Session I.2 - Computational Number Theory
Talks
Monday, June 12, 14:00 ~ 14:30
Modular Galois representations, congruences and entanglement
Samuele Anni
Aix-Marseille Université, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
The theory of congruences of modular forms is a central topic in contemporary number theory, lying at the basis of the proof of Mazur's theorem on torsion in elliptic curves, Fermat's Last Theorem, and Sato-Tate, amongst others. Congruences are a display of the interplay between geometry and arithmetic.
In this talk, I will not only explain how to study and test congruences relations between modular forms, but also how to study isomorphisms of modular Galois representations.
In particular, I will explain how to test efficiently and effectively whether two odd modular Galois representations of the absolute Galois group of the rational numbers are isomorphic and present new optimal bounds on the number of traces to be checked (joint work with Peter Bruin, University of Leiden). I will also briefly discuss new results concerning entanglement of such representations (joint work with Luis Dieulefait, Universitat de Barcelona, and Gabor Wiese, Université du Luxembourg).
Monday, June 12, 14:30 ~ 15:00
Rational points and intersecting lines on del Pezzo surfaces
Rosa Winter
King's College London, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.
Del Pezzo surfaces are classified by their degree~$d$, which is an integer between 1 and 9 (for $d\geq3$, these are the smooth surfaces of degree $d$ in $\mathbb{P}^d$). Over algebraically closed fields they are rational, and contain a fixed number of `lines' (exceptional curves), depending on $d$. The set of rational points over non-algebraically closed fields is not fully understood, with more open questions as $d$ goes down. A long-standing open problem is whether every del Pezzo surface of degree 1 has a dense set of rational points. Partial results are known, and often, the configuration of the lines on the surface plays a role in these results. In this talk I will show how the lines come in to play, and go over several computational results on the configuration of the 240 lines on a del Pezzo surface of degree 1. This is based on joint results, as well as work in progress, with Julie Desjardins, Yu Fu, Kelly Isham, and Ronald van Luijk.
Monday, June 12, 16:30 ~ 17:00
Algebraic curves from their translation surfaces
Türkü Özlüm Çelik
Koç University, Turkey - This email address is being protected from spambots. You need JavaScript enabled to view it.
We study constructing an algebraic curve from a Riemann surface given via a translation surface, which is a collection of finitely many polygons in the plane with sides identified by translation. We use the theory of discrete Riemann surfaces to give an algorithm for approximating the Jacobian variety of a translation surface whose polygon can be decomposed into squares. We first implement the algorithm in the case of L-shaped polygons where the algebraic curve is already known. The algorithm is also implemented in any genus for specific examples of Jenkins-Strebel representatives, a dense family of translation surfaces that, until now, lived on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves. Using Riemann theta functions, we give numerical experiments and resulting conjectures up to genus 5.
Joint work with Samantha Fairchild (MPI MiS, Germany) and Yelena Mandelshtam (UC Berkeley, USA).
Monday, June 12, 17:00 ~ 17:30
Algorithms for abelian surfaces
Jeroen Sijsling
Universität Ulm, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
This talk will give an overview of algorithms for working with abelian surfaces, both analytically and arithmetically.
In the first part of the talk, we describe the reconstruction of principally polarized abelian surfaces (ppas) from their period matrices; for simple ppas, this comes down to reconstructing a genus-2 curve, which can be done both geometrically (over $\mathbb{C}$) and arithmetically (over the natural field of definition of the ppas). We discuss this theme, the link with theta functions and their derivatives, and currently available algorithms in various systems.
The second part of the talk describes how to recover principally polarizations on a given lattice in $\mathbb{C}^2$. We also consider a Prym variety that is a natural example of a non-principally polarized abelian surface over $\mathbb{Q}$ and for which a principal polarization over $\mathbb{Q}$ is not available.
We conclude by discussion explicit modularity results on abelian surfaces and the link with the L-Functions and Modular Forms Database (LMFDB).
