The semistable ChowHa of a quiver with stability is defined as an analog of the cohomological Hall algebra of Kontsevich and Soibelman via convolution in equivariant Chow groups of semistable loci in representation varieties of quivers. We prove several structural results on the semistable ChowHa, namely isomorphism of the cycle map, a tensor product decomposition, and a tautological presentation. For symmetric quivers, this leads to an identification of their quantized Donaldson–Thomas invariants with the Chow–Betti numbers of moduli spaces.

An abelian surface $A_{∕\mathbb{Q}}$ of prime conductor $N$ is favorable if its 2-division field $F$ is an ${\mathcal{S}}_5$-extension over $\mathbb{Q}$ with ramification index 5 over ${\mathbb{Q}}_2$. Let $A$ be favorable and let $B$ be a semistable abelian variety of dimension $2d$ and conductor $N^d$ with $B\left[2\right]$ filtered by copies of $A\left[2\right]$. We give a sufficient class field theoretic criterion on $F$ to guarantee that $B$ is isogenous to $A^d$.

As expected from our paramodular conjecture, we conclude that there is one isogeny class of abelian surfaces for each conductor in $\left\{277,349,461,797,971\right\}$. The general applicability of our criterion is discussed in the data section.

Kollár (2008) introduced the surfaces

$$\left(x_1^{a_1}{x}_2+x_2^{a_2}{x}_3+x_3^{a_3}{x}_4+x_4^{a_4}{x}_1=0\right)\subset \mathbb{P}\left(w_1,w_2,w_3,w_4\right)$$

where $w_i=W_i∕w^{\ast }$, $W_i=a_{i+1}{a}_{i+2}{a}_{i+3}-a_{i+2}{a}_{i+3}+a_{i+3}-1$, and $w^{\ast }=gcd\left(W_1,\dots ,W_4\right)$. The aim was to give many interesting examples of $\mathbb{Q}$-homology projective planes. They occur when $w^{\ast }=1$. For that case, we prove that Kollár surfaces are Hwang–Keum (2012) surfaces. For $w^{\ast }>1$, we construct a geometrically explicit birational map between Kollár surfaces and cyclic covers $z^{w^{\ast }}=l_1^{a_2{a}_3{a}_4}{l}_2^{-a_3{a}_4}{l}_3^{a_4}{l}_4^{-1}$, where $\left\{l_1,l_2,l_3,l_4\right\}$ are four general lines in ${\mathbb{P}}^2$. In addition, by using various properties on classical Dedekind sums, we prove that:

1. For any $w^{\ast }>1$, we have $p_g=0$ if and only if the Kollár surface is rational. This happens when $a_{i+1}\equiv 1$ or $a_i{a}_{i+1}\equiv -1\left(mod{w}^{\ast }\right)$ for some $i$.

2. For any $w^{\ast }>1$, we have $p_g=1$ if and only if the Kollár surface is birational to a K3 surface. We classify this situation.

3. For $w^{\ast }\gg 0$, we have that the smooth minimal model $S$ of a generic Kollár surface is of general type with $K_S^2∕e\left(S\right)\to 1$.

Soit $G$ un groupe $SO\left(2n+1\right)$ défini sur un corps $p$-adique. Nous calculons le front d’onde des représentations irréductibles anti-tempérées de $G\left(F\right)$ qui sont de réduction unipotente. Le front d’onde d’une telle représentation est l’orbite orthogonale duale à l’orbite symplectique qui intervient dans le paramètre d’Arthur de cette représentation.

Let $G$ be a group $SO\left(2n+1\right)$ defined over a $p$-adic field. We compute the wave front set of the antitempered irreducible representations of $G\left(F\right)$ which are of unipotent reduction. The wave front set of such representations is the orthogonal orbit dual to the symplectic orbit appearing in the Arthur’s parametrization of the representation.

We study the formal properties of correspondences of curves without a core, focusing on the case of étale correspondences. The motivating examples come from Hecke correspondences of Shimura curves. Given a correspondence without a core, we construct an infinite graph ${\mathcal{G}}_{gen}$ together with a large group of “algebraic” automorphisms $A$. The graph ${\mathcal{G}}_{gen}$ measures the “generic dynamics” of the correspondence. We construct specialization maps ${\mathcal{G}}_{gen}\to {\mathcal{G}}_{phys}$ to the “physical dynamics” of the correspondence. Motivated by the abstract structure of the supersingular locus, we also prove results on the number of bounded étale orbits, in particular generalizing a recent theorem of Hallouin and Perret. We use a variety of techniques: Galois theory, the theory of groups acting on infinite graphs, and finite group schemes.

We prove that any complex or real analytic set or function germ is topologically equivalent to a germ defined by polynomial equations whose coefficients are algebraic numbers.

Working over an infinite field of positive characteristic, an upper bound is given for the nilpotency index of a finitely generated nil algebra of bounded nil index $n$ in terms of the maximal degree in a minimal homogenous generating system of the ring of simultaneous conjugation invariants of tuples of $n$-by-$n$ matrices. This is deduced from a result of Zubkov. As a consequence, a recent degree bound due to Derksen and Makam for the generators of the ring of matrix invariants yields an upper bound for the nilpotency index of a finitely generated nil algebra that is polynomial in the number of generators and the nil index. Furthermore, a characteristic free treatment is given to Kuzmin’s lower bound for the nilpotency index.

We consider the variance of sums of arithmetic functions over random short intervals in the function field setting. Based on the analogy between factorizations of random elements of ${\mathbb{F}}_q\left[T\right]$ into primes and the factorizations of random permutations into cycles, we give a simple but general formula for these variances in the large $q$ limit for arithmetic functions that depend only upon factorization structure. From this we derive new estimates, quickly recover some that are already known, and make new conjectures in the setting of the integers.

In particular we make the combinatorial observation that any function of this sort can be explicitly decomposed into a sum of functions $u$ and $v$, depending on the size of the short interval, with $u$ making a negligible contribution to the variance, and $v$ asymptotically contributing diagonal terms only.

This variance evaluation is closely related to the appearance of random matrix statistics in the zeros of families of $L$-functions and sheds light on the arithmetic meaning of this phenomenon.

Consider a field $k$ of characteristic $p>0$, the $r$-th Frobenius kernel ${\mathbb{G}}_{\left(r\right)}$ of a smooth algebraic group $\mathbb{G}$, the Drinfeld double $D{\mathbb{G}}_{\left(r\right)}$ of ${\mathbb{G}}_{\left(r\right)}$, and a finite dimensional $D{\mathbb{G}}_{\left(r\right)}$-module $M$. We prove that the cohomology algebra $H^{\ast }\left(D{\mathbb{G}}_{\left(r\right)},k\right)$ is finitely generated and that $H^{\ast }\left(D{\mathbb{G}}_{\left(r\right)},M\right)$ is a finitely generated module over this cohomology algebra. We exhibit a finite map of algebras ${\theta }_r:H^{\ast }\left({\mathbb{G}}_{\left(r\right)},k\right)\otimes S\left(\mathtt{g}\right)\to H^{\ast }\left(D{\mathbb{G}}_{\left(r\right)},k\right)$, which offers an approach to support varieties for $D{\mathbb{G}}_{\left(r\right)}$-modules. For many examples of interest, ${\theta }_r$ is injective and induces an isomorphism of associated reduced schemes. For $M$ an irreducible $D{\mathbb{G}}_{\left(r\right)}$-module, ${\theta }_r$ enables us to identify the support variety of $M$ in terms of the support variety of $M$ viewed as a ${\mathbb{G}}_{\left(r\right)}$-module.