Starting from the reciprocity law for Gaussian sums attached to an integral quadratic form we prove functional equations for a new kind of Dirichlet series in two variables. For special values of one variable they are of Hecke type with respect to the other variable. With Weil's converse theorem we derive automorphic functions which generalize Siegel's genus invariant and the automorphic functions of Cohen and Zagier.

Since J.Wahl ([27]), it is known that degree $d$ plane curves having some fixed numbers of nodes and cusps as its only singularities can be represented by a scheme, let say $H$, which can be singular. In Wahl's example, $H$ is singular along a subscheme $F$ but the induced reduced scheme $H_{\rm{red}}$ is smooth along $F$. In this work, we construct explicitly a family of plane curves with nodes and cusps which are represented by singular points of $H_{\rm{red}}$.

To this end, we begin to show that the Hilbert scheme of smooth and connected space curves of degree 12 and genus 15 is irreducible and generically smooth. It follows that it is singular along a hypersurface (3.10). This example is minimal in the sense that the Hilbert scheme of smooth and connected space curves is regular in codimension 1 for $d<2$ % (B.2). Finally we construct our plane curves from the space curves represented by points of this hypersurface (4.7).

On sait depuis J.Wahl ([27]) que les courbes planes de degré fixé ayant pour seules singularités des nœuds et des cusps en nombres imposés sont représenté es par un schéma $H$ qui peut ètre singulier. Wahl exhibe une famille de courbes comme ci-dessus dont les points correspondants sont singuliers dans $H$ mais lisses dans $H_{\rm{red}}$, la structure réduite sous-jacente. Ici, on construit explicitement une famille de courbes planes à nœuds et à cusps représenté par des points singuliers de $H_{\rm{red}}$.

Pour ce faire, on montre tout d'abord que le schéma de Hilbert des courbes lisses et connexes de degré 12 et de genre 15 de l'espace projectif complexe est irréductible et génériquement lisse; puis qu'il est singulier le long d'une hypersurface (3.10). Cet exemple est minimal dans le sens où le schéma de Hilbert des courbes lisses et connexes de degré $d$ et genre $g$ est lisse en codimension 1 pour $d<12$ (B.2). Enfin, on construit les courbes planes à partir des courbes gauches représentés par cette hypersurface (4.7).

We study the behaviour of motivic zeta functions of prehomogeneous vector spaces under castling transformations. In particular we deduce how the motivic Milnor fibre and the Hodge spectrum at the origin behave under such transformations.

Let $D$ be a bounded pseudoconvex domain in $\mathbb{C}^n$ and $\zeta \in \mathrm {D}_{\!\!\!.}$. By $K_D$ and $B_D$ we denote the Bergman kernel and metric of $D$, respectively. Given a ball $B=B(\zeta, R)$, we study the behavior of the ratio $K_D/K_{D\cap B} (w)$ when $w\in D\cap B$ tends towards $\zeta$. It is well-known, that it remains bounded from above and below by a positive constant. We show, that the ratio tends to $1$, as $w$ tends to $\zeta$, under an additional assumption on the pluricomplex Green function $\mathcal{G}_D(\cdot , w)$ of $D$ with pole at $w$, the pluricomplex Green function $\mathcal{G}_D(\cdot , w)$ of $D$ with pole at $w$, namely that the diameter of the sublevel sets $A_w := \{z\in D \,\,|\,\, \mathcal{G}_D(z,w)<-1\}$ tends to zero, as $w\rightarrow \zeta$. A similar result is obtained also for the Bergman metric. In this case we also show that the extremal function associated to the Bergman kernel has the concentration of mass property introduced in {DiOh1}, where the question was discussed how to recognize a weight function from the associated Bergman space. The hypothesis concerning the set $A_w$ is satisfied for example, if the domain is regular in the sense of Diederich-Fornæss, ([DiFo2]).

We consider isomonodromic deformations of second-order Fuchsian differential equations on elliptic curves. The isomonodromic deformations are described as a completely integrable Hamiltonian system.

We find invariants for the differential systems of rank $2n$ in $n^{2}$ variables with $n$ unknowns under the linear changes of the unknowns with variable coefficients. We look for a set of coefficients that determines the other coefficients, and give transformation rules under the linear changes above and coordinate changes. These can be considered as a generalization of the Schwarzian derivative, which is the invariant for second order ordinary differential equations under the change of the unknown by multiplying a non-zero function. Special treatment is done when $n = 2$: the conformal structure obtained through the Plücker embedding is studied, and a relation with line congruences is discussed.

Let $X_{\mathbb R} \subset {\mathbb R}^{N}$ a real analytic set such that its complexification $X_{\mathbb C} \subset {\mathbb C}^{N}$ is normal with an isolated singularity at $0$. Let $f_{\mathbb R} : X_{\mathbb R} \rightarrow {\mathbb R}$ a real analytic function such that its complexification $f_{\mathbb C} : X_{\mathbb C} \rightarrow {\mathbb C}$ has an isolated singularity at $0$ in $X_{\mathbb C}$. Assuming an orientation given on $X_{\mathbb R}^{*}$, to a connected component $A$ of $X_{\mathbb R}^{*}$ we associate a compact cycle $\Gamma (A)$ in the Milnor fiber of $f_{\mathbb C}$ which determines completely the poles of the meromorphic extension of $\int_{A} f^{\lambda} \square$ or equivalently the asymptotics when $\tau \rightarrow \pm \infty$ of the oscillating integrals $\int_{A} e^{i \tau f} \square$. A topological construction of $\Gamma(A)$ is given. This completes the results of [BM] paragraph 6.

This paper is a contribution to the "revision" project of Gorenstein, Lyons and Solomon, whose goal is to produce a unified proof of the Classification of Finite Simple Groups.