Joint work with Andrew Booker (University of Bristol), Edgar Costa (Massachusetts Institute of Technology), Nicolas Mascot (Trinity College Dublin), Andrew Sutherland (Massachusetts Institute of Technology), John Voight (Dartmouth College), Dan Yasaki (University of North Carolina at Greensboro).
Monday, June 12, 17:30 ~ 18:00
A database of basic numerical invariants of Hilbert modular surfaces
Avi Kulkarni
Dartmouth College, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Hilbert modular surfaces are two dimensional analogies of modular curves, in that they are moduli spaces of abelian surfaces with prescribed structure. In this talk, we describe our software package to compute some invariants of Hilbert modular surfaces including the Chern numbers and Kodaira dimension. Time permitting, I will also discuss future goals for the project.
Joint work with Eran Assaf (Dartmouth College), Angelica Babei (McMaster University), Ben Breen (Clemson University), Edgar Costa (MIT), Juanita Duque-Rosero (Dartmouth College), Aleksander Horawa (Oxford University), Jean Kieffer (Harvard), Grant Molnar (Dartmouth College), Sam Schiavone (MIT) and John Voight (Dartmouth College).
Monday, June 12, 18:00 ~ 18:30
Finding the Kawamata locus of subvarieties of Abelian varieties
Natalia Garcia-Fritz
Pontificia Universidad Catolica de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.
From the recently proved uniform Mordell-Lang theorem, we know that the number of rational points outside the Kawamata locus of subvarieties of abelian varieties is bounded in terms of the dimension of the abelian variety, the degree of the subvariety, and the Mordell-Weil rank. From this, it is an interesting problem to explicitly determine the Kawamata locus, and in fact this step is crucial in several arithmetic applications. In this talk we will focus on the case of surfaces by means of a technique coming from value distribution theory. Using this approach we will discuss applications to problems on points in elliptic curves.
Joint work with Hector Pasten (Pontificia Universidad Catolica de Chile, Chile).
Tuesday, June 13, 14:00 ~ 15:00
Computing Selmer sets
Jennifer Balakrishnan
Boston University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Kim's nonabelian Chabauty program for studying rational points on hyperbolic curves yields a series of refined Selmer sets at depth n, cut out by n-fold iterated p-adic integrals. We discuss computations that have been done of these sets in explicit cases, including punctured elliptic curves and smooth projective curves of genus 2 and 3.
Tuesday, June 13, 15:00 ~ 15:30
Exceptional Units from Geometry
Nicholas Triantafillou
Center for Communications Research - Princeton, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We describe a geometric approach to construct explicit infinite families of number fields with large Lenstra constant (i.e. containing large sets of integers whose pairwise differences are units) and whose rings of integers contain many exceptional units. For example, by considering integral points on an appropriate model of the modular curve $X_1(11)$ punctured at its 5 rational points, we produce 80 infinite families of sextic number fields, each with Lenstra constant at least 6 and each containing at least 60 exceptional units. We will also discuss applications to studying torsion subgroups on elliptic curves over number fields.
Joint work with Dino Lorenzini (University of Georgia, USA).
Tuesday, June 13, 15:30 ~ 16:00
Explicit uniform bounds for Brauer groups of singular K3 surfaces
Rachel Newton
King's College London, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.
By analogy with Merel's theorem on torsion groups of elliptic curves, Várilly-Alvarado has conjectured that Brauer groups (modulo constants) of K3 surfaces over number fields are bounded by a number that only depends on degree of the field and the isomorphism class of the Néron-Severi lattice. Orr and Skorobogatov proved this conjecture for K3 surfaces of CM type, showing the existence of a bound that only depends on the degree of the number field. I will present joint work with Francesca Balestrieri and Alexis Johnson in which we re-prove Várilly-Alvarado’s conjecture for singular K3 surfaces, this time with an explicit bound. This bound is very large in general but can be improved dramatically in certain cases, e.g. if the geometric Picard group is generated by divisors defined over the base field. When combined with results of Kresch–Tschinkel and Poonen–Testa–van Luijk, this shows that the Brauer–Manin sets for these varieties are effectively computable.
