Schedule for: 17w5065 - Arithmetic Aspects of Explicit Moduli Problems

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

General methods from diophantine geometry have been very successful in proving finiteness results for points of algebraic curves with values in number fields. Those results however are generally not effective, for deep reasons, and this prevents from proving triviality (and not only finiteness) of relevant sets of rational points. In this talk I will explain how the situation can be much better in the case of modular curves because of the only, deep, but simple feature that in many cases their jacobian possesses a non-trivial quotient with rank zero over the rationals. From that we can derive effective upper bounds for the height of rational points by using specific arakelovian tools.

Let $E$ be an elliptic curve over a number field $K$ of degree $d$ with a point of order $n$. What properties is $E$ forced to have? For appropriate choices of $(n,d)$ the answer can be: 1) have even rank; 2) be a $\mathbb{Q}$-curve; 3) the field $K$ over which $E$ is defined has to be of certain type; 4) be a base change of an elliptic curve defined over $\mathbb{Q}$; 5) have Tamagawa numbers of specific form. In this talk we will sketch why these kinds of results are true and show that they come from the geometric properties of modular curves and maps between modular curves and their moduli interpretations. We will describe in a bit more detail new results regarding Tamagawa numbers of elliptic curves with a point of order $13$ over quadratic fields.

We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed or fully minimal. The type of A depends on the normalized Weil numbers of A and its twists over its minimal field of definition. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic. For supersingular curves of genus 3 in characteristic 2, we use a parametrization of a moduli space of such curves by Viana and Rodriguez to determine the L-polynomial and the type of each. This is joint work with Valentijn Karemaker.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

We describe how to explicitly compute the natural action of $SL_2(\mathbb{Z})$ on the space of cusp forms $S_k(\Gamma(N))$. This action is useful for finding equations of modular curves via their canonical embedding. It will reduce to studying Atkin-Lehner involutions and the arithmetic of the underlying modular curves.

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A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I’ll discuss recent progress on Mazur’s ”Program B”, including my own recent work with Jeremy Rouse which completely classifies the possibilities for the 2-adic image of Galois associated to an elliptic curve over the rationals. I will also discuss a large number of other very recent results by many authors.

We study the relationship between the p-rank of a curve and the $p$-ranks of the Prym varieties of its unramified cyclic covers in characteristic $p > 0$. For arbitrary primes $p$ and $\ell$ with $\ell \ne p$ and integers $g \ge 3$ and $0 \le f \le g$, we generalize a result of Nakajima by proving that the Prym varieties of all the unramified $\mathbb{Z}/\ell$-covers of a generic curve $X$ of genus $g$ and $p$-rank $f$ are ordinary. Furthermore, when $p \ge 5$ and $\ell \ne 2$, we prove that there exists a curve of genus $g$ and $p$-rank $f$ having an unramified double cover whose Prym has p-rank $f^\prime$ for each $g/2 − 1 \le f^\prime \le g − 2$; (these Pryms are not ordinary). Using work of Raynaud, we use these two theorems to prove results about the (non)-intersection of the \ell-torsion group scheme with the theta divisor of the Jacobian of a generic curve $X$ of genus $g$ and $p$-rank $f$. The proofs involve geometric results about the $p$-rank stratification of the moduli space of unramified cyclic covers of curves. This is joint work with Rachel Pries.

Humbert surfaces are certain surfaces embedded in the moduli space $A_2$ of principally polarized abelian surfaces. In this talk I will explain the connection between the components of the intersection of two Humbert surfaces and classes of certain binary quadratic forms. More precisely, for each positive quadratic form $q$ in $r$ variables one can associate a closed subvariety $H(q)$ of $A_2$ (which depends only on the equivalence class of the form). If $r = 1$, then we recover the Humbert surfaces. For $r = 2$ we get curves which can be used to describe the intersection of two Humbert surfaces. (Using the reduction theory of binary quadratic forms, this can be done quite explicitly.) If q is a primitive binary quadratic form, then $H(q)$ is irreducible, but in the general case $H(q)$ is a union of the images of modular curves (modular correspondences) lying on $X(N) x X(N)$ (or on Hilbert modular surfaces). By studying conjugacy classes of matrices mod $N$, the irreducible components of $H(q)$ can be identified. Thus, one gets an explicit description of all irreducible components of the intersection of two Humbert surfaces.

Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $\mathrm{Br}{Y}/ \mathrm{Br}_1{Y}$ is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric N\'eron-Severi lattices. Over a field of characteristic 0, we prove that the existence of a strong uniform bound on the size of the odd-torsion of $\mathrm{Br}{Y}/ \mathrm{Br}_1{Y}$ is equivalent to the existence of a strong uniform bound on integers $n$ for which there exist non-CM elliptic curves with abelian $n$-division fields. Using the same methods we show that, for a fixed prime $p$, a number field $k$ of fixed degree $r$, and a fixed discriminant of the geometric N\'eron-Severi lattice, $(\mathrm{Br}{Y} / \mathrm{Br}_1{Y})[p^\infty]$ is bounded by a constant that depends only on $p$, $r$, and the discriminant. This is joint work with Anthony Várilly-Alvarado.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In 2006, Mazur, Stein, and Tate gave an algorithm to compute p-adic heights and regulators on elliptic curves, motivated by the p-adic Birch and Swinnerton-Dyer conjecture. Their work on these explicit methods has led to a new use of p-adic heights, namely as a tool to find rational points on curves. I'll give a survey of these methods (joint with A. Besser, N. Dogra, J. S. Mueller) and describe the landscape of future work for families of curves and curves with extra structure.

I will discuss joint work with J. Balakrishnan and N. Dogra on the computation of the rational points on a curve of genus 2 over the rationals whose Jacobian has rank 2 and real multiplication. The method we used is based on recent work of Balakrishnan and Dogra on non-abelian Chabauty and generalizes to hyperelliptic curves whose Jacobians have RM and rank equal to the genus of the curve.

Let $A$ be an abelian surface defined over $\Q$. It is well known that the $\Q$-algebra $End(A) \otimes \Q$ is either $\Q$, $\Q \times \Q$, $M_2(\Q)$, a real quadratic field $E$, the product of $\Q$ and a quadratic imaginary field of class number 1, a quartic CM field $L$, a definite division quaternion algebra $B$, or $M_2(K)$, where $K$ is a quadratic imaginary field. It is conjectured that there are only finitely many possibilities for $End(A) \otimes \Q$. While almost nothing is known about the sets of possibilities for $E$ and $B$, Murabashy and Umegaki have proved that there are precisely 19 possibilities for the quartic CM field $L$. In this talk I will report on a joint work with Xevi Guitart in which we show that there are at most 52 possibilities for the quadratic imaginary field $K$. An equivalent way to formulate the result is by saying that there are at most 52 $\bar\Q$-isogeny classes of abelian surfaces defined over $\Q$ that are isogenous to the square of an elliptic curve with CM.

The motivation for this talk comes from the inverse Galois problem for finite linear groups and its variations involving ramification conditions. Given an n-dimensional abelian variety $A/\Q$ which is principally polarised, we consider for each prime number the representation of the absolute Galois group of the rational numbers, $\rho_{A,\ell}: G_\Q \rightarrow \GSp(2n,\ell)$ attached to the $\ell$-torsion points of $A$. Provided the representation is surjective, we obtain a realisation of $\GSp(2n,\ell)$ as the Galois group of the finite extension $\Q(A[\ell])/\Q$, and the ramification type of a prime $p$ in this extension can be read off from the type of reduction of $A$ at $p$. This approach motivates our interest for the existence of abelian varieties over $\Q$ satisfying local conditions at a finite set of primes, which can be rephrased as the problem of the existence of a rational point in the intersection of a finite number of $p$-adic open sets of a suitable moduli space. In practice, it is very difficult to find such points, and a good strategy is to consider the Jacobian of curves in a family, which can easily be deformed p-adically. Recently there have appeared several constructions of curves defined over $\Q$ satisfying that the $\ell$-torsion representation attached to its Jacobian is surjective (e.g. Anni et al., Arias-de-Reyna et al. and Zywina for genus 3, more recently Anni and Dokchitser for any genus). In principle, these constructions allow one to carry out the above strategy in an explicit way to address the problem of producing realisations of $\GSp(2n,\ell)$ with prefixed ramification conditions.

