相关论文: Murmurations, Mestre--Nagao sums, and Convolutiona…

Determining the rank of an elliptic curve E/Q is a hard problem, and in some applications (e.g. when searching for curves of high rank) one has to rely on heuristics aimed at estimating the analytic rank (which is equal to the rank under…

This paper investigates the detection of the rank of elliptic curves with ranks 0 and 1, employing a heuristic known as the Mestre-Nagao sum \[ S(B) = \frac{1}{\log{B}} \sum_{\substack{p<B \\ \textrm{good reduction}}}…

Determining the rank of an elliptic curve E/Q remains a central challenge in number theory. Heuristics such as Mestre--Nagao sums are widely used to estimate ranks, but there is considerable room for improving their predictive power. This…

Random matrices provide a well-established statistical model for a range of arithmetic phenomena. In this paper, we investigate the extent to which one- and two-dimensional convolutional neural networks (CNNs) can distinguish between…

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…

He, Lee, Oliver, and Pozdnyakov~\cite{HLOP} have empirically observed that the average of the $p$th coefficients of the $L$-functions of elliptic curves of particular ranks in a given range of conductors $N$ appears to approximate a…

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…

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…

We introduce a new method for studying murmurations, based on random matrix theory. With this method, we exhibit murmurations or similar phenomena: assuming ratios conjectures, for elliptic curves ordered by height, quadratic twists of a…

This article presents a comprehensive data-scientific investigation into the arithmetic statistics of congruent number elliptic curves, leveraging a dataset of square-free integers up to $3$ million. We analyze the Mordell-Weil ranks,…

We compute the first murmurations for elliptic curves over function fields F_q(t): oscillatory patterns in average Frobenius traces that separate rank-0 from rank-1 curves, with z-scores up to 256. For the family E_D: y^2 = x^3 + x + D(t)…

The representations of atmospheric moist convection in general circulation models have been one of the most challenging tasks due to its complexity in physical processes, and the interaction between processes under different time/spatial…

In this article, we propose a novel technique for classification of the Murmurs in heart sound. We introduce a novel deep neural network architecture using parallel combination of the Recurrent Neural Network (RNN) based Bidirectional Long…

We present the multiplicative recurrent neural network as a general model for compositional meaning in language, and evaluate it on the task of fine-grained sentiment analysis. We establish a connection to the previously investigated…

Trained on the works of an artist under study and visually comparable works of other artists, convolutional neural networks can identify forgeries and provide attributions. They can also assign classification probabilities within a…

The prediction of salient areas in images has been traditionally addressed with hand-crafted features based on neuroscience principles. This paper, however, addresses the problem with a completely data-driven approach by training a…

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…

Estimation of model uncertainty can help improve the explainability of Graph Convolutional Networks and the accuracy of the models at the same time. Uncertainty can also be used in critical applications to verify the results of the model by…

Nagao's conjecture relates the rank of an elliptic surface to a limit formula arising from a weighted average of fibral Frobenius traces, and it is further generalized for smooth irreducible projective surfaces by M. Hindry and A. Pacheco.…

We present a description of the function space and the smoothness class associated with a convolutional network using the machinery of reproducing kernel Hilbert spaces. We show that the mapping associated with a convolutional network…