Joint work with Francesca Balestrieri (The American University of Paris) and Alexis Johnson (DryvIQ).
Tuesday, June 13, 16:30 ~ 17:00
Frobenius distributions on K3 surfaces
David Kohel
Institut de Mathématiques de Marseille, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $S$ be a K3 surface over $\mathbb{Q}$. The zeta function of the reduction of $S$ over $\mathbb{F}_p$ takes the form $$ \frac{1}{(1-T)(1-pT)(1-p^2T)p^{-1}L(pT)} $$ where $L(T) \in \mathbb{Z}[T]$ is of degree $21$, with $L(0) = p$ and all roots on the unit circle. The polynomial $L(T)$ factors as $$ L(T) = L_{\mathrm{alg}}(T)L_{\mathrm{trc}}(T) $$ into the algebraic and transcendental parts, where the roots of former are roots of unity. The polynomial $L(T)$ carries the information of the image of Frobenius in the Sato-Tate group associated to $S$. We describe the characterization of this Galois representation associated to $S$, in terms of character theory of orthogonal groups.
Joint work with Kiran Kedlaya.
Tuesday, June 13, 17:00 ~ 17:30
Algebraic Geometry codes in the sum-rank metric
Elena Berardini
Université de Bordeaux, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
Linear codes in the Hamming metric have been playing a central role in the theory of error correction since the 50s and were extensively studied. In reverse, the theory of codes in the sum-rank metric is still in its beginnings, and to date, only a few constructions are known.
Algebraic Geometry codes in the Hamming metric allow overcoming the main drawback of Reed-Solomon codes, which is that their length is bounded by the cardinality of the finite field we work on, while benefiting from good parameters. The counterpart of Reed-Solomon codes in the sum-rank metric are linearized Reed-Solomon codes. They have optimal parameters but suffer from the same limitation as Reed-Solomon codes. However, in contrast with the situation of codes in the Hamming metric, no geometric construction has been proposed so far.
In this talk, we will present the first geometric construction of sum-rank metric codes, called linearized Algebraic Geometry codes. After introducing some background on codes in the sum-rank metric, we will develop the theory of Riemann-Roch spaces over Ore polynomials rings with coefficients in the function field of a curve, by exploiting the classical theory of divisors and Riemann-Roch spaces on algebraic curves. With this theory at hand, we will study the parameters of linearized Algebraic Geometry codes and give lower bounds for their dimension and minimum distance. Notably, we will show that our new codes exhibit quite good parameters, respecting a similar bound to Goppa’s bound for Algebraic Geometry codes in the Hamming metric.
Joint work with Xavier Caruso (Université de Bordeaux).
Tuesday, June 13, 17:30 ~ 18:00
Limit periods on curves and arithmetic heights
Emre Sertöz
Leibniz University Hannover, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
We recently proved that in a nodal degeneration of smooth curves, the periods of the resulting limit mixed Hodge structure (LMHS) contain arithmetic information. For instance, if the nodal fiber is identified with a smooth curve C glued at two points p and q then the LMHS relates to the Neron--Tate height of p-q in the Jacobian of C. In making this relation precise, we observed that a "tropical correction term" is required that is based on finite reductions of the degenerate fiber. In this talk, I will explain this circle of ideas with a focus on the explicit and computable aspects.
Joint work with Spencer Bloch (University of Chicago) and Robin de Jong (Leiden University).
Wednesday, June 14, 14:00 ~ 15:00
Counting points on modular curves
Andrew Sutherland
Massachusetts Institute of Technology, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $H$ be an open subgroup of $\mathrm{GL}_2(\hat{\mathbb{Z}})$, let $X_H$ be the corresponding modular curve that parametrizes elliptic curves with $H$-level structure, and let~$\mathbb{F}_q$ be a finite field whose characteristic does not divide the level of $H$.
