Related papers: Brauer's Height Zero Conjecture for metacyclic def…

We study blocks all of whose height zero ordinary characters have the same degree. We suspect that these might be the Broue-Puig nilpotent blocks.

This paper treats what we call `weak geometric liftings' of Galois representations associated to abelian varieties. This notion can be seen as a generalization of the idea of lifting a Galois representation along an isogeny of algebraic…

Let $G$ be a finite group with Sylow $2$-subgroup $P \leqslant G$. Navarro-Tiep-Vallejo have conjectured that the principal $2$-block of $N_G(P)$ contains exactly one irreducible Brauer character if and only if all odd-degree ordinary…

We determine the blocks of the walled Brauer algebra in characteristic zero. These can be described in terms of orbits of the action of a Weyl group of type $A$ on a certain set of weights. In positive characteristic we give a linkage…

We prove the Breuil-M\'ezard conjecture for 2-dimensional potentially Barsotti-Tate representations of the absolute Galois group G_K, K a finite extension of Q_p, for any p>2 (up to the question of determining precise values for the…

Let $p$ be an odd prime and let $B$ be a $p$-block of a finite group which has cyclic defect groups. We show that all exceptional characters in $B$ have the same Frobenius-Schur indicators. Moreover the common indicator can be computed,…

Let $p$ be a prime, $k$ an algebraic closure of $\mathbb{F}_p$ and $\Gamma$ the Galois group ${\rm Gal}(k/\mathbb{F}_p)$. Let $\mathcal{C}$ be a finite category and $\mathcal{O}_{\mathcal{C}}$ the $p$-orbit category of $\mathcal{C}$ defined…

The Atiyah conjecture for a discrete group G states that the $L^2$-Betti numbers of a finite CW-complex with fundamental group G are integers if G is torsion-free and are rational with denominators determined by the finite subgroups of G in…

We show that the $p$-part of the degree of an irreducible character of a symmetric group is completely determined by the set of vanishing elements of $p$-power order. As a corollary we deduce that the set of zeros of prime power order…

We give a new local proof of Breuil-M\'ezard conjecture for two dimensional representations of the absolute Galois group of $\mathbb{Q}_p$, when $p\ge 5$ and the representation has scalar endomorphisms.

Let B be a p-block of a finite group G with abelian defect group D such that S\unlhd G, S'=S, G/Z(S)\le\Aut(S) and S/Z(S) is a sporadic simple group. We show that B is isotypic to its Brauer correspondent in N_G(D) in the sense of Brou\'e.…

The Bounded Height Conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian $\overline{\mathbb{Q}}$-variety $G$ there is an upper bound on the Weil height of the points…

In this paper, we investigate some relations between the Loewy lengths of the centers of blocks of group algebras and its defect groups. In particular, we give a new upper bound of the Loewy length and determine the structure of blocks with…

Hans Zassenhaus conjectured that every torsion unit of the integral group ring of a finite group $G$ is conjugate within the rational group algebra to an element of the form $\pm g$ with $g\in G$. This conjecture has been disproved recently…

In this paper we study the relationship between degeneracy and decomposability in abelian crossed products. In particular we construct an indecomposable abelian crossed product division algebra of exponent $p$ and index $p^2$ for $p$ an odd…

In this article, we consider the finite exceptional groups of Lie type $E_6$ and $^2E_6$. We prove the inductive blockwise Alperin weight condition holds for unipotent $l$-blocks of $E_6^{\varepsilon}(q)$ if $2,3\nmid q$, $l\geq 5$

We prove a strengthening of the "reciprocity conjecture" of Khare and Wintenberger. The input to the original conjecture is an odd prime p, a CM number field F containing the pth roots of unity, and a pair of primes of the maximal totally…

We give sufficient conditions on $p$-blocks of $p$-nilpotent groups over $\mathbb{F}_p$ to be splendidly Rickard equivalent and $p$-permutation equivalent to their Brauer correspondents. The paper also contains Galois descent results on…

Let p be a fixed prime number. Let K be a totally real number field of discriminant D\_K and let T\_K be the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of K (under Leopoldt's conjecture). We…

We show that the 3-block of the sporadic simple Janko group J_4 with defect group C_3 x C_3, and the principal 3-block of the alternating group A_8 are Puig equivalent, answering a question posed in earlier work of Koshitani-Kunugi-Waki. To…