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Edgar Costa

Publications and source records attributed to Edgar Costa.

At least 19 recordsLinked to original sources
arXiv2026

Murmurations, Mestre--Nagao sums, and Convolutional Neural Networks for elliptic curves

We apply one-dimensional convolutional neural networks to the Frobenius traces of elliptic curves over $\mathbb{Q}$ and evaluate and interpret their predictive capacity. In keeping with similar experiments by Kazalicki--Vlah, Bujanovi\'{c}--Kazalicki--Novak, and Pozdnyakov, we observe high accuracy predictions for the analytic rank across a range of conductors. We interpret the prediction using saliency curves and explore the interesting interplay between murmurations and Mestre--Nagao sums, the details of which vary with the conductor and the (predicted) rank.

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arXiv2026

Ordinary abelian varieties: isogeny graphs and polarizations

Given an integer $D$ and an ordinary isogeny class of abelian varieties defined over a finite field $\mathbb{F}_q$ with commutative $\mathbb{F}_q$-endomorphism algebra, we provide algorithms for computing all isogenies of degree dividing $D$ and polarizations of degree dividing $D$. We discuss phenomena that arise for higher dimension abelian varieties but not elliptic curves, bounds on the diameter of the graph of minimal isogenies, and decompositions of isogeny graphs into orbits for the Picard group of the Frobenius order.

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arXiv2025

Curve equations from expansions of 1-forms at a nonrational point

We exhibit an algorithm to compute equations of an algebraic curve over a computable characteristic 0 field from the power series expansions of its regular 1-forms at a nonrational point of the curve, extending a 2005 algorithm of Baker, Gonz\'alez-Jim\'enez, Gonz\'alez, and Poonen for expansions at a rational point. If the curve is hyperelliptic, the equations present it as an explicit double cover of a smooth plane conic, or as a double cover of the projective line when possible. If the curve is nonhyperelliptic, the equations cut out the canonical model. The algorithm has been used to compute equations over $\mathbb{Q}$ for many hyperelliptic modular curves without a rational cusp in the L-functions and Modular Forms Database.

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arXiv2025

Explicit modularity of K3 surfaces with complex multiplication of large degree

We consider the transcendental motive of three K3 surfaces $X$ conjectured to have complex multiplication (CM). Under this assumption, we match these to explicit algebraic Hecke quasi-characters $\psi_X$, and CM abelian threefolds $A$. This provides substantial evidence that a power of $A$ corresponds to $X$ under the Kuga-Satake correspondence.

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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.

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arXiv2025

Machine learning the vanishing order of rational L-functions

In this paper, we study the vanishing order of rational $L$-functions from a data scientific perspective. Each $L$-function is represented in our data by finitely many Dirichlet coefficients, the normalisation of which depends on the context. We observe murmuration-like patterns in averages across our dataset, find that PCA clusters rational $L$-functions by their vanishing order, and record that LDA and neural networks may accurately predict this quantity.

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arXiv2025

Labeling abelian varieties over finite fields

We describe a deterministic process to associate a practical, permanent label to isomorphism classes of abelian varieties defined over finite fields with commutative endomorphism algebra as long as they are ordinary or defined over a prime field. In the ordinary case, we also provide labels for the polarizations they admit.

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arXiv2025

Learning Fricke signs from Maass form Coefficients

In this paper, we conduct a data-scientific investigation of Maass forms. We find that averaging the Fourier coefficients of Maass forms with the same Fricke sign reveals patterns analogous to the recently discovered "murmuration" phenomenon, and that these patterns become more pronounced when parity is incorporated as an additional feature. Approximately 43% of the forms in our dataset have an unknown Fricke sign. For the remaining forms, we employ Linear Discriminant Analysis (LDA) to machine learn their Fricke sign, achieving 96% (resp. 94%) accuracy for forms with even (resp. odd) parity. We apply the trained LDA model to forms with unknown Fricke signs to make predictions. The average values based on the predicted Fricke signs are computed and compared to those for forms with known signs to verify the reasonableness of the predictions. Additionally, a subset of these predictions is evaluated against heuristic guesses provided by Hejhal's algorithm, showing a match approximately 95% of the time. We also use neural networks to obtain results comparable to those from the LDA model.

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arXiv2024

The constructive inverse Galois problem via Hilbert modular forms: realizing the transitive group 17T7

We show how Hilbert modular forms can be used in the constructive inverse Galois problem over the rationals. In particular, we prove that the transitive permutation group 17T7, isomorphic to a split extension of C_2 by PSL_2(FF_16), is a Galois group over the rationals and exhibit an explicit degree 17 polynomial with this Galois group. The group arises from the field of definition of the 2-torsion on an abelian fourfold with real multiplication defined over a real quadratic field; we find such a fourfold attached to a Hilbert modular form. Building upon work of Dembele, we describe a method for reconstructing a period matrix attached to a Hilbert modular form, and we use it to construct the 2-isogeny polynomial. We also rigorously identify the relevant fourfold as the Jacobian of a genus 4 Shimura curve and compute explicit equations for this curve.

