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Authors
Mauricio Garay
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In this short note, I explain how the non-degeneracy condition of the KAM can be bypassed. The first version of the note has been submitted for publication back in 2010 and this version in 2012.
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The structure of anticyclic operad on the Dendriform operad defines in particular a matrix of finite order acting on the vector space spanned by planar binary trees. We compute its characteristic polynomial and propose a (compatible) conjecture for the characteristic polynomial of the Coxeter transformation for the Tamari lattice, which is mostly a square root of this matrix.
Let ${rm F}$ be a rank-2 semi-stable sheaf on the projective plane, with Chern classes $c_{1}=0,c_{2}=n$. The curve $beta_{rm F}$ of jumping lines of ${rm F}$, in the dual projective plane, has degree $n$. Let ${rm M}_{n}$ be the moduli space of equivalence classes of semi-stables sheaves of rank 2 and Chern classes $(0,n)$ on the projective plane and ${cal C}_{n}$ be the projective space of curves of degree $n$ in the dual projective plane. The Barth morphism $$beta: {rm M}_{n}longrightarrow{cal C}_{n}$$ associates the point $beta_{rm F}$ to the class of the sheaf ${rm F}$. We prove that this morphism is generically injective for $ngeq 4.$ The image of $beta$ is a closed subvariety of dimension $4n-3$ of ${cal C}_{n}$; as a consequence of our result, the degree of this image is given by the Donaldson number of index $4n-3$ of the projective plane.
The aim of this article is to prove a Thom-Sebastiani theorem for the asymptotics of the fiber-integrals. This means that we describe the asymptotics of the fiber-integrals of the function $f oplus g : (x,y) to f(x) + g(y)$ on $(mathbb{C}^ptimes mathbb{C}^q, (0,0))$ in term of the asymptotics of the fiber-integrals of the holomorphic germs $f : (mathbb{C}^p,0) to (mathbb{C},0)$ and $g : (mathbb{C}^q,0) to (mathbb{C},0)$. This reduces to compute the asymptotics of a convolution $Phi_*Psi$ from the asymptotics of $Phi$ and $Psi$ modulo smooth terms. To obtain a precise theorem, giving the non vanishing of expected singular terms in the asymptotic expansion of $foplus g$, we have to compute the constants coming from the convolution process. We show that they are given by rational fractions of Gamma factors. This enable us to show that these constants do not vanish.
On the rank of Jacobians over function fields.} Let $f:mathcal{X}to C$ be a projective surface fibered over a curve and defined over a number field $k$. We give an interpretation of the rank of the Mordell-Weil group over $k(C)$ of the jacobian of the generic fibre (modulo the constant part) in terms of average of the traces of Frobenius on the fibers of $f$. The results also give a reinterpretation of the Tate conjecture for the surface $mathcal{X}$ and generalizes results of Nagao, Rosen-Silverman and Wazir.
We reinterpret a conjecture of Breuil on the locally analytic $mathrm{Ext}^1$ in a functorial way using $(varphi,Gamma)$-modules (possibly with $t$-torsion) over the Robba ring, making it more accurate. Then we prove several special or partial cases of this improved conjecture, notably for ${rm GL}_3(mathbb{Q}_p)$.
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