We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are
m
m
distances between distinct points in it and it is a spherical
(
2
m
−
1
)
(2m-1)
-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the
E
8
E_8
and Leech lattices. We also prove the same result for the vertices of the
600
600
-cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we conclude with an extension to Euclidean space.
Building on Viazovska's recent solution of the sphere packing problem in
eight dimensions, we prove that the Leech lattice is the densest packing of
congruent spheres in twenty-four dimensions and that it is the unique optimal
periodic packing. In particular, we find an optimal auxiliary function for the
linear programming bounds, which is an analogue of Viazovska's function for the
eight-dimensional case.Comment: 17 page
We prove that the Leech lattice is the unique densest lattice in ޒ 24 . The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in ޒ 24 can exceed the Leech lattice's density by a factor of more than 1 C 1:65 10 30 , and we give a new proof that E 8 is the unique densest lattice in ޒ 8 .
We outline a method to compute rational models for the Hilbert modular
surfaces Y_{-}(D), which are coarse moduli spaces for principally polarized
abelian surfaces with real multiplication by the ring of integers in
Q(sqrt{D}), via moduli spaces of elliptic K3 surfaces with a Shioda-Inose
structure. In particular, we compute equations for all thirty fundamental
discriminants D with 1 < D < 100, and analyze rational points and curves on
these Hilbert modular surfaces, producing examples of genus-2 curves over Q
whose Jacobians have real multiplication over Q.Comment: 83 pages. Final versio
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