COULANGEON, Renaud

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COULANGEON, Renaud
Institut de Mathématiques de Bordeaux [IMB]

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International Mathematics Research Notices. 2006p. Art. ID 49620, 16 pp.

Oxford University Press (OUP)

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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 D4 and E8, as well as the Leech lattice, 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).< Réduire

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Spherical designs and zeta functions of lattices

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