The Epstein ζ function ζ(Γ,s) of a lattice Γ is defined by a series which converges for any compl... more The Epstein ζ function ζ(Γ,s) of a lattice Γ is defined by a series which converges for any complex number s such that ℜ s > n/2, and admits a meromorphic continuation to the complex plane, with a simple pole at s = n/2. The question as to which Γ, for a fixed s, minimizes ζ(Γ,s), has a long history, dating back to Sobolev's work on numerical integration, and subsequent papers by Delone and Ryshkov among others. This was also investigated more recently by Sarnak and Strombergsson. The present paper is concerned with similar questions for positive definite quadratic forms over number fields, also called Humbert forms. We define Epstein zeta functions in that context and study their meromorphic continuation and functional equation, this being known in principle but somewhat hard to find in the literature. Then, we give a general criterion for a Humbert form to be finally ζ extreme, which we apply to a family of examples in the last section.
We apply Voronoi's algorithm to compute representatives of the conjugacy classes of maximal finit... more We apply Voronoi's algorithm to compute representatives of the conjugacy classes of maximal finite subgroups of the unit group of a maximal order in some simple $\QQ $-algebra. This may be used to show in small cases that non-conjugate orders have non-isomorphic unit groups.
The Epstein ζ function ζ(Γ,s) of a lattice Γ is defined by a series which converges for any compl... more The Epstein ζ function ζ(Γ,s) of a lattice Γ is defined by a series which converges for any complex number s such that ℜ s > n/2, and admits a meromorphic continuation to the complex plane, with a simple pole at s = n/2. The question as to which Γ, for a fixed s, minimizes ζ(Γ,s), has a long history, dating back to Sobolev's work on numerical integration, and subsequent papers by Delone and Ryshkov among others. This was also investigated more recently by Sarnak and Strombergsson. The present paper is concerned with similar questions for positive definite quadratic forms over number fields, also called Humbert forms. We define Epstein zeta functions in that context and study their meromorphic continuation and functional equation, this being known in principle but somewhat hard to find in the literature. Then, we give a general criterion for a Humbert form to be finally ζ extreme, which we apply to a family of examples in the last section.
We apply Voronoi's algorithm to compute representatives of the conjugacy classes of maximal finit... more We apply Voronoi's algorithm to compute representatives of the conjugacy classes of maximal finite subgroups of the unit group of a maximal order in some simple $\QQ $-algebra. This may be used to show in small cases that non-conjugate orders have non-isomorphic unit groups.
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Papers by Renaud Coulangeon