We study energy minimization for pair potentials among periodic sets in Euclidean spaces. We derive some sufficient conditions under which a point lattice locally minimizes the energy associated to a large class of potential functions. This allows in particular to prove a local version of Cohn and Kumar's conjecture that A 2 , D 4 , E 8 and the Leech lattice are globally universally optimal, regarding energy minimization, and among periodic sets of fixed point density.2000 Mathematics Subject Classification. 82B, 52C, 11H.
ABSTRACT. We set up a connection between the theory of spherical designs and the question of minima of Epstein's zeta function. More precisely, we prove that a Euclidean lattice, all layers of which hold a 4-design, achieves a local minimum of the Epstein's zeta function, at least at any real s > n 2 . We deduce from this a new proof of Sarnak and Strömbergsson's theorem asserting that the root lattices D 4 and E 8 , as well as the Leech lattice Λ 24 , achieve a strict local minimum of the Epstein's zeta function at any s > 0. Furthermore, our criterion enables us to extend their theorem to all the so-called extremal modular lattices (up to certain restrictions) using a theorem of Bachoc and Venkov, and to other classical families of lattices (e.g. the Barnes-Wall lattices).INTRODUCTION.
International audienceWe describe an algorithm, meant to be very general, to compute a presentation of the group of units of an order in a (semi-)simple algebra over QQ. Our method is based on a generalisation of Voronoï's algorithm for computing perfect forms, combined with Bass–Serre theory. It differs essentially from previously known methods to deal with such questions, e.g. for units in quaternion algebras. We illustrate this new algorithm by a series of examples where the computations are carried out completely
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