Consider a smooth projective variety over a number field. The image of the associated (complex) Abel-Jacobi map inside the (transcendental) intermediate Jacobian is a complex abelian variety. We show that this abelian variety admits a distinguished model over the original number field, and use it to address a problem of Mazur on modeling the cohomology of an arbitrary smooth projective variety by that of an abelian variety. (We also recover an old theorem of Deligne on intermediate Jacobians of complete intersection varieties.) In special cases, our construction gives a way to compare certain arithmetic moduli spaces to moduli spaces of abelian varieties. We expect that more such applications exist. This is joint work with Sebastian Casalaina-Martin and Charles Vial.

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For each open subgroup $G$ of $\mathrm{GL}_2(\hat{\mathbb{Z}})$ containing $-I$ with full determinant, let $X_G/\mathbb{Q}$ denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the Galois action on its torsion points, has image contained in $G$. Up to conjugacy, we determine a complete list of the $248$ such groups $G$ of prime power level for which $X_G(\mathbb{Q})$ is infinite. For each $G$, we also construct explicit maps from each $X_G$ to the $j$-line. This list consists of $220$ modular curves of genus $0$ and $28$ modular curves of genus $1$. For each prime $\ell$, these results provide an explicit classification of the possible images of $\ell$-adic Galois representations arising from elliptic curves over $\mathbb{Q}$ that is complete except for a finite set of exceptional $j$-invariants. This is joint work with David Zywina.

This talk is about continuous representations of absolute Galois groups of number fields on finite Abelian groups. Typical sources of such representations include torsion subschemes of elliptic curves and Hecke eigenforms. Any such representation can be described by a finite amount of data. I will discuss theoretical and practical aspects of the following questions: what are suitable ways to explicitly write down Galois representations, what information can be extracted once this has been done, and how can the required computations be done efficiently?

We give examples of smooth plane quartics over $\mathbb{Q}$ with complex multiplication over $\bar{\mathbb{Q}}$ by a maximal order with primitive CM type. We describe the required algorithms as we go, these involve the reduction of period matrices, the fast computation of Dixmier-Ohno invariants, and reconstruction from these invariants. Finally, we discuss some of the reduction properties of the curves that we obtain.

An important tool for bounding the number of rational or torsion points on a curve is to find a function that vanishes at those points and bounding its zeroes. This is essential to Coleman's effective Chabauty and Buium's p-adic differential characters. Buium's work, while appearing quite different, is based on Manin's proof of the function field Mordell conjecture which made use of a Picard-Fuchs differential operator annihilating the periods of an Abelian variety. In this talk, we will discuss a project with Dupuy, Rabinoff, and Zureick-Brown to unify Buium's and Coleman's work in hope of finding more functions vanishing on points of arithmetic interest. This involves constructing differential characters using the p-adic integration theories of Coleman and Colmez, understanding the periods of Abelian varieties, and connections with p-adic Hodge theory.

In Bouw et al. 15 and Kilicer et al. 17, we give a bound for the primes of bad reduction of curves of genus 3 with CM. This bound does not directly translates into a bound for the primes appearing in the denominators of class polynomials of curves of genus 3 because the bad reduction locus for genus 3 has co-dimension 2. However, for special subfamilies in which the bad reduction locus has co-dimension 1, this bound do translate into a bound for primes in the denominators. The family of Picard curves is an example of such subfamily. The bound obtained in these works in huge and not computationally practical. In this new work we give a better set of invariants for Picard curves and a sharper bound for the corresponding denominators. We do it by studying not only primes of bad reduction but also primes of a very particular type of good reduction.

The determination of which finite abelian groups can occur as the torsion subgroup of an elliptic curve over a number field has a long history starting with Barry Mazur who proved that there are exactly 15 groups that can occur as the torsion subgroup of an elliptic curve over the rational numbers. It is a theorem due to Loïc Merel that for every integer d the set of isomorphism classes of groups occurring as the torsion subgroup of a number field of degree d is finite. If a torsion subgroup occurs for a certain degree, then one can also ask for how many distinct pairwise non-isomorphic elliptic curves this happens. The question which torsion groups can occur for infinitely many non-isomorphic elliptic curves of a fixed degree is studied during this talk. The main result is a complete classification of the torsion subgroups that occur infinitely often for degree 5 and 6. This is joint work with Andrew Sutherland and heavily builds on previous joint work with Mark van Hoeij.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.