Schedule for: 20w5005 - Modern Breakthroughs in Diophantine Problems (Online)

A brief introduction video from BIRS staff with important logistical information

By reformulating and extending results of Elkies, we prove some results on $\mathbb Q$-curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees~$\ell$ which an elliptic curve without CM may have are those degrees which are already possible over~$\mathbb Q$ itself (in particular, $\ell\le37$), and we show the existence of a bound on the degrees of cyclic isogenies between $\mathbb Q$-curves depending only on the degree of the field. We also prove that the only possible torsion groups of $\mathbb Q$-curves over number fields of degree not divisible by a prime $\ell\leq 7$ are the $15$ groups that appear as torsion groups of elliptic curves over $\mathbb Q$. This is joint work with John Cremona.

In this talk we present work in progress on a new highly optimized solver for general and constraint S-unit equations over number fields. It has diophantine applications including asymptotic Fermat theorems, Siegel's method for computing integral points, and most strikingly for computing large tables of elliptic curves over number fields with good reduction outside given sets of primes S. For the latter, we improved on the method of Koutsianas (Parshin, Shafarevich, Elkies).

Please turn on your cameras for the "group photo" -- a screenshot in Zoom's Gallery view.

Let C be a curve defined over a number field $k$, and suppose $C(k)$ is nonempty. We say a closed point $x$ on $C$ of degree $d$ is isolated if it does not belong to an infinite family of degree d points parametrized by the projective line or a positive rank abelian subvariety of the curve's Jacobian. In this talk we will identify the non-CM elliptic curves with rational $j$-invariant which give rise to an isolated point of odd degree on $X_1(N)$ for some positive integer $N$. This is joint work with David Gill, Jeremy Rouse, and Lori D. Watson.

Consider a family of degree m cyclic covers of the projective line, with any number of branch points and inertia type. The Jacobians of the curves in this family are abelian varieties having an automorphism of order m with a prescribed signature. For each such family, the signature determines a PEL-type Shimura variety. Under a condition on the class number of m, we determine the Hermitian form and Shimura datum of the component of the Shimura variety containing the Torelli locus. For the proof, we study the boundary of Hurwitz spaces, investigate narrow class numbers of real cyclotomic fields, and build on an algorithm of Van Wamelen about principal polarizations on abelian varieties with complex multiplication. This is joint work with Li, Mantovan, and Tang.

The question that at most how many squares one can find among $N$ consecutive terms of an arithmetic progression, has attracted a lot of attention. An old conjecture of Erd\H{o}s predicted that this number $P_N(2)$ is at most $o(N)$; it was proved by Szemer\'edi. Later, using various deep tools, Bombieri, Granville and Pintz showed that $P_N(2) < O(N^{2/3+o(1)})$, which bound was refined to $O(N^{3/5+o(1)})$ by Bombieri and Zannier. There is a conjecture due to Rudin which predicts a much stronger behavior of $P_N(2)$, namely, that $P_N(2)=O(\sqrt{N})$ should be valid. An even stronger form of this conjecture says that we have $$P_2(N)=P_{24,1;N}(2)=\sqrt{\frac{8}{3}N}+O(1)$$ for $N\geq 6$, where $P_{24,1;N}(2)$ denotes the number of squares in the arithmetic progression $24n+1$ for $0 \leq n < N$. This stronger form has been recently proved for $N \leq 52$ by Gonz\'alez-Jim\'enez and Xarles. In the talk we take up the problem for arbitrary $\ell$-th powers. First we characterize those arithmetic progressions which contain the most $\ell$-th powers asymptotically. In fact, we can give a complete description, and it turns out that basically the 'best' arithmetic progression is unique for any $\ell$. Then we formulate analogues of Rudin's conjecture for general powers $\ell$, and we prove these conjectures for $\ell=3$ and $4$ up to $N=19$ and $5$, respectively. The new results presented are joint with Sz. Tengely.

