The Gauss circle problem concerns the difference $P_2\left(n\right)$ between the area of a circle of radius $\sqrt{n}$ and the number of lattice points it contains. In this paper, we study the Dirichlet series with coefficients $P_2{\left(n\right)}^2$, and prove that this series has meromorphic continuation to $ℂ$. Using this series, we prove that the Laplace transform of $P_2{\left(n\right)}^2$ satisfies ${\int }_0^{\infty }{P}_2{\left(t\right)}^2{e}^{-t∕X}dt=C{X}^{3∕2}-X+O\left(X^{1∕2+𝜖}\right)$, which gives a power-savings improvement to a previous result of Ivić (1996).

Similarly, we study the meromorphic continuation of the Dirichlet series associated to the correlations $r_2\left(n+h\right)r_2\left(n\right)$, where $h$ is fixed and $r_2\left(n\right)$ denotes the number of representations of $n$ as a sum of two squares. We use this Dirichlet series to prove asymptotics for ${\sum }_{n\ge 1}{r}_2\left(n+h\right)r_2\left(n\right)e^{-n∕X}$, and to provide an additional evaluation of the leading coefficient in the asymptotic for ${\sum }_{n\le X}{r}_2\left(n+h\right)r_2\left(n\right)$.

In this article, we construct ${SL}_k$-friezes using Plücker coordinates, making use of the cluster structure on the homogeneous coordinate ring of the Grassmannian of $k$-spaces in $n$-space via the Plücker embedding. When this cluster algebra is of finite type, the ${SL}_k$-friezes are in bijection with the so-called mesh friezes of the corresponding Grassmannian cluster category. These are collections of positive integers on the AR-quiver of the category with relations inherited from the mesh relations on the category. In these finite type cases, many of the ${SL}_k$-friezes arise from specializing a cluster to 1. These are called unitary. We use Iyama–Yoshino reduction to analyze the nonunitary friezes. With this, we provide an explanation for all known friezes of this kind. An appendix by Cuntz and Plamondon proves that there are 868 friezes of type $E_6$.

Motivated by questions from the study of relative trace formulae, we construct a generalization of Grothendieck’s simultaneous resolution over the regular locus of certain symmetric pairs. We use this space to prove a relative version of results of Donagi and Gaitsgory about the automorphism sheaf of regular stabilizers. We also obtain partial results toward applications in Springer theory for symmetric spaces.

We study the cancellation property of projective modules of rank $2$ with a trivial determinant over Noetherian rings of dimension $\le 4$. If $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $k$ such that $6\in k^{\times }$, then we prove that stably free $R$-modules of rank $2$ are free if and only if a Hermitian $K$-theory group ${\tilde{V}}_{SL}\left(R\right)$ is trivial.

The purpose of this article is to prove a “Newton over Hodge” result for exponential sums on curves. Let $X$ be a smooth proper curve over a finite field ${\mathbb{𝔽}}_q$ of characteristic $p\ge 3$ and let $V\subset X$ be an affine curve. For a regular function $\bar{f}$ on $V$, we may form the $L$-function $L\left(\bar{f},V,s\right)$ associated to the exponential sums of $\bar{f}$. In this article, we prove a lower estimate on the Newton polygon of $L\left(\bar{f},V,s\right)$. The estimate depends on the local monodromy of $f$ around each point $x\in X-V$. This confirms a hope of Deligne that the irregular Hodge filtration forces bounds on $p$-adic valuations of Frobenius eigenvalues. As a corollary, we obtain a lower estimate on the Newton polygon of a curve with an action of $\mathbb{Z}∕p\mathbb{Z}$ in terms of local monodromy invariants.

We relate the existence of linear models on central simple algebras to local root numbers.

An algebraic framework for noncommutative bundles with (quantum) homogeneous fibres is proposed. The framework relies on the use of principal coalgebra extensions which play the role of principal bundles in noncommutative geometry which might be additionally equipped with a Hopf algebra symmetry. The proposed framework is supported by two examples of noncommutative $ℂP_q^1$-bundles: the quantum flag manifold viewed as a bundle with a generic Podleś sphere as a fibre, and the quantum twistor bundle viewed as a bundle over the quantum 4-sphere of Bonechi, Ciccoli and Tarlini.

We find new lower bounds on the torsion orders of very general Fano hypersurfaces over (uncountable) fields of arbitrary characteristic. Our results imply that unirational parametrizations of most Fano hypersurfaces need to have very large degree. Our results also hold in characteristic two, where they solve the rationality problem for hypersurfaces under a logarithmic degree bound, thereby extending a previous result of the author from characteristic different from two to arbitrary characteristic.

We construct a basis of the space $S_{14}\left({Sp}_{12}\left(\mathbb{Z}\right)\right)$ of Siegel cusp forms of degree $6$ and weight $14$ consisting of harmonic theta series. One of these functions has vanishing order $2$ at the boundary which implies that the Kodaira dimension of ${\mathcal{𝒜}}_6$ is nonnegative.

We generalize the gcd results of Corvaja and Zannier and of Levin on ${\mathbb{𝔾}}_m^n$ to more general settings. More specifically, we analyze the height of a closed subscheme of codimension at least $2$ inside an $n$-dimensional Cohen–Macaulay projective variety, and show that this height is small when evaluated at integral points with respect to a divisor $D$ when $D$ is a sum of $n+1$ effective divisors which are all numerically equivalent to some multiples of a fixed ample divisor. Our method is inspired by Silverman’s gcd estimate, but instead of his usage of Vojta’s conjecture, we use the recent result of Ru and Vojta.

We correct an error in the statement of Theorem 3.12 of the article in question.