Given a locally finite leafless tree $T$, various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$. We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$. This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $2$, the Tits index of simple algebraic subgroups of $\operatorname{Aut}(T)$ is not always preserved under Chabauty limits.

Let $T$ be a locally finite tree without vertices of degree $1$. We show that among the closed subgroups of $\text{Aut}(T)$ acting with a bounded number of orbits, the Chabauty-closure of the set of topologically simple groups is the set of groups without proper open subgroup of finite index. Moreover, if all vertices of $T$ have degree ${\geqslant}3$, then the set of isomorphism classes of topologically simple closed subgroups of $\text{Aut}(T)$ acting doubly transitively on $\unicode[STIX]{x2202}T$ carries a natural compact Hausdorff topology inherited from Chabauty. Some of our considerations are valid in the context of automorphism groups of locally finite connected graphs. Applications to Weyl-transitive automorphism groups of buildings are also presented.

Let $M$ be a $\text{II}_{1}$ factor and let ${\mathcal{F}}(M)$ denote the fundamental group of $M$. In this article, we study the following property of $M$: for any $\text{II}_{1}$ factor $B$, we have ${\mathcal{F}}(M\,\overline{\otimes }\,B)={\mathcal{F}}(M){\mathcal{F}}(B)$. We prove that for any subgroup $G\leqslant \mathbb{R}_{+}^{\ast }$ which is realized as a fundamental group of a $\text{II}_{1}$ factor, there exists a $\text{II}_{1}$ factor $M$ which satisfies this property and whose fundamental group is $G$. Using this, we deduce that if $G,H\leqslant \mathbb{R}_{+}^{\ast }$ are realized as fundamental groups of $\text{II}_{1}$ factors, then so are groups $G\cdot H$ and $G\cap H$.

We consider a meromorphic family of endomorphisms of degree at least 2 of a complex projective space that is parameterized by the unit disk. We prove that the measure of maximal entropy of these endomorphisms converges to the equilibrium measure of the associated non-Archimedean dynamical system when the system degenerates. The convergence holds in the hybrid space constructed by Berkovich and further studied by Boucksom and Jonsson. We also infer from our analysis an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of endomorphisms.

We investigate interesting connections between Mizohata type vector fields and microlocal regularity of nonlinear first-order PDEs, establishing results in Denjoy–Carleman classes and real analyticity results in the linear case.

Let $(G,\unicode[STIX]{x1D707})$ be a pair of a reductive group $G$ over the $p$-adic integers and a minuscule cocharacter $\unicode[STIX]{x1D707}$ of $G$ defined over an unramified extension. We introduce and study ‘$(G,\unicode[STIX]{x1D707})$-displays’ which generalize Zink’s Witt vector displays. We use these to define certain Rapoport–Zink formal schemes purely group theoretically, i.e. without $p$-divisible groups.

In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as ‘Birch sums’. Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the distribution of large values of this maximum, that hold in a wide uniform range. This improves a recent result of Kowalski and Sawin. The proofs use a blend of probabilistic methods, harmonic analysis techniques, and deep tools from algebraic geometry. The results can also be generalized to other types of $\ell$-adic trace functions. In particular, the lower bound of our result also holds for partial sums of Kloosterman sums. As an application, we show that there exist $x\in [1,p]$ and $a\in \mathbb{F}_{p}^{\times }$ such that $|\sum _{n\leqslant x}\exp (2\unicode[STIX]{x1D70B}i(n^{3}+an)/p)|\geqslant (2/\unicode[STIX]{x1D70B}+o(1))\sqrt{p}\log \log p$. The uniformity of our results suggests that this bound is optimal, up to the value of the constant.

We construct the $p$-adic standard $L$-functions for ordinary families of Hecke eigensystems of the symplectic group $\operatorname{Sp}(2n)_{/\mathbb{Q}}$ using the doubling method. We explain a clear and simple strategy of choosing the local sections for the Siegel Eisenstein series on the doubling group $\operatorname{Sp}(4n)_{/\mathbb{Q}}$, which guarantees the nonvanishing of local zeta integrals and allows us to $p$-adically interpolate the restrictions of the Siegel Eisenstein series to $\operatorname{Sp}(2n)_{/\mathbb{Q}}\times \operatorname{Sp}(2n)_{/\mathbb{Q}}$.

We finalize the analysis of the trace formula initiated in S. A. Altuğ [Beyond endoscopy via the trace formula-I: Poisson summation and isolation of special representations, Compos. Math.151(10) (2015), 1791–1820] and developed in S. A. Altuğ [Beyond endoscopy via the trace formula-II: asymptotic expansions of Fourier transforms and bounds toward the Ramanujan conjecture. Submitted, preprint, 2015, Available at: arXiv:1506.08911.pdf], and calculate the asymptotic expansion of the beyond endoscopic averages for the standard $L$-functions attached to weight $k\geqslant 3$ cusp forms on $\mathit{GL}(2)$ (cf. Theorem 1.1). This, in particular, constitutes the first example of beyond endoscopy executed via the Arthur–Selberg trace formula, as originally proposed in R. P. Langlands [Beyond endoscopy, in Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 611–698 (The Johns Hopkins University Press, Baltimore, MD, 2004), chapter 22]. As an application we also give a new proof of the analytic continuation of the $L$-function attached to Ramanujan’s $\unicode[STIX]{x1D6E5}$-function.

This paper contains two results on Hodge loci in $\mathsf{M}_{g}$. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibers and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fiber is contained in a proper Hodge locus. The second result deals with divisors in $\mathsf{M}_{g}$. It is proved that the image under the period map of a divisor in $\mathsf{M}_{g}$ is not contained in a proper totally geodesic subvariety of $\mathsf{A}_{g}$. It follows that a Hodge locus in $\mathsf{M}_{g}$ has codimension at least 2.