arXiv subjects

Alexey Pozdnyakov

Publications and source records attributed to Alexey Pozdnyakov.

10 recordsLinked to original sources

arXiv2026

Murmurations of quadratic and cubic characters over function fields

We compute the murmuration density for the family of quadratic characters over function fields. We show that the expectation of this density coincides with a correction term arising in the traces of high powers of the Frobenius class in this family, for which we determine large-genus asymptotics. This correction term admits a description in terms of the stable homology of the moduli space of hyperelliptic curves of genus $g$ with one marked Weierstrass point. We further identify this correction as a lower-order term in the one-level density and we obtain a non-vanishing result at the central point for quadratic $L$-functions over function fields, via an alternative approach to using the ratios conjecture. We also compute the murmuration density for the thin family of primitive cubic characters in the Kummer setting.

math.NT↗

arXiv2026

Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves

We investigate the extent to which the reduced minimal Weierstrass coefficients of an elliptic curve over $\mathbb{Q}$ may be computed from it's Frobenius traces. Decision tree models reveal that the first two reduced minimal Weierstrass coefficients can be recovered with perfect accuracy from the Frobenius traces at the primes $2$ and $3$, and the third by supplementing these two traces with the conductor parity. We subsequently prove explicit formulae for these coefficients using the Frobenius traces and conductor parity. These formulae appear to be new. In particular, we deduce that the first three reduced minimal Weierstrass coefficients of an elliptic curve are determined by its isogeny class.

math.NT↗

arXiv2026

On (not) learning the M\"obius function

We prove lower bounds on learning the M\"obius or Liouville function with a variety of standard learning techniques, including kernel methods, noisy gradient methods, and correlational statistical query algorithms. These results follow from quantitative bounds on the correlation of M\"obius with digital characters of various finite abelian groups, where the group is dictated by the type of input data the algorithm is given. Using residues mod $p$ for many different primes corresponds to a cyclic group, and using the base $p$ expansion for a fixed prime corresponds to an elementary abelian $p$-group. We also note that lower bounds of this form are closely related to certain types of digital prime number theorems.

math.NT↗

arXiv2026

Murmurations: a case study in AI-assisted mathematics

We report the emergence of a striking new phenomenon in arithmetic, which we call murmurations. First observed experimentally through averages over large arithmetic datasets, murmurations can be detected and analyzed using standard interpretability tools from machine learning, including principal component weightings, saliency curves, and convolutional filters. Although discovered computationally, they constitute a genuinely new and intriguing phenomenon in arithmetic that can be formulated and investigated using established tools of number theory. In particular, murmurations encode subtle information about Frobenius traces and naturally belong to the framework of arithmetic statistics. More precisely, murmurations connect to central themes surrounding the conjecture of Birch and Swinnerton-Dyer and perspectives from random matrix theory. In this paper, we present an overview of murmurations, contextualizing them within number theory and AI.

math.NT↗

arXiv2025

Learning Euler Factors of Elliptic Curves

We apply transformer models and feedforward neural networks to predict Frobenius traces $a_p$ from elliptic curves given other traces $a_q$. We train further models to predict $a_p \bmod 2$ from $a_q \bmod 2$, and cross-analysis such as $a_p \bmod 2$ from $a_q$. Our experiments reveal that these models achieve high accuracy, even in the absence of explicit number-theoretic tools like functional equations of $L$-functions. We also present partial interpretability findings.

math.NT↗

arXiv2024

Murmurations of elliptic curves

We investigate the average value of the Frobenius trace at p over elliptic curves in a fixed conductor range with given rank. Plotting this average as p varies over the primes yields a striking oscillating pattern, the details of which vary with the rank. Based on this observation, we perform various data-scientific experiments with the goal of classifying elliptic curves according to their ranks.

math.NT↗

arXiv2024

Predicting root numbers with neural networks

We report on two machine learning experiments in search of statistical relationships between Dirichlet coefficients and root numbers or analytic ranks of certain low-degree $L$-functions. The first experiment is to construct interpretable models based on murmurations, a recently discovered correlation between Dirichlet coefficients and root numbers. We show experimentally that these models achieve high accuracy by learning a combination of Mestre-Nagao type heuristics and murmurations, noting that the relative importance of these features varies with degree. The second experiment is to search for a low-complexity statistic of Dirichlet coefficients that can be used to predict root numbers in polynomial time. We give experimental evidence and provide heuristics that suggest this can not be done with standard machine learning techniques.

math.NT↗

arXiv2023

Murmurations of Dirichlet characters

We calculate murmuration densities for two families of Dirichlet characters. The first family contains complex Dirichlet characters normalized by their Gauss sums. Integrating the first density over a geometric interval yields a murmuration function compatible with experimental observations. The second family contains real Dirichlet characters weighted by a smooth function with compact support. We show that the second density exhibits a universality property analogous to Zubrilina's density for holomorphic newforms, and it interpolates the phase transition in the the $1$-level density for a symplectic family of $L$-functions.

math.NT↗

arXiv2022

Fair pricing and hedging under small perturbations of the numéraire on a finite probability space

We consider the problem of fair pricing and hedging under small perturbations of the numéraire. We show that for replicable claims, the change of numéraire affects neither the fair price nor the hedging strategy. For non-replicable claims, we demonstrate that is not the case. By reformulating the key stochastic control problem in a more tractable form, we show that both the fair price and optimal strategy are stable with respect to small perturbations of the numéraire. Further, our approach allows for explicit asymptotic formulas describing the fair price and hedging strategy's leading order correction terms. Mathematically, our results constitute stability and asymptotic analysis of a stochastic control problem under certain perturbations of the integrator of the controlled process, where constraints make this problem hard to analyze.

q-fin.PR↗

arXiv2022

Stochastic Scaling in Loss Functions for Physics-Informed Neural Networks

Differential equations are used in a wide variety of disciplines, describing the complex behavior of the physical world. Analytic solutions to these equations are often difficult to solve for, limiting our current ability to solve complex differential equations and necessitating sophisticated numerical methods to approximate solutions. Trained neural networks act as universal function approximators, able to numerically solve differential equations in a novel way. In this work, methods and applications of neural network algorithms for numerically solving differential equations are explored, with an emphasis on varying loss functions and biological applications. Variations on traditional loss function and training parameters show promise in making neural network-aided solutions more efficient, allowing for the investigation of more complex equations governing biological principles.

cs.LG↗