I will discuss improvements to the moduli-theoretic approach for computing $\#X_H(\mathbb{F}_q)$ that lead to an algorithm that is practically and asymptotically faster than existing approaches as $q$, the genus of $X_H$, and the level of $H$ vary.
Wednesday, June 14, 15:00 ~ 15:30
Computing Euler factors of curves
Celine Maistret
University of Bristol, UK - This email address is being protected from spambots. You need JavaScript enabled to view it.
L-functions of abelian varieties are objects of great interest. In particular, they are believed (and known in some cases) to carry key arithmetic information of the variety via the Birch and Swinnerton-Dyer conjecture. As such, it is useful to be able to compute them in practice. In this talk, we will address the case of a genus 2 curve $C/\mathbb{Q}$ with bad reduction at an odd prime p where the Jacobian of $C$ has good reduction. Our approach relies on counting points on the special fibre of the minimal regular model of the curve, which we extract using the theory of cluster pictures of hyperelliptic curves. Our method yields a fast algorithm in the sense that all computations occur in at most quadratic extensions of $\mathbb{Q}$ or finite fields.
Joint work with Andrew Sutherland (MIT).
Wednesday, June 14, 15:30 ~ 16:00
Distribution of the number of points on curves over finite fields: new results and conjectures
Christophe Ritzenthaler
Université Rennes 1 and Université Côte d'Azur, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, we will see how to go beyond Katz-Sarnak theory on the distribution of curves over finite fields according to their number of rational points. In particular, we present a formula for the limits of the moments measuring the asymmetry of this distribution for curves of genus $g \geq 3$. Then, with some experimental data, we will try to convince the audience that there may be a stronger notion of convergence than the one provided by the Katz–Sarnak framework for all curves of genus $\geq 3$. This was not observed before because for elliptic curves and for hyperelliptic curves of every genus, this stronger convergence cannot occur.
Joint work with JONAS BERGSTRÖM (Stockholms Universitet), EVERETT W. HOWE and ELISA LORENZO GARCÍA (université de Neuchâtel).
Wednesday, June 14, 16:30 ~ 17:00
Explicit computation of regular symmetric differentials on singular surfaces
Nils Bruin
Simon Fraser University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
For a variety of general type over a number field, the Bombieri-Lang conjecture predicts that the rational points are not dense. For a surface of general type, this means that all but finitely many rational points lie on curves of genus 0 and 1. Hence, it is of Diophantine interest to be able to find the genus 0 and 1 curves on a surface. A global section of the sheaf of differentials, or a symmetric power thereof, gives significant information on such curves.
We discuss some techniques for computing such differentials and present some examples where these can be used to determine all genus 0 and 1 curves on the surface. This talk is based on [Nils Bruin, Jordan Thomas and Anthony Várilly-Alvarado, Explicit computation of symmetric differentials and its application to quasihyperbolicity, Algebra and Number Theory 16-6, 2022] and includes some interesting updates stemming from joint work with Nathan Ilten and Zhe Xu.
Wednesday, June 14, 17:00 ~ 17:30
Primitive points on elliptic curves
Peter Stevenhagen
Universiteit Leiden, Netherlands - This email address is being protected from spambots. You need JavaScript enabled to view it.
Given a point $P$ of infinite order on an elliptic curve $E$ defined over a number field $K$, one may ask, after Lang and Trotter, whether the set of primes $\frak p$ of $K$ for which the reduction of $P$ generates the point group over the residue class field of $\frak p$ possesses a density. Unlike the density for the set of primes of cyclic reduction of $E$, the heuristical density in this case has not been proven to be correct, not even under GRH.
We will focus on the vanishing of the heuristical density. This is a question that can be answered without assuming GRH. It has more subtleties than the density of the set of primes of cyclic reduction of $E$.
Joint work with Francesco Campagna (Leibniz Universitaet Hannover, Germany), Francesco Pappalardi (Roma 3, Italy) and Nathan Jones (University of Illinois at Chicago, USA).
