We investigate the arithmetic of algebraic curves on coarse moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. We prove a structure theorem for morphisms from the affine line into the moduli space. We show that the set of integral points on any nondegenerate algebraic curve on the moduli space can be effectively determined.

We describe two conjectures, one strictly stronger than the other, that give descriptions of the integral Bernstein center for ${GL}_n\left(F\right)$ (that is, the center of the category of smooth $W\left(k\right)\left[{GL}_n\left(F\right)\right]$-modules, for $F$ a $p$-adic field and $k$ an algebraically closed field of characteristic $\ell$ different from $p$) in terms of Galois theory. Moreover, we show that the weak version of the conjecture (for $m\le n$), together with the strong version of the conjecture for $m<n$, implies the strong conjecture for ${GL}_n$. In a companion paper (Invent. Math. 214:2 (2018), 999–1022) we show that the strong conjecture for $n-1$ implies the weak conjecture for $n$; thus the two papers together give an inductive proof of both conjectures. The upshot is a description of the Bernstein center in purely Galois theoretic terms; previous work of the author shows that this description implies the conjectural “local Langlands correspondence in families” of (Ann. Sci. Éc. Norm. Supér. $\left(4\right)$ 47:4 (2014), 655–722).

We realize the moduli spaces of cubic fourfolds with specified group actions as arithmetic quotients of complex hyperbolic balls or type $IV$ symmetric domains, and study their compactifications. We prove the geometric ($GIT$) compactifications are naturally isomorphic to the Hodge theoretic (Looijenga, in many cases Baily–Borel) compactifications. The key ingredients of the proof are the global Torelli theorem by Voisin, the characterization of the image of the period map given by Looijenga and Laza independently, and the functoriality of Looijenga compactifications proved in the Appendix.

We establish a short exact sequence about depth-graded motivic double zeta values of even weight relative to ${\mu }_2$. We find a basis for the depth-graded motivic double zeta values relative to ${\mu }_2$ of even weight and a basis for the depth-graded motivic triple zeta values relative to ${\mu }_2$ of odd weight. As an application of our main results, we prove Kaneko and Tasaka’s conjectures about the sum odd double zeta values and the classical double zeta values. We also prove an analogue of Kaneko and Tasaka’s conjecture in depth three. Finally, we formulate a conjecture which is related to sum odd multiple zeta values in higher depth.

Let $F$ be the maximal totally real subfield of $\mathbb{Q}\left({\zeta }_{32}\right)$, the cyclotomic field of $32$-nd roots of unity. Let $D$ be the quaternion algebra over $F$ ramified exactly at the unique prime above $2$ and 7 of the real places of $F$. Let $\mathcal{𝒪}$ be a maximal order in $D$, and $X_0^D\left(1\right)$ the Shimura curve attached to $\mathcal{𝒪}$. Let $C=X_0^D\left(1\right)∕⟨w_D⟩$, where $w_D$ is the unique Atkin–Lehner involution on $X_0^D\left(1\right)$. We show that the curve $C$ has several striking features. First, it is a hyperelliptic curve of genus $16$, whose hyperelliptic involution is exceptional. Second, there are $34$ Weierstrass points on $C$, and exactly half of these points are CM points; they are defined over the Hilbert class field of the unique CM extension $E∕F$ of class number $17$ contained in $\mathbb{Q}\left({\zeta }_{64}\right)$, the cyclotomic field of $64$-th roots of unity. Third, the normal closure of the field of $2$-torsion of the Jacobian of $C$ is the Harbater field $N$, the unique Galois number field $N∕\mathbb{Q}$ unramified outside $2$ and $\infty$, with Galois group $Gal\left(N∕\mathbb{Q}\right)\simeq F_{17}=\mathbb{Z}∕17\mathbb{Z}⋊{\left(\mathbb{Z}∕17\mathbb{Z}\right)}^{\times }$. In fact, the Jacobian $Jac\left(X_0^D\left(1\right)\right)$ has the remarkable property that each of its simple factors has a $2$-torsion field whose normal closure is the field $N$. Finally, and perhaps the most striking fact about $C$, is that it is also hyperelliptic over $\mathbb{Q}$.

For a new class of Shimura varieties of orthogonal type over a totally real number field, we construct special cycles and show the modularity of Kudla’s generating series in the cohomology group.

Let $k$ be a perfect field of characteristic $p>2$, and let $K$ be a finite totally ramified extension over $W\left(k\right)\left[\frac{1}{p}\right]$ of ramification degree $e$. Let $R_0$ be a relative base ring over $W\left(k\right)⟨t_1^{\pm 1},\dots ,t_m^{\pm 1}⟩$ satisfying some mild conditions, and let $R=R_0{\otimes }_{W\left(k\right)}{\mathcal{𝒪}}_K$. We show that if $e<p-1$, then every crystalline representation of ${\pi }_1^{\acute{e}t}\left(SpecR\left[\frac{1}{p}\right]\right)$ with Hodge–Tate weights in $\left[0,1\right]$ arises from a $p$-divisible group over $R$.

We consider two group actions on $m$-tuples of $n\times n$ matrices with entries in the field $K$. The first is simultaneous conjugation by ${GL}_n$ and the second is the left-right action of ${SL}_n\times {SL}_n$. Let $\overline{K}$ be the algebraic closure of the field $K$. Recently, a polynomial time algorithm was found to decide whether $0$ lies in the Zariski closure of the ${SL}_n\left(\overline{K}\right)\times {SL}_n\left(\overline{K}\right)$-orbit of a given $m$-tuple by Garg, Gurvits, Oliveira and Wigderson for the base field $K=\mathbb{Q}$. An algorithm that also works for finite fields of large enough cardinality was given by Ivanyos, Qiao and Subrahmanyam. A more general problem is the orbit closure separation problem that asks whether the orbit closures of two given $m$-tuples intersect. For the conjugation action of ${GL}_n\left(\overline{K}\right)$ a polynomial time algorithm for orbit closure separation was given by Forbes and Shpilka in characteristic $0$. Here, we give a polynomial time algorithm for the orbit closure separation problem for both the conjugation action of ${GL}_n\left(\overline{K}\right)$ and the left-right action of ${SL}_n\left(\overline{K}\right)\times {SL}_n\left(\overline{K}\right)$ in arbitrary characteristic. We also improve the known bounds for the degree of separating invariants in these cases.