Mod pq Galois representations and Serre's conjecture

We consider linear representations of the Galois groups of number fields in two different characteristics and examine conditions under which they arise simultaneously from a motive.     

Semigroups and weights for group representations

Let be a finite group. Consider a pair of linear characters of subgroups of with and agreeing on . Naturally associated with is a finite monoid . Semigroup representation theory then yields a representation of . If is irreducible, we say that is a weight for . When the underlying field is the field of complex numbers, we obtain a formula for the character of in terms of and . We go on to construct weights for some familiar group representations.

     

Integral representations of functions and embeddings of sobolev spaces on cuspidal domains

Some classes of cuspidal domainsG ⊂ ℝ ⁿ are introduced, and embeddings of the formW _p ^(l) (G)↪L_q(G),l ∈ ℕ, for sobolev spaces are established. To this end, estimates of some integral operators are needed. These operators cannot be estimated via Riesz potentials or their anisotropic analogs. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 201–219, February, 1997. Translated by V. E. Nazaikinskii     

Sums of fractions modulo p

Let \({\mathbb{F}_p}\) be the field of residue classes modulo a large prime p. The present paper is devoted to the problem of representability of elements of \({\mathbb{F}_p}\) as sums of fractions of the form x/y with x, y from short intervals of \({\mathbb{F}_p}\).     

On the semisimplicity conjecture and Galois representations

The semisimplicity conjecture says that for any smooth projective scheme over a finite field , the Frobenius correspondence acts semisimply on , where is an algebraic closure of . Based on the works of Deligne and Laumon, we reduce this conjecture to a problem about the Galois representations of function fields. This reduction was also achieved by Laumon a few years ago (unpublished).

     

Weights in Serre's conjecture for via the Bernstein–Gelfand–Gelfand complex

Let be an n-dimensional mod p Galois representation. If ρ is modular for a weight in a certain class, called p-minute, then we restrict the Fontaine–Laffaille numbers of ρ; in other words, we specify the possibilities for the restriction of ρ to inertia at p. Our result agrees with the Serre-type conjectures for GL_n formulated by Ash, Doud, Pollack, Sinnott, and Herzig; to our knowledge, this is the first unconditional evidence for these conjectures for arbitrary n.     

On cuspidal representations of general linear groups over discrete valuation rings

We define a new notion of cuspidality for representations of GL _n over a finite quotient o _k of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups G _λ of torsion o-modules. When n is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of GL _n (F). We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations of GL _n (o _k ) for k ≥ 2 for all n is equivalent to the construction of the representations of all the groups G _λ . A functional equation for zeta functions for representations of GL _n (o _k ) is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for GL₄(o₂) are constructed. Not all these representations are strongly cuspidal.     

A proof of Freud's conjecture for exponential weights

LetW (x) be a function nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ²(x) are finite. Let {p _n (W ²;x)} ₀ denote the sequence of orthonormal polynomials with respect to the weightW ²(x), and let {A _n } ₁ and {B _n } ₁ denote the coefficients in the recurrence relation

Proof of a conjecture on a congruence modulo 243 for overpartitions

Periodica Mathematica Hungarica - Let $$\bar{p}(n)$$ denote the number of overpartitions of n. Recently, numerous congruences modulo powers of 2, 3 and 5 were established regarding $$\bar{p}(n)$$ ....     

Formal degree and existence of stable arithmetic lattices of cuspidal representations ofp-adic reductive groups

Research at MSRI supported in part by NSF Grant DMS-812079-05     

Pseudo-K-theories modulo p and higher Milnor operations

Translated from Matematicheskie Zametki, Vol. 43, No. 1, pp. 12–24, January, 1988.     

Average of Dirichlet coefficients of cuspidal representations related to GL(2)

The Ramanujan Journal - Let $$\pi $$ be a cuspidal representation on $${{\,\mathrm{GL}\,}}(2,\mathbb {A}_{\mathbb {Q}}).$$ We give nontrivial lower and upper bounds for average of absolute values...     

Freud's conjecture and approximation of reciprocals of weights by polynomials

LetW(x) be a function that is nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ² (x) are finite. Let {p _n (W ²;x)} ₀ ^∞ denote the sequence of orthonormal polynomials with respect to the weightW ², and let {α _n } ₁ ^∞ and {β _n } ₁ ^∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = \alpha _{n + 1} p_{n + 1} (W^2 ,x) + \beta _n p_n (W^2 ,x) + \alpha _n p_{n - 1} (W^2 ,x).$$ We obtain a sufficient condition, involving mean approximation ofW ^?1 by reciprocals of polynomials, for $$\mathop {\lim }\limits_{n \to \infty } {{\alpha _n } \mathord{\left/ {\vphantom {{\alpha _n } {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{\beta _n } \mathord{\left/ {\vphantom {{\beta _n } {c_{n + 1} }}} \right. \kern-\nulldelimiterspace} {c_{n + 1} }} = 0,$$ wherec _{n 1} ^∞ is a certain increasing sequence of positive numbers. In particular, we obtain a sufficient condition for Freud's conjecture associated with weights onR.     

Multiplicities of weights of some representations and convex polytopes

Moscow Institute of Engineering, Electronics and Automation. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 28, No. 4, pp. 88–90, October–December, 1994.     

Galois representations modulo p and cohomology of Hilbert modular varieties

The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let us mention:

•

control of the image of Galois representations modulo p,

•

Hida's congruence criterion outside an explicit set of primes,

•

freeness of the integral cohomology of a Hilbert modular variety over certain local components of the Hecke algebra and Gorenstein property of these local algebras.

We study the arithmetic properties of Hilbert modular forms by studying their modulo p Galois representations and our main tool is the action of inertia groups at primes above p. In order to determine this action, we compute the Hodge-Tate (resp. Fontaine-Laffaille) weights of the p-adic (resp. modulo p) étale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine on the cohomology of Siegel modular varieties and builds upon geometric constructions of Tilouine and the author.     

Solutions of KZ differential equations modulo p

The Ramanujan Journal - We construct polynomial solutions of the KZ differential equations over a finite field $${\mathbb F}_p$$ as analogs of hypergeometric solutions.     

Reduction of quasidifferentials and minimal representations

Some criterias for the non-minimality of pairs of compact convex sets of a real locally convex topological vector space are proved, based on a reduction technique via cutting planes and excision of compact convex subsets. Following an example of J. Grzybowski, we construct a class of equivalent minimal pairs of compact convex sets which are not connected by translations.Corresponding author.