Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

On Representation Zeta Functions for Special Linear Groups

Abstract: We prove that the numbers of irreducible n-dimensional complex continuous representations of the special linear groups over p-adic integers grow slower than the square of n. We deduce that the abscissas of convergence of the representation zeta functions of the special linear groups over the ring of integers are bounded above by 2. In order to show these results we prove also that if G is a connected, simply connected, semi-simple algebraic group defined over the field of rational numbers, then the G-representation variety of the fundamental group of a compact Riemann surface of genus n has rational singularities if and only if the G-character variety has rational singularities.

The regular representations of $\mathrm{GL}_{N}$ over finite local principal ideal rings

Abstract: Let $\mathfrak{o}$ be the ring of integers in a non-Archimedean local field with finite residue field, $\mathfrak{p}$ its maximal ideal, and $r\geq2$ an integer. An irreducible representation of the finite group $G_{r}=\mathrm{GL}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ is called regular if its restriction to the principal congruence kernel $K^{r-1}=1+\mathfrak{p}^{r-1}\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ consists of representations whose stabilisers modulo $K^{1}$ are centralisers of regular elements in $\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p})$. The regular representations form the largest class of representations of $G_{r}$ which is currently amenable to explicit construction. Their study, motivated by constructions of supercuspidal representations, goes back to Shintani, but the general case remained open for a long time. In this paper we give an explicit construction of all the regular representations of $G_{r}$.

The endomorphism ring of projectives and the Bernstein centre

Abstract: Let $F$ be a local non-archimedean field and $\mathcal{O}_F$ its ring of integers. Let $\Omega$ be a Bernstein component of the category of smooth representations of $GL_n(F)$, let $(J, \lambda)$ be a Bushnell-Kutzko $\Omega$-type, and let $\mathfrak{Z}_{\Omega}$ be the centre of the Bernstein component $\Omega$. This paper contains two major results. Let $\sigma$ be a direct summand of $\mathrm{Ind}_J^{GL_n(\mathcal{O}_F)} \lambda$. We will begin by computing $\mathrm{c\text{--} Ind}_{GL_n(\mathcal{O}_F)}^{GL_n(F)} \sigma\otimes_{\mathfrak{Z}_{\Omega}}\kappa(\mathfrak{m})$, where $\kappa(\mathfrak{m})$ is the residue field at maximal ideal $\mathfrak{m}$ of $\mathfrak{Z}_{\Omega}$, and the maximal ideal $\mathfrak{m}$ belongs to a Zariski-dense set in $\mathrm{Spec}\: \mathfrak{Z}_{\Omega}$. This result allows us to deduce that the endomorphism ring $\mathrm{End}_{GL_n(F)}(\mathrm{c\text{--} Ind}_{GL_n(\mathcal{O}_F)}^{GL_n(F)} \sigma)$ is isomorphic to $\mathfrak{Z}_{\Omega}$, when $\sigma$ appears with multiplicity one in $\mathrm{Ind}_J^{GL_n(\mathcal{O}_F)} \lambda$.

p-rigidity and Iwasawa μ-invariants

Abstract: Let F be a totally real field with ring of integers O and p be an odd prime unramified in F. Let p be a prime above p. We prove that a mod p Hilbert modular form associated to F is determined by its restriction to the partial Serre–Tate deformation space G m ⊗ Op (p-rigidity). Let K∕F be an imaginary quadratic CM extension such that each prime of F above p splits in K and λ a Hecke character of K. Partly based on p-rigidity, we prove that the μ-invariant of the anticyclotomic Katz p-adic L-function of λ equals the μ-invariant of the full anticyclotomic Katz p-adic L-function of λ. An analogue holds for a class of Rankin–Selberg p-adic L-functions. When λ is self-dual with the root number − 1, we prove that the μ-invariant of the cyclotomic derivatives of the Katz p-adic L-function of λ equals the μ-invariant of the cyclotomic derivatives of the Katz p-adic L-function of λ. Based on previous works of the authors and Hsieh, we consequently obtain a formula for the μ-invariant of these p-adic L-functions and derivatives. We also prove a p-version of a conjecture of Gillard, namely the vanishing of the μ-invariant of the Katz p-adic L-function of λ.