In Aizenbud et al. (2010, Annals of Mathematics 172, 1407–1434), a multiplicity one theorem is proved for general linear groups, orthogonal groups, and unitary groups ( $GL, O,$ and U) over p-adic local fields. That is to say that when we have a pair of such groups $G_n{\subseteq } G_{n+1}$ , any restriction of an irreducible smooth representation of $G_{n+1}$ to $G_n$ is multiplicity-free. This property is already known for $GL$ over a local field of positive characteristic, and in this paper, we also give a proof for $O,U$ , and $SO$ over local fields of positive odd characteristic. These theorems are shown in Gan, Gross, and Prasad (2012, Sur les Conjectures de Gross et Prasad. I, Société Mathématique de France) to imply the uniqueness of Bessel models, and in Chen and Sun (2015, International Mathematics Research Notice 2015, 5849–5873) to imply the uniqueness of Rankin–Selberg models. We also prove simultaneously the uniqueness of Fourier–Jacobi models, following the outlines of the proof in Sun (2012, American Journal of Mathematics 134, 1655–1678).

By the Gelfand–Kazhdan criterion, the multiplicity one property for a pair $H\leq G$ follows from the statement that any distribution on G invariant to conjugations by H is also invariant to some anti-involution of G preserving H. This statement for $GL, O$ , and U over p-adic local fields is proved in Aizenbud et al. (2010, Annals of Mathematics 172, 1407–1434). An adaptation of the proof for $GL$ that works over of local fields of positive odd characteristic is given in Mezer (2020, Mathematische Zeitschrift 297, 1383–1396). In this paper, we give similar adaptations of the proofs of the theorems on orthogonal and unitary groups, as well as similar theorems for special orthogonal groups and for symplectic groups. Our methods are a synergy of the methods used over characteristic 0 (Aizenbud et al. [2010, Annals of Mathematics 172, 1407–1434]; Sun [2012, American Journal of Mathematics 134, 1655–1678]; and Waldspurger [2012, Astérisque 346, 313–318]) and of those used in Mezer (2020, Mathematische Zeitschrift 297, 1383–1396).

We study the free metabelian group $M(2,n)$ of prime power exponent n on two generators by means of invariants $M(2,n)'\to \mathbb {Z}_n$ that we construct from colorings of the squares in the integer grid $\mathbb {R} \times \mathbb {Z} \cup \mathbb {Z} \times \mathbb {R}$ . In particular, we improve bounds found by Newman for the order of $M(2,2^k)$ . We study identities in $M(2,n)$ , which give information about identities in the Burnside group $B(2,n)$ and the restricted Burnside group $R(2,n)$ .

Let G be a p-adic classical group. The representations in a given Bernstein component can be viewed as modules for the corresponding Hecke algebra—the endomorphism algebra of a pro-generator of the given component. Using Heiermann’s construction of these algebras, we describe the Bernstein components of the Gelfand–Graev representation for $G=\mathrm {SO}(2n+1)$, $\mathrm {Sp}(2n)$, and $\mathrm {O}(2n)$.

In Kiukas, Lahti, and Ylinen (2006, Journal of Mathematical Physics 47, 072104), the authors asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in Pietrzycki and Stochel (2021, Journal of Functional Analysis 280, 109001). Let T be a self-adjoint operator, and let F be a Borel semispectral measure on the real line with compact support. For which positive integers $p< q$ do the equalities $T^k =\int _{\mathbb {R}} x^k F(\mathrm {d\hspace {.1ex}} x)$, $k=p, q$, imply that F is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if $p$ is odd and $q$ is even, and negative otherwise. The case $(p,q)=(1,2)$ closely related to intrinsic noise operator was solved by several authors including Kruszyński and de Muynck, as well as Kiukas, Lahti, and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on $C^*$-algebras is also provided.

The notion of the truncated Euler characteristic for Iwasawa modules is a generalization of the the usual Euler characteristic to the case when the Selmer groups are not finite. Let p be an odd prime, $E_{1}$ and $E_{2}$ be elliptic curves over a number field F with semistable reduction at all primes $v|p$ such that the $\operatorname {Gal}(\overline {F}/F)$ -modules $E_{1}[p]$ and $E_{2}[p]$ are irreducible and isomorphic. We compare the Iwasawa invariants of certain imprimitive multisigned Selmer groups of $E_{1}$ and $E_{2}$ . Leveraging these results, congruence relations for the truncated Euler characteristics associated to these Selmer groups over certain $\mathbb {Z}_{p}^{m}$ -extensions of F are studied. Our results extend earlier congruence relations for elliptic curves over $\mathbb {Q}$ with good ordinary reduction at p.

We obtain a nontrivial upper bound for the multiplicative energy of any sufficiently large subset of a subvariety of a finite algebraic group. We also find some applications of our results to the growth of conjugates classes, estimates of exponential sums, and restriction phenomenon.