Let M be a closed oriented d-dimensionnal manifold and let LM be the space of free loops on M . In this paper, we give a geometrical interpretation of the loop-coproduct and we study it’s compatibility with the Serre spectral sequence associated to the fibration ΩM → LM ev(0) → M . Then, we show that the spectral sequence associated to the free loop fibration LN → LX → LM of some Serre fibratio...

A conjecture of Serre predicts a precise from of Torelli Theorem for genus 3 curves, namely, an indecomposable principally polarized abelian threefold is a Jacobian if and only if some specific invariant is a square. We study here a three dimensional family of such threefolds, introduced by Howe, Leprevost and Poonen. By a new formulation, we link their results to the conjecture of Serre. Then,...

the unique normalized cusp form of weight 12 with respect to the full modular group. Although Lehmer’s speculation that τ(n) 6= 0 for every positive n remains open, Serre [S] has made substantial progress on the basic question regarding the number of Fourier coefficients of a modular form which can be zero. He shows (see [p. 179, S]) that τ(n) is non-zero for the vast majority of n. In the same...

In this paper, we study the André-Quillen homology of simplicial commutative l-algebras, l a field, having certain vanishing properties. When l has non-zero characteristic, we obtain an algebraic version of a theorem of J.-P. Serre and Y. Umeda that characterizes such simplicial algebras having bounded homotopy groups. We further discuss how this theorem fails in the rational case and, as an ap...

Let F be a totally real field, and v a place of F dividing an odd prime p. We study the weight part of Serre’s conjecture for continuous totally odd representations ρ : GF → GL2(Fp) that are reducible locally at v. Let W be the set of predicted Serre weights for the semisimplification of ρ|GFv . We prove that when ρ|GFv is generic, the Serre weights inW for which ρ is modular are exactly the on...

Let F be a totally real eld with ring of integers O and p be an odd prime unrami ed 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 Ĝm⊗Op. Using this p-rigidity and a linear independence of mod p Hilbert modular forms restricted to the partial Serre-Tate deformation space Ĝm⊗...

Following Bass–Milnor–Serre, we prove that SLn(Z) has the congruence subgroup property for n ≥ 3. This was originally proved by Mennicke and Bass–Lazard–Serre. Let Γn = SLn(Z). The congruence subgroup problem for Γn (solved independently by Mennicke [Me] and Bass–Lazard–Serre [BLS]) seeks to classify all finite-index subgroups of Γn. For l ≥ 2, the level l principal congruence subgroup of Γn, d...

We describe examples motivated by the work of Serre and Abhyankar. Subject Class: Primary 11F22, 11F03. Secondary 30F35, 20C34.

Let k be a field and Q a finite quiver without oriented cycles. Let kQ be the path algebra of Q and mod kQ the category of k-finite-dimensional right kQ-modules. The cluster category CQ was introduced in [1] (for general Q) and, independently, in [4] (for Q of type An). It is defined as the orbit category of the bounded derived category D(mod kQ) under the action of the automorphism Σ−1 ◦ S, wh...

We begin a study of the intersection theory of the moduli spaces of degree two stable maps from two-pointed rational curves to arbitrary-dimensional projective space. First we compute the Betti numbers of these spaces using Serre polynomial and equivariant Serre polynomial methods developed by E. Getzler and R. Pandharipande. Then, via the excision sequence, we compute an additive basis for the...