The generalized Catalan equation in positive characteristic

Let $K = \mathbb{F}_p(z_1, \ldots, z_r)$ be a finitely generated field over $\mathbb{F}_p$. In this article we study the generalized Catalan equation $ax^m + by^n = 1$ in $x, y \in K$ and integers $m, n > 1$ coprime with $p$.Our main result corrects earlier work of Silverman

Higher genus theory

In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss's genus theory. In this paper we extend Gauss's work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith \cite{smith2} in his recent breakthrough on Goldfeld's conjecture and the Cohen--Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides an explicit description for the group $\text{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives a sharp upper bound for the size of $\text{Cl}^{+}(K)[2]$. For every positive integer $n$, we find infinitely many multi-quadratic number fields $K$ such that $[K:\mathbb{Q}]$ equals $2^n$ and $\text{Gal}(L/\mathbb{Q})$ is a universal expansion group. Such fields $K$ are obtained using Smith's notion of additive systems and their basic Ramsey-theoretic behavior

Unit equations and Fermat surfaces in positive characteristic

In this article we study the three-variable unit equation $x + y + z = 1$ to be solved in $x, y, z \in \mathcal{O}_S^\ast$, where $\mathcal{O}_S^\ast$ is the $S$-unit group of some global function field. We give upper bounds for the height of solutions and the number of solutions. We also apply these techniques to study the Fermat surface $x^N + y^N + z^N = 1$

A sharp upper bound for the 22-torsion of class groups of multiquadratic fields

Let $K$ be a multiquadratic extension of $\mathbb{Q}$ and let $\text{Cl}^{+}(K)$ be its narrow class group. Recently, the authors \cite{KP} gave a bound for $|\text{Cl}^{+}(K)[2]|$ only in terms of the degree of $K$ and the number of ramifying primes. In the present work we show that this bound is sharp in a wide number of cases. Furthermore, we extend this to ray class groups

Vinogradov's three primes theorem with primes having given primitive roots

The first purpose of our paper is to show how Hooley's celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy-Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series. In particular, using results of Lenstra, Stevenhagen and Moree, we provide a partial factorisation as an Euler product and prove that this does not extend to a complete factorisation

On the 44-rank of class groups of Dirichlet biquadratic fields

We show that for 100\% of the odd, squarefree integers $n > 0$ the $4$-rank of $\text{Cl}(\mathbb{Q}(i, \sqrt{n}))$ is equal to $\omega_3(n) - 1$, where $\omega_3$ is the number of prime divisors of $n$ that are $3$ modulo $4$

On the 16-rank of class groups of Q( √ −2p) for primes p ≡ 1 mod 4

We use Vinogradov’s method to prove equidistribution of a spin symbol governing the 16-rank of class groups of quadratic number fields Q( √ −2p), where p ≡ 1 mod 4 is a prime

Spins of prime ideals and the negative Pell equation x(2)-2py(2) =-1

Let p ≡ 1 mod 4 be a prime number. We use a number field variant of Vinogradov’s method to prove density results about the following four arithmetic invariants: (i) 16- rank of the class group Cl(−4p) of the imaginary quadratic number field Q( √ −4p); (ii) 8-rank of the ordinary class group Cl(8p) of the real quadratic field Q( √ 8p); (iii) the solvability of the negative Pell equation x 2 − 2py2 = −1 over the integers; (iv) 2-part of the Tate-Šafarevič group X(Ep) of the congruent number elliptic curve Ep : y 2 = x 3 − p 2x. Our results are conditional on a standard conjecture about short character sums