For any abelian group $A$, we prove an asymptotic formula for the number of $A$-extensions $K/\mathbb {Q}$ of bounded discriminant such that the associated norm one torus $R_{K/\mathbb {Q}}^1 \mathbb {G}_m$ satisfies weak approximation. We are also able to produce new results on the Hasse norm principle and to provide new explicit values for the leading constant in some instances of Malle's conjecture.

Let K be a number field, let A be a finite-dimensional K-algebra, let $\operatorname {\mathrm {J}}(A)$ denote the Jacobson radical of A and let $\Lambda $ be an $\mathcal {O}_{K}$ -order in A. Suppose that each simple component of the semisimple K-algebra $A/{\operatorname {\mathrm {J}}(A)}$ is isomorphic to a matrix ring over a field. Under this hypothesis on A, we give an algorithm that, given two $\Lambda $ -lattices X and Y, determines whether X and Y are isomorphic and, if so, computes an explicit isomorphism $X \rightarrow Y$ . This algorithm reduces the problem to standard problems in computational algebra and algorithmic algebraic number theory in polynomial time. As an application, we give an algorithm for the following long-standing problem: Given a number field K, a positive integer n and two matrices $A,B \in \mathrm {Mat}_{n}(\mathcal {O}_{K})$ , determine whether A and B are similar over $\mathcal {O}_{K}$ , and if so, return a matrix $C \in \mathrm {GL}_{n}(\mathcal {O}_{K})$ such that $B= CAC^{-1}$ . We give explicit examples that show that the implementation of the latter algorithm for $\mathcal {O}_{K}=\mathbb {Z}$ vastly outperforms implementations of all previous algorithms, as predicted by our complexity analysis.

For any $n>1$ we determine the uniform and nonuniform lattices of the smallest covolume in the Lie group $\operatorname {\mathrm {Sp}}(n,1)$ . We explicitly describe them in terms of the ring of Hurwitz integers in the nonuniform case with n even, respectively, of the icosian ring in the uniform case for all $n>1$ .

Let K/k be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for K/k and the defect of weak approximation for the norm one torus \[R_{K/k}^1{\mathbb{G}_m}\] . We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure of K/k has symmetric or alternating Galois group.

Given a commutative unital ring R, we show that the finiteness length of a group G is bounded above by the finiteness length of the Borel subgroup of rank one $\textbf {B}_2^{\circ }(R)=\left ( \begin {smallmatrix} * & * \\ 0 & * \end {smallmatrix}\right )\leq \operatorname {\textrm {SL}}_2(R)$ whenever G admits certain R-representations with metabelian image. Combined with results due to Bestvina–Eskin–Wortman and Gandini, this gives a new proof of (a generalization of) Bux’s equality on the finiteness length of S-arithmetic Borel groups. We also give an alternative proof of an unpublished theorem due to Strebel, characterizing finite presentability of Abels’ groups $\textbf {A}_n(R) \leq \operatorname {\textrm {GL}}_n(R)$ in terms of n and $\textbf {B}_2^{\circ }(R)$ . This generalizes earlier results due to Remeslennikov, Holz, Lyul’ko, Cornulier–Tessera, and points out to a conjecture about the finiteness length of such groups.

For an integer $n\geq 8$ divisible by $4$ , let $R_n={\mathbb Z}[\zeta _n,1/2]$ and let $\operatorname {\mathrm {U_{2}}}(R_n)$ be the group of $2\times 2$ unitary matrices with entries in $R_n$ . Set $\operatorname {\mathrm {U_2^\zeta }}(R_n)=\{\gamma \in \operatorname {\mathrm {U_{2}}}(R_n)\mid \det \gamma \in \langle \zeta _n\rangle \}$ . Let $\mathcal {G}_n\subseteq \operatorname {\mathrm {U_2^\zeta }}(R_n)$ be the Clifford-cyclotomic group generated by a Hadamard matrix $H=\frac {1}{2}[\begin {smallmatrix} 1+i & 1+i\\1+i &-1-i\end {smallmatrix}]$ and the gate $T_n=[\begin {smallmatrix}1 & 0\\0 & \zeta _n\end {smallmatrix}]$ . We prove that $\mathcal {G}_n=\operatorname {\mathrm {U_2^\zeta }}(R_n)$ if and only if $n=8, 12, 16, 24$ and that $[\operatorname {\mathrm {U_2^\zeta }}(R_n):\mathcal {G}_n]=\infty $ if $\operatorname {\mathrm {U_2^\zeta }}(R_n)\neq \mathcal {G}_n$ . We compute the Euler–Poincaré characteristic of the groups $\operatorname {\mathrm {SU_{2}}}(R_n)$ , $\operatorname {\mathrm {PSU_{2}}}(R_n)$ , $\operatorname {\mathrm {PU_{2}}}(R_n)$ , $\operatorname {\mathrm {PU_2^\zeta }}(R_n)$ , and $\operatorname {\mathrm {SO_{3}}}(R_n^+)$ .