Posters
Pseudorandom numbers from curves of genus 2
Vishnupriya Anupindi
RICAM, Austrian Academy of Sciences, Austria - This email address is being protected from spambots. You need JavaScript enabled to view it.
Pseudorandom sequences, that is, sequences which are generated with deterministic algorithms but look random, have many applications, for example in cryptography, in wireless communication or in numerical methods.
In this poster, we briefly recall the group law on genus 2 hyperelliptic curves and discuss some properties of pseudorandomness of sequences derived from these curves. In particular, we look at two different ways of generating sequences, that is, the linear congruential generator and the Frobenius endomorphism generator over hyperelliptic curves of genus 2. We show that these sequences possess good pseudorandom properties in terms of linear complexity.
Joint work with László Mérai (RICAM, Austrian Academy of Sciences).
Multiplication polynomials for elliptic curves over finite local rings
Riccardo Invernizzi
KU Leuven, Belgium - This email address is being protected from spambots. You need JavaScript enabled to view it.
For a given elliptic curve $E$ over a finite local ring, we denote by $E^{\infty}$ its subgroup at infinity. Every point $P \in E^{\infty}$ can be described solely in terms of its $x$-coordinate $P_x$, which can be therefore used to parameterize all its multiples $nP$. We refer to the coefficient of $(P_x)^i$ in the parameterization of $(nP)_x$ as the $i$-th multiplication polynomial.
We show that this coefficient is a degree-$i$ rational polynomial without a constant term in $n$. We also prove that no primes greater than $i$ may appear in the denominators of its terms. As a consequence, for every finite field $\mathbb{F}_q$ and any $k\in\mathbb{N}^*$, we prescribe the group structure of a generic elliptic curve defined over $\mathbb{F}_q[X]/(X^k)$, and we show that their ECDLP on $E^{\infty}$ may be efficiently solved.
Joint work with Daniele Taufer (KU Leuven).
Applications and Constructions of Trisections of Low Genus on Elliptic Surfaces
Vojin Jovanovic
University of Toronto, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
Consider a rational elliptic surface over a field $k$ with char $0$ given by $\mathcal{E}: y^2 = x^3 + f(t)x + g(t)$, with $f,g\in k[t]$, $\deg(f) \leq 4$ and $\deg(g) \leq 6$. If all the bad fibres are irreducible, such a surface comes from the blow-up of a del Pezzo surface of degree 1. We are interested in studying multisections, i.e. curves which intersect each fibre a fixed number of times; more precisely trisections (three times). Many configurations of singularities on a trisection lead to a lower genus; here we focus on one of them. By specifying conditions on the coefficients $f,g$ of the surface $\mathcal{E}$, and looking at trisections which pass through a given point three times, we obtain a pencil of cubics on such surfaces. Our construction could have interesting applications in proving the Zariski density of the rational points. It is especially interesting since the results in this regard are partial for del Pezzo surfaces of degree 1. Further reduction of the genus of a trisection with a triple singularity is possible, however this process leads to an isolated curve of genus zero instead of a family.
Joint work with Julie Desjardins (University of Toronto, Canada).
A $p$-adic Descartes solver: the Strassman solver
Josue Tonelli-Cueto
The University of Texas at San Antonio, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Solving polynomials is a fundamental computational problem in mathematics. In the real setting, we can use Descartes' rule of signs to efficiently isolate the real roots of a square-free real polynomial. In this poster, we translate this method into the $p$-adic worlds. We show how the $p$-adic analog of Descartes' rule of signs, Strassman's theorem, leads to an algorithm to isolate the roots of a square-free $p$-adic polynomial. Moreover, we show that this algorithm runs in $\mathcal{O}(d^2\log^3d)$-time for a random $p$-adic polynomial of degree $d$. To perform this analysis, we introduce the condition-based complexity framework from real/complex numerical algebraic geometry into $p$-adic numerical algebraic geometry.