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Register a new userSur les fonctions `a singularite de dimension 1
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Authors
Daniel Barlet
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In this article we show that all results proved for a large class of holomorphic germs $f : (mathbb{C}^{n+1}, 0) to (mathbb{C}, 0)$ with a 1-dimension singularity in [B.II] are valid for an arbitrary such germ.
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This paper contains results concerning a conjecture made by Lang and Silverman predicting a lower bound for the canonical height on abelian varieties of dimension 2 over number fields. The method used here is a local height decomposition. We derive as corollaries uniform bounds on the number of torsion points on families of abelian surfaces and on the number of rational points on families of genus 2 curves.
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.
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