We develop a general method for computing integral points on modular curves, based on Baker’s inequality. As an illustration, we show that for , the only integral points on the curve $X_{ns}^{+}\left(p\right)$ are the CM points.

A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of covering systems with distinct moduli was initiated by Erdős in 1950, and over the following decades numerous problems were posed regarding their properties. One particularly notorious question, due to Erdős, asks whether there exist covering systems whose moduli are distinct and all odd. We show that if in addition one assumes the moduli are square-free, then there must be an even modulus.

Let $A$ be an abelian variety over a number field $F$ and let $p$ be a prime. Cohen–Lenstra–Delaunay-style heuristics predict that the Tate–Shafarevich group III$\left(A_s\right)$ should contain an element of order $p$ for a positive proportion of quadratic twists $A_s$ of $A$. We give a general method to prove instances of this conjecture by exploiting independent isogenies of $A$. For each prime $p$, there is a large class of elliptic curves for which our method shows that a positive proportion of quadratic twists have nontrivial $p$-torsion in their Tate–Shafarevich groups. In particular, when the modular curve $X_0\left(3p\right)$ has infinitely many $F$-rational points, the method applies to “most” elliptic curves $E$ having a cyclic $3p$-isogeny. It also applies in certain cases when $X_0\left(3p\right)$ has only finitely many rational points. For example, we find an elliptic curve over $\mathbb{Q}$ for which a positive proportion of quadratic twists have an element of order $5$ in their Tate–Shafarevich groups.

The method applies to abelian varieties of arbitrary dimension, at least in principle. As a proof of concept, we give, for each prime $p\equiv 1\left(mod9\right)$, examples of CM abelian threefolds with a positive proportion of quadratic twists having elements of order $p$ in their Tate–Shafarevich groups.

We investigate in a statistical fashion the smallest height of a rational point on a Fano hypersurface defined over the field of rational numbers. Along the way, we establish an average version of Manin’s conjecture about the number of rational points of bounded height on Fano varieties for the complete family of Fano hypersurfaces of fixed degree and dimension.

We show, in the large $q$ limit, that the average size of $n$-Selmer groups of elliptic curves of bounded height over ${\mathbb{𝔽}}_q\left(t\right)$ is the sum of the divisors of $n$. As a corollary, again in the large $q$ limit, we deduce that $100\%$ of elliptic curves of bounded height over ${\mathbb{𝔽}}_q\left(t\right)$ have rank $0$ or $1$.

We show that if a rational map is constant on each isomorphism class of unpolarized abelian varieties of a given dimension, then it is a constant map. Our results shed light on a question raised by Boneh et al. (J. Math. Cryptol. 14:1 (2020), 5–14) concerning a proposal for multiparty noninteractive key exchange.

Let $k$ be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers $h^{i,j}\left(X\right)=dim{H}^j\left(X,{\mathrm{\Omega }}_X^i\right)$ that hold for every smooth proper variety $X$ over $k$ are the ones given by Serre duality. We also show that the only linear combinations of Hodge numbers that are birational invariants of $X$ are given by the span of the $h^{i,0}\left(X\right)$ and the $h^{0,j}\left(X\right)$ (and their duals $h^{i,n}\left(X\right)$ and $h^{n,j}\left(X\right)$). The corresponding statements for compact Kähler manifolds were proven by Kotschick and Schreieder.

It is known that if is a prime number and $E∕\mathbb{Q}$ is an elliptic curve without complex multiplication, then the image of the mod $p$ Galois representation

${\bar{\rho }}_{E,p}:Gal\left(\overline{\mathbb{Q}}∕\mathbb{Q}\right)\to GL\left(E\left[p\right]\right)$

of $E$ is either the whole of $GL\left(E\left[p\right]\right)$, or is contained in the normaliser of a nonsplit Cartan subgroup of $GL\left(E\left[p\right]\right)$. In this paper, we show that when , the image of ${\bar{\rho }}_{E,p}$ is either $GL\left(E\left[p\right]\right)$, or the full normaliser of a nonsplit Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For $d\ge 1$, let $I\left(d\right)$ denote the set of primes $p$ for which there exists an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication admitting a degree $p$ isogeny defined over a number field of degree $\le d$. We show that, for $d\ge 1.4\times 10^7$, we have

$I\left(d\right)=\left\{p\text{ prime} : p\le d-1\right\}.$

Given a finite dimensional Lie algebra $L$, let $I$ be the augmentation ideal in the universal enveloping algebra $U\left(L\right)$. We study the conditions on $L$ under which the $Ext$-groups $Ext\left(k,k\right)$ for the trivial $L$-module $k$ are the same when computed in the category of all $U\left(L\right)$-modules or in the category of $I$-torsion $U\left(L\right)$-modules. An application to cohomology of equivariant sheaves is given.

We show that the non-Archimedean skeleton of the Prym variety associated to an unramified double cover of an algebraic curve is naturally isomorphic (as a principally polarized tropical abelian variety) to the tropical Prym variety of the associated tropical double cover. This confirms a conjecture by Jensen and the first author. We prove a new upper bound on the dimension of the Prym–Brill–Noether locus for a generic unramified double cover in a dense open subset in the moduli space of unramified double covers of curves with fixed even gonality on the base. Our methods also give a new proof of the classical Prym–Brill–Noether theorem for generic unramified double covers that is originally due to Welters and Bertram.