We study short sums of algebraic trace functions via the $q$-analogue of the van der Corput method, and develop theory of arithmetic exponent pairs that coincide with the classical case when the moduli have sufficiently good factorizations. As an application, we prove a quadratic analogue of the Brun–Titchmarsh theorem on average, bounding the number of primes $p⩽X$ such that $p^2+1\equiv 0\text{mod}q$. The other two applications include a larger level of distribution of divisor functions in arithmetic progressions and a sub-Weyl subconvex bound of Dirichlet $L$-functions studied previously by Irving.

Using the $p$-adic local Langlands correspondence for ${\text{GL}}_2\left({\mathbb{Q}}_p\right)$, we prove that the support of the patched modules $M_{\infty }\left(\sigma \right)\left[1∕p\right]$ constructed by Caraiani et al. (Compos. Math. 154:3 (2018), 503–548) meets every irreducible component of the potentially semistable deformation ring $R_{\overline{r}}^{\square }(\sigma )[1∕p]$. This gives a new proof of the Breuil–Mézard conjecture for 2-dimensional representations of the absolute Galois group of ${\mathbb{Q}}_p$ when , which is new for $p=3$ and $\overline{r}$ a twist of an extension of the trivial character by the mod $p$ cyclotomic character. As a consequence, a local restriction in the proof of the Fontaine–Mazur conjecture by Kisin (J. Amer. Math. Soc. 22:3 (2009), 641–690) is removed.

We formulate and prove an analog of Poonen’s finite-field Bertini theorem with Taylor conditions that holds in the Grothendieck ring of varieties. This gives a broad generalization of the work of Vakil and Wood, who treated the case of smooth hypersurface sections, and is made possible by the use of motivic Euler products to write down candidate motivic probabilities. As applications, we give motivic analogs of many results in arithmetic statistics that have been proven using Poonen’s sieve, including work of Bucur and Kedlaya on complete intersections and Erman and Wood on semiample Bertini theorems.

Let $F$ be a finitely generated regular field extension of transcendence degree $\ge 2$ over a perfect field $k$. We show that the multiplicative group $F^{\times }∕k^{\times }$ endowed with the equivalence relation induced by algebraic dependence on $F$ over $k$ determines the isomorphism class of $F$ in a functorial way. As a special case of this result, we obtain that the isomorphism class of the graded Milnor K-ring $K_{\ast }^M\left(F\right)$ determines the isomorphism class of $F$, when $k$ is algebraically closed or finite.

We establish the analytic Hasse principle for Diophantine systems consisting of one diagonal form of degree $k$ and one general form of degree $d$, where $d$ is smaller than $k$. By employing a hybrid method that combines ideas from the study of general forms with techniques adapted to the diagonal case, we are able to obtain bounds that grow exponentially in $d$ but only quadratically in $k$, reflecting the growth rates typically obtained for both problems separately. We also discuss some of the most interesting generalisations of our approach.

Deformations of ordinary varieties of K3 type can be described in terms of displays, by recent work of Langer and Zink. We extend this to the general (nonordinary) case using displays with $G$-structure for a reductive group $G$. This is based on a modified definition of the tensor category of displays and variants of these which is related to the Frobenius gauges of Fontaine and Jannsen.

We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $ℍ$. The angles of lattice points arising from the orbit of the modular group ${\text{PSL}}_2\left(\mathbb{Z}\right)$, and lying on hyperbolic circles, are shown to be equidistributed for generic radii. However, the angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing multiplicity. Surprisingly, the distribution of angles on hyperbolic circles turns out to be related to the angular distribution of ${\mathbb{Z}}^2$-lattice points (with certain parity conditions) lying on circles in ${ℝ}^2$, along a thin subsequence of radii. A notable difference is that measures in the hyperbolic setting can break symmetry; on very thin subsequences they are not invariant under rotation by $\pi ∕2$, unlike in the Euclidean setting where all measures have this invariance property.

We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight $2k$ associated to positive definite integral binary quadratic forms. For example, we show that if there are no nontrivial cusp forms of weight $2k$, then the $n$-th coefficients of these meromorphic modular forms are divisible by $n^{k-1}$ for every natural number $n$. Moreover, we prove that their coefficients are nonvanishing and have either constant or alternating signs. Finally, we obtain a relation between the Fourier coefficients of meromorphic modular forms, the coefficients of the $j$-function, and the partition function.