Arithmetic Geometry: Theory and Computation

Session Organizers

Sho Tanimoto, Nagoya University

Bianca Viray, University of Washington

8 December, 2022

Abstract

Let $d\geq 2$ and let $H$ denote the absolute multiplicative Weil height on $\bar{\mathbb{Q}}$. Let $f(z)\in \bar{\mathbb{Q}}[z]$ of degree $d$ and let $a\in\bar{\mathbb{Q}}$, the multiplicative and logarithmic canonical heights of $a$ with respect to $f$ are defined as

$$ \hat{H}_f (a) =\lim H(f^n(a))^{1/d^n} \quad \text{and}\quad \hat{h}_f(a)=\log \hat{H}_f(a). $$

Let $n$ be a positive integer. For $1\leq i\leq n$, let $f_i\in\bar{\mathbb{Q}}[z]$ of degree $d$ and let $a_i\in\bar{\mathbb{Q}}$. In this talk, we provide a complete characterization of when the $\hat{H}_{f_i}(a_i)$’s are multiplicatively dependent modulo constant meaning there exist integers $m_1,\ldots,m_n$ not all of which are $0$ and $a\in \bar{\mathbb{Q}}$ such that:

$$ \hat{H}_{f_1}(a_1)^{m_1} \cdots \hat{H}_{f_n}(a_n)^{m_n}=a. $$

As an immediate consequence, we characterize all the pairs $(f,a)$ such that $\hat{H}_f(a)$ is an algebraic number and proves the existence of $(f,a)$ such that $\hat{h}_f(a)$ is an irrational number. The proof uses the Medvedev-Scanlon classification of preperiodic subvarieties under the dynamics of a split polynomial map and the construction of a certain auxiliary polynomial. This is joint work with Jason Bell.

Abstract

In 1900, Hilbert posed the following problem: “Given a Diophantine equation with integer coefficients: to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in (rational) integers.”

Building on the work of several mathematicians, in 1970, Matiyasevich proved that this problem has a negative answer, i.e., such a general `process’ (algorithm) does not exist.

In the late 1970’s, Denef–Lipshitz formulated an analogue of Hilbert’s 10th problem for rings of integers of number fields.

In recent years, techniques from arithmetic geometry have been used extensively to attack this problem. One such instance is the work of García-Fritz and Pasten (from 2019) which showed that the analogue of Hilbert’s 10th problem is unsolvable in the ring of integers of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and $q$. In joint work with Lei and Sprung, we improve their proportions and extend their results in several directions.

Abstract

We study Campana’s orbifold conjecture for finite ramified covers of $\mathbb P^2$ with three components admitting sufficiently large multiplicities. We also prove a truncated second main theorem of level one for analytic maps into $\mathbb P^2$ intersecting the coordinate lines in sufficiently high multiplicities. In particular, the exceptional set for the later result can be described explicitly.

This is joint work with Ji Guo.

Abstract

In this talk, I will explain how the Batyrev-Manin conjecture on rational points can be generalized to Deligne-Mumford stacks by using twisted sectors. In the original conjecture, the so-called a- and b-invariants are determined by positions of the ample line bundle in question and the canonical divisor in the Néron-Severi space relative to the pseudo-effective cone. In generalization to stacks, we introduce orbifold versions of these algebro-geometric notions. Once we define them suitably, the generalized conjecture is formulated more or less in the same way as the original conjecture was. The Malle conjecture on Galois extensions of a number field is then regarded as a special case of it. This is a joint work with Ratko Darda.

9 December, 2022

Abstract

Manin’s conjecture is a conjectural asymptotic formula for the counting function of rational points on Fano varieties, and mainly with Brian Lehmann, I have been studying exceptional sets arising in this conjecture. In this talk I would like to discuss my joint work with Brian Lehmann and Akash Sengupta on birational geometry of exceptional sets, then I will discuss applications of this study to understand the geometry of moduli spaces of curves on Fano varieties which is joint work with Brian Lehmann and Eric Riedl.

Abstract

I will discuss local-global principles for two notions of semi-integral points, termed Campana points and Darmon points. In particular, I will introduce a semi-integral version of the Brauer-Manin obstruction interpolating between Manin’s classical version for rational points and the integral version developed by Colliot-Thélène and Xu. Lastly, we will apply these tools to study semi-integral points on quadric orbifolds.

Abstract

The characteristic cycle of a constructible sheaf on a projective smooth algebraic variety is an algebraic cycle on the cotangent bundle that computes the Euler characteristic of the sheaf. In this talk, we consider a rank 1 sheaf on the variety. For a computation of the characteristic cycle of a rank 1 sheaf, we introduce a general theory called partially logarithmic ramification theory, and construct an algebraic cycle using several invariants measuring the ramification of the sheaf, which is compared with the characteristic cycle.

Abstract

The construction of Kummer K3 surfaces from abelian surfaces can be generalized to yield higher dimensional varieties known as hyperk"ahler varieties of Kummer type. Hassett and Tschinkel showed that a portion of the middle cohomology of generalized Kummer 4-folds may be understood as fixed loci of symplectic involutions corresponding to the three-torsion points of the abelian surface. In recent work with Sarah Frei, we have extended this result, allowing us to characterize the Galois action on the cohomology when working over non-closed fields.