We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent orbit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger orbits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analogous results for Whittaker–Fourier coefficients of automorphic representations. For $\text{GL}_{n}(\mathbb{F})$ this implies that a smooth admissible representation $\unicode[STIX]{x1D70B}$ has a generalized Whittaker model ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ corresponding to a nilpotent coadjoint orbit ${\mathcal{O}}$ if and only if ${\mathcal{O}}$ lies in the (closure of) the wave-front set $\operatorname{WF}(\unicode[STIX]{x1D70B})$. Previously this was only known to hold for $\mathbb{F}$ non-archimedean and ${\mathcal{O}}$ maximal in $\operatorname{WF}(\unicode[STIX]{x1D70B})$, see Moeglin and Waldspurger [Modeles de Whittaker degeneres pour des groupes p-adiques, Math. Z. 196 (1987), 427–452]. We also express ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ as an iteration of a version of the Bernstein–Zelevinsky derivatives [Bernstein and Zelevinsky, Induced representations of reductive p-adic groups. I., Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 441–472; Aizenbud et al., Derivatives for representations of$\text{GL}(n,\mathbb{R})$and$\text{GL}(n,\mathbb{C})$, Israel J. Math. 206 (2015), 1–38]. This enables us to extend to $\text{GL}_{n}(\mathbb{R})$ and $\text{GL}_{n}(\mathbb{C})$ several further results by Moeglin and Waldspurger on the dimension of ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ and on the exactness of the generalized Whittaker functor.

Let T be an algebraic torus over ℚ such that T(ℝ) is compact. Assuming the generalized Riemann hypothesis, we give a lower bound for the size of the class group of T modulo its n-torsion in terms of a small power of the discriminant of the splitting field of T. As a corollary, we obtain an upper bound on the n-torsion in that class group. This generalizes known results on the structure of class groups of complex multiplication fields.

An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

Dans ce papier nous étudions une correspondance de Jacquet–Langlands locale pour toutes les représentations lisses irréductibles. La correspondance est caractérisée par le fait qu’elle respecte la correspondance de Jacquet–Langlands classique et commute avec le foncteur d’induction parabolique. Elle est compatible dans un sens à préciser au foncteur de Jacquet et à l’involution d’Aubert–Schneider–Stuhler. Nous utilisons cette correspondance pour montrer qu’une certaine classe de représentations d’une forme intérieure de $\mathrm{GL}_n$ sur un corps $p$-adique sont unitarisables. C’est le premier pas dans la preuve de la conjecture U1 de Tadić.

We study a local Jacquet–Langlands correspondence for all smooth irreducible representations. This correspondence is characterized by the fact that it respects the classical Jacquet–Langlands correspondence and it commutes with the parabolic induction functor. It has good behavior with respect to the Jacquet’s functor and the involution of Aubert–Schneider–Stuhler. Using this correspondence, we prove some particular cases of the global Jacquet–Langlands correspondence and we deduce that a certain class of representations of an inner form of $\mathrm{GL}_n$ over a $p$-adic field are unitarizable. This is the first step towards the proof of Conjecture U1 of Tadić.

Let $K$ be a function field of genus $g$ with a finite constant field ${\mathbb{F}}_q$. Choose a place $\infty$ of $K$ of degree $\delta$ and let ${\mathbb{C}}$ be the arithmetic Dedekind domain consisting of all elements of $K$ that are integral outside $\infty$. An explicit formula is given (in terms of $q$, $g$ and $\delta$) for the minimum index of a non-congruence subgroup in SL$_2({\mathcal{C}})$. It turns out that this index is always equal to the minimum index of an arbitrary proper subgroup in SL$_2({\mathcal{C}})$. The minimum index of a normal non-congruence subgroup is also determined.