We show that, for any integer $N\ge 1$, there are only finitely many cuspidal algebraic automorphic representations of ${\mathrm{GL}}_n$ over $\mathbb{Q}$, with $n$ varying, whose conductor is $N$ and whose weights are in the interval $\{0,\dots ,23\}$. More generally, we define an explicit sequence $\left(\mathrm{r}\right(w){)}_{w\ge 0}$ such that, for any number field $E$ whose root discriminant is less than $\mathrm{r}\left(w\right)$ and any ideal $\mathcal{N}$ in the ring of integers of $E$, there are only finitely many cuspidal algebraic automorphic representations of ${\mathrm{GL}}_n$ over $E$, with $n$ varying, whose conductor is $\mathcal{N}$ and whose weights are in the interval $\{0,\dots ,w\}$. We also show that, assuming a version of the generalized Riemann hypothesis, we may replace $\mathrm{r}\left(w\right)$ with $8\pi e^{-\psi (1+w)}$ in this statement. The proofs here are based on some new positivity properties of certain real quadratic forms which occur in the study of the Weil explicit formula for Rankin–Selberg $\mathrm{L}$-functions. Both the effectiveness and the optimality of the methods are discussed.

We prove Mahler’s conjecture concerning the volume product of centrally symmetric, convex bodies in ${\mathbb{R}}^n$ in the case where $n=3$. More precisely, we show that, for every $3$-dimensional, centrally symmetric, convex body $K\subset {\mathbb{R}}^3$, the volume product $\left|K\right|\left|K^{\circ }\right|$ is greater than or equal to $32/3$ with equality if and only if $K$ or $K^{\circ }$ is a parallelepiped.

Fake projective planes are smooth, complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the $2$-dimensional ball by explicitly written arithmetic subgroups. In the following, we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order $7$ automorphism.

Let $\{{\rho }_{\ell }{\}}_{\ell }$ be the system of $\ell$-adic representations arising from the $i$th $\ell$-adic cohomology of a proper smooth variety $X$ defined over a number field $K$. Let ${\Gamma }_{\ell }$ and ${\mathbf{G}}_{\ell }$ be, respectively, the image and the algebraic monodromy group of ${\rho }_{\ell }$. We prove that the reductive quotient of ${\mathbf{G}}_{\ell }^{\circ }$ is unramified over every degree $12$ totally ramified extension of ${\mathbb{Q}}_{\ell }$ for all sufficiently large $\ell$. We give a necessary and sufficient condition $(*)$ on $\{{\rho }_{\ell }{\}}_{\ell }$ such that, for all sufficiently large $\ell$, the subgroup ${\Gamma }_{\ell }$ is in some sense maximal compact in ${\mathbf{G}}_{\ell }\left({\mathbb{Q}}_{\ell }\right)$. This is used to deduce Galois maximality results for $\ell$-adic representations arising from abelian varieties (for all $i$) and hyper-Kähler varieties ($i=2$) defined over finitely generated fields over $\mathbb{Q}$.