let $g$ be a $p$-group of order $p^n$ and $phi$=$phi(g)$ be the frattini subgroup of $g$. it is shown that the nilpotency class of $autf(g)$, the group of all automorphisms of $g$ centralizing $g/ fr(g)$, takes the maximum value $n-2$ if and only if $g$ is of maximal class. we also determine the nilpotency class of $autf(g)$ when $g$ is a finite abelian $p$-group.

We relate a part of the abelian étale fundamental group of curves over local fields to the component group of the Néron model of the jacobian. We apply the result to the modular curve X0(p)/Qp to show that the unramified abelian covering X1(p) → X0(p) (Shimura covering) uses up all the possible ramification over the special fiber of X0(p).

We describe a particularly easy way of evaluating the modular irreducible matrix representations of the symmetric group. It shows that Specht’s approach to the ordinary irreducible representations, along Specht polynomials, can be unified with Clausen’s approach to the modular irreducible representations using symmetrized standard bideterminants. The unified method, using symmetrized Specht pol...

. The constants are effectively computable. Proof. Part (a) follows from the fact that the algebraic group GSp2g has dimension 2g +g+1. Part (b) is obvious. Combined with Theorem 1.1 of [Ghi04], Theorem 1 gives Corollary 3 (Algebraic modular forms). Let B/Q be the quaternion algebra ramified at p and ∞. The number of systems of Hecke eigenvalues coming from algebraic modular forms (mod p) of le...

Based on Kleshchev's branching theorems for the p-modular irreducible representations of the symmetric group and on the recent proof of the Mullineux Conjecture, we investigate in this article the corresponding branching problem for the p-modular irreducible representations of the alternating group A n. We obtain information on the socle of the restrictions of such A n-representations to A n?1 ...

In this paper, we study modular categories whose Galois group action on their simple objects are transitive. We show that such admit unique factorization into prime transitive factors. The representations of $${\text {SL}}_2({\mathbb {Z}})$$ associated with proven to be minimal and irreducible. Using the Verlinde formula, characterize as conjugates adjoint subcategory quantum category $${\mathc...

We prove the indecomposability of the Galois representation restricted to the p-decomposition group attached to a non CM nearly p-ordinary weight two Hilbert modular form over a totally real field F under the assumption that either the degree of F over Q is odd or the automorphic representation attached to the Hilbert modular form is square integrable at some finite place of F .

Here H denotes the complex upper half plane and wτ the order of the stabilizer of τ in PSL2(Z). One purpose of this note is to give a generalization of this formula to modular forms on the orthogonal group O(2, p), which only depends on the Baily-Borel compactification of the arithmetic quotient in question. Moreover, we show that certain integrals, occurring in Arakelov intersection theory, as...

let $g$ be a $p$-group of order $p^n$ and $phi$=$phi(g)$ be the frattini subgroup of $g$. it is shown that the nilpotency class of $autf(g)$, the group of all automorphisms of $g$ centralizing $g/ fr(g)$, takes the maximum value $n-2$ if and only if $g$ is of maximal class. we also determine the nilpotency class of $autf(g)$ when $g$ is a finite abelian $p$-group.