with real-world scenarios in which users dictate their privacy preferences individually. This work considers the problem of mean estimation, where each user can impose their own distinct privacy level. The algorithm we propose is shown to be minimax optimal and has a near-linear run-time. Our results elicit an interesting saturation phenomenon that occurs. Namely, the privacy requirements of the most stringent users dictate the overall error rates. As a consequence, users with less but differing privacy requirements are all given more privacy than they require, in equal amounts. In other words, these privacy-indifferent users are given a nontrivial degree of privacy for free, without any sacrifice in the performance of the estimator.

cs.CR↗

arXiv2022

Automorphy lifting with adequate image

Let $F$ be a CM number field. We generalize existing automorphy lifting theorems for regular residually irreducible $p$-adic Galois representations over $F$ by relaxing the big image assumption on the residual representation.

math.NT↗