We compute all the moments of the $p$-torsion in the first step of a filtration of the class group defined by Gerth (1987) for cyclic fields of degree $p$, unconditionally for $p=3$ and under GRH in general. We show that it satisfies a distribution which Gerth conjectured as an extension of the Cohen–Lenstra–Martinet conjectures. In the $p=3$ case this gives the distribution of the $3$-torsion of the class group modulo the Galois invariant part. We follow the strategy used by Fouvry and Klüners (2007) in their proof of the distribution of the $4$-torsion in quadratic fields.

Let $F$ be a field of characteristic zero admitting a biquadratic field extension. We give an example of a torus $G$ over $F$ whose classifying stack $BG$ is stably rational and such that $\left\{BG\right\}\ne {\left\{G\right\}}^{-1}$ in the Grothendieck ring of algebraic stacks over $F$. We also give an example of a finite étale group scheme $A$ over $F$ such that $BA$ is stably rational and $\left\{BA\right\}\ne 1$.

We propose a new heuristic approach to integral moments of $L$-functions over function fields, which we demonstrate in the case of Dirichlet characters ramified at one place (the function field analogue of the moments of the Riemann zeta function, where we think of the character $n^{it}$ as ramified at the infinite place). We represent the moment as a sum of traces of Frobenius on cohomology groups associated to irreducible representations. Conditional on a hypothesis on the vanishing of some of these cohomology groups, we calculate the moments of the $L$-function and they match the predictions of the Conry, Farmer, Keating, Rubinstein, and Snaith recipe (Proc. Lond. Math. Soc. $\left(3\right)$ 91 (2005), 33–104).

In this case, the decomposition into irreducible representations seems to separate the main term and error term, which are mixed together in the long sums obtained from the approximate functional equation, even when it is dyadically decomposed. This makes our heuristic statement relatively simple, once the geometric background is set up. We hope that this will clarify the situation in more difficult cases like the $L$-functions of quadratic Dirichlet characters to squarefree modulus. There is also some hope for a geometric proof of this cohomological hypothesis, which would resolve the moment problem for these $L$-functions in the large degree limit over function fields.

Let $X$ be a complete $\mathbb{Q}$-factorial toric variety. We explicitly describe the space $H^2\left(X,{\mathcal{𝒯}}_X\right)$ and the cup product map $H^1\left(X,{\mathcal{𝒯}}_X\right)\times H^1\left(X,{\mathcal{𝒯}}_X\right)\to H^2\left(X,{\mathcal{𝒯}}_X\right)$ in combinatorial terms. Using this, we give an example of a smooth projective toric threefold for which the cup product map does not vanish, showing that in general, smooth complete toric varieties may have obstructed deformations.

In this paper we prove the equidistribution of the restriction of the mass of automorphic newforms to a nonsplit torus in the depth aspect. This result is better than the current known results on the similar problem in the eigenvalue aspect. The method is relatively elementary and makes use of the known effective QUE result in the depth aspect.

We show how a natural constant introduced by Jiang and Pareschi for a polarized abelian variety encodes information about the syzygies of the section ring of the polarization. As a particular case this gives a quick and characteristic-free proof of Lazarsfeld’s conjecture on syzygies of abelian varieties, originally proved by Pareschi in characteristic zero.

We study the Hodge–Tate period domain associated to a quaternionic Shimura curve at a prime of bad reduction, and give an explicit description of its Ekedahl–Oort stratification.

We prove a moving lemma for the additive and ordinary higher Chow groups of relative $0$-cycles of regular semilocal $k$-schemes essentially of finite type over an infinite perfect field. From this, we show that the cycle classes can be represented by cycles that possess certain finiteness, surjectivity, and smoothness properties. It plays a key role in showing that the crystalline cohomology of smooth varieties can be expressed in terms of algebraic cycles.