In this talk we study the Hasse principle for the family of "diagonal K3 surfaces of degree 2", given by the explicit equations: $$w^2 = A_1 x_1^6 + A_2 x_2^6 + A_3 x_3^6.$$ I will explain how many such surfaces, when ordered by their coefficients, have a Brauer-Manin obstruction to the Hasse principle. This is joint work with Damián Gvirtz and Masahiro Nakahara.

Paranjape showed that K3 surfaces that are double covers of P^2 branched along six lines are dominated by the square of a curve of genus 5. In this talk, we describe a somewhat analogous construction and use it to show that K3 surfaces in P^4 with 15 nodes are dominated by the square of a curve of genus 7. We will explain a birational equivalence between the moduli space of a related family of K3 surfaces and a moduli space of covers of rational curves with additional data. This is joint work with Colin Ingalls and Owen Patashnick.

K3 surfaces have been extensively studied over the past decades for several reasons. For once, they have a rich and yet tractable geometry and they are the playground for several open arithmetic questions. Moreover, they form the only class which might admit more than one elliptic fibration with section. A natural question is to ask if one can classify such fibrations, and indeed that has been done by several authors, among them Nishiyama, Garbagnati and Salgado. The particular setting that we were interested in studying is when a K3 surface arises as a double cover of an extremal rational elliptic surface with a unique reducible fiber. This K3 surface will have a non-symplectic involution τ fixing two smooth Galois-conjugate genus 1 curves. In this joint work we provide a list of all elliptic fibrations on those K3 surfaces together with the degree of a field extension over which each genus one fibration is defined and admits a section. We show that the latter depends, in general, on the action of the cover involution τ on the fibers of the genus 1 fibration. This is a joint work with Alice Garbagnati, Cecília Salgado, Antonela Trbovíc and Rosa Winter.

We discuss a version of Ekedahl's geometric sieve for integral quadratic forms of rank at least five. This can be used to address some natural questions to do with strong approximation and local solubility in families.

This talk introduces Campana points, an arithmetic notion, first studied by Campana and Abramovich, that interpolates between the notions of rational and integral points. Campana points are expected to satisfy suitable analogs of Lang's conjecture, Vojta's conjecture and Manin's conjecture, and their study introduce new number theoretic challenges of a computational nature.

In this talk we will present some new cases of Vojta's conjecture for surfaces with truncated counting functions in the function field setting, with an explicit description of the exceptional set. As an application, these new estimates allow us to study certain systems of Diophantine equations with unspecified exponents. The method concerns a local study of omega-integral curves and global estimates for intersection numbers. This builds on our earlier work regarding the explicit computation of the exceptional set in the context of the Bombieri-Lang conjecture.

Following Wiles's breakthrough work, it has been shown in recent years that elliptic curves over each totally real field of degree 2 (Freitas-Le Hung-Siksek) or 3 (Derickx-Najman-Siksek) are modular. We study the degree 4 case and show that if K is a totally real quartic field in which 5 is not a square, then every elliptic curve over K is modular. Thanks to strong results of Thorne and Kalyanswami, this boils down to the determination of all quartic points on a few modular curves. Some of these curves have infinitely many quartic points. In this talk I will discuss how Chabauty's method and sieving can nevertheless be used to describe such points.

For any positive integer N, we propose a conjecture on the rational torsion points on J_0(N). Also, we prove this conjecture up to finitely many primes. More precisely, we prove that the prime-to-m parts of the rational torsion subgroup of J_0(N) and the rational cuspidal divisor class group of X_0(N) coincide, where m is the largest perfect square dividing 12N.

We will give a bound for the number of rational points in a hyperbolic surface contained in an abelian threefold of Mordell-Weil rank $1$ over $\mathbb{Q}$. The form of the estimate is analogous to the classical Chabauty-Coleman bound for curves, although the proof uses a completely different approach. The new method concerns w-integral schemes, especially in positive characteristic. This is joint work with Jerson Caro.