Spherical designs from norm-3 shell of integral lattices

A set of vectors all of which have a constant (non-zero) norm value in an Euclidean lattice is called a shell of the lattice. Venkov classified strongly perfect lattices of minimum 3 (R\'{e}seaux et "designs" sph\'{e}rique, 2001), whose minimal shell is a spherical 5-design. This note considers the classification of integral lattices whose shells of norm 3 are 5-designs.Comment: 10 pages, http://www2.math.kyushu-u.ac.jp/~j.shigezumi

An elementary approach to toy models for D. H. Lehmer's conjecture

In 1947, Lehmer conjectured that the Ramanujan's tau function $\tau (m)$ never vanishes for all positive integers $m$, where $\tau (m)$ is the $m$-th Fourier coefficient of the cusp form $\Delta_{24}$ of weight 12. The theory of spherical $t$-design is closely related to Lehmer's conjecture because it is shown, by Venkov, de la Harpe, and Pache, that $\tau (m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is a spherical 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. However, Bannai-Miezaki showed that none of the nonempty shells of the integer lattice \ZZ^2 in \RR^2 is a spherical 4-design, and that none of the nonempty shells of the hexagonal lattice $A_2$ is a spherical 6-design. Moreover, none of the nonempty shells of the integer lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for \QQ(\sqrt{-1}) and \QQ(\sqrt{-3}) is a spherical 2-design. In the proof, the theory of modular forms played an important role. Recently, Yudin found an elementary proof for the case of \ZZ^{2}-lattice which does not use the theory of modular forms but uses the recent results of Calcut. In this paper, we give the elementary (i.e., modular form free) proof and discuss the relation between Calcut's results and the theory of imaginary quadratic fields.Comment: 18 page

Construction of spherical cubature formulas using lattices

We construct cubature formulas on spheres supported by homothetic images of shells in some Euclidian lattices. Our analysis of these cubature formulas uses results from the theory of modular forms. Examples are worked out on the sphere of dimension n-1 for n=4, 8, 12, 14, 16, 20, 23, and 24, and the sizes of the cubature formulas we obtain are compared with the lower bounds given by Linear Programming

Spherical designs from norm-3 shell of integral lattices

A set of vectors all of which have a constant (non-zero) norm value in an Euclidean lattice is called a shell of the lattice. Venkov classified strongly perfect lattices of minimum 3 (R\'{e}seaux et "designs" sph\'{e}rique, 2001), whose minimal shell