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arXiv2023

Hypergeometric $L$-functions in average polynomial time, II

For a fixed positive integer $e$, we describe an algorithm for computing, for all primes $p \leq X$, the mod-$p^e$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometric motive over $\mathbb{Q}$ in time quasilinear in $X$. This extends our previous work for the mod-$p$ reduction, again combining the Beukers--Cohen--Mellit trace formula with average polynomial time techniques of Harvey and Harvey--Sutherland; the key new ingredient is an expanded version of Harvey's "generic prime" construction, making it possible to incorporate certain $p$-adic transcendental functions into the computation. One of these is the $p$-adic Gamma function, whose average polynomial time computation is an intermediate step which may be of independent interest. We also provide an implementation in Sage and discuss the remaining computational issues around computing hypergeometric $L$-series.

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arXiv2023

Computing isogeny classes of typical principally polarized abelian surfaces over the rationals

We describe an efficient algorithm which, given a principally polarized (p.p.) abelian surface $A$ over $\mathbb{Q}$ with geometric endomorphism ring equal to $\mathbb{Z}$, computes all the other p.p. abelian surfaces over $\mathbb{Q}$ that are isogenous to $A$. This algorithm relies on explicit open image techniques for Galois representations, and we employ a combination of analytic and algebraic methods to efficiently prove or disprove the existence of isogenies. We illustrate the practicality of our algorithm by applying it to 1 440 894 isogeny classes of Jacobians of genus 2 curves.

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arXiv2022

Counting points on smooth plane quartics

We present efficient algorithms for counting points on a smooth plane quartic curve $X$ modulo a prime $p$. We address both the case where $X$ is defined over $\mathbb F_p$ and the case where $X$ is defined over $\mathbb Q$ and $p$ is a prime of good reduction. We consider two approaches for computing $\#X(\mathbb F_p)$, one which runs in $O(p\log p\log\log p)$ time using $O(\log p)$ space and one which runs in $O(p^{1/2}\log^2\!p)$ time using $O(p^{1/2}\log p)$ space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $X/\mathbb Q$ that compute $\#X(\mathbb F_p)$ for good primes $p\le N$ in $O(N\log^3\! N)$ time using $O(N)$ space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $\mathbb P^1$, which in combination with previous results addresses all curves of genus $g\le 3$. Our algorithms also compute Cartier-Manin/Hasse-Witt matrices that may be of independent interest.

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arXiv2021

Abelian varieties of prescribed order over finite fields

Given a prime power $q$ and $n \gg 1$, we prove that every integer in a large subinterval of the Hasse--Weil interval $[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$ is $#A(\mathbb{F}_q)$ for some geometrically simple ordinary principally polarized abelian variety $A$ of dimension $n$ over $\mathbb{F}_q$. As a consequence, we generalize a result of Howe and Kedlaya for $\mathbb{F}_2$ to show that for each prime power $q$, every sufficiently large positive integer is realizable, i.e., $#A(\mathbb{F}_q)$ for some abelian variety $A$ over $\mathbb{F}_q$. Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse--Weil interval. A separate argument determines, for fixed $n$, the largest subinterval of the Hasse--Weil interval consisting of realizable integers, asymptotically as $q \to \infty$; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if $q \le 5$, then every positive integer is realizable, and for arbitrary $q$, every positive integer $\ge q^{3 \sqrt{q} \log q}$ is realizable.

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arXiv2020

Hypergeometric L-functions in average polynomial time

We describe an algorithm for computing, for all primes $p \leq X$, the mod-$p$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometric motive in time quasilinear in $X$. This combines the Beukers--Cohen--Mellit trace formula with average polynomial time techniques of Harvey et al.

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arXiv2020

Effective obstruction to lifting Tate classes from positive characteristic

We give an algorithm that takes a smooth hypersurface over a number field and computes a $p$-adic approximation of the obstruction map on the Tate classes of a finite reduction. This gives an upper bound on the "middle Picard number" of the hypersurface. The improvement over existing methods is that our method relies only on a single prime reduction and gives the possibility of cutting down on the dimension of Tate classes by two or more. The obstruction map comes from $p$-adic variational Hodge conjecture and we rely on the recent advancement by Bloch-Esnault-Kerz to interpret our bounds.

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arXiv2020

Computing classical modular forms

We discuss practical and some theoretical aspects of computing a database of classical modular forms in the L-functions and Modular Forms Database (LMFDB).

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arXiv2020

Restrictions on Weil polynomials of Jacobians of hyperelliptic curves

Inspired by experimental data, this paper investigates which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by demonstrating that the Weil polynomial of a hyperelliptic Jacobian must have a particular form modulo 2. For fixed ${g\geq1}$, the proportion of isogeny classes of $g$ dimensional abelian varieties defined over $\mathbb{F}_q$ which fail this condition is $1 - Q(2g + 2)/2^g$ as $q\to\infty$ ranges over odd prime powers, where $Q(n)$ denotes the number of partitions of $n$ into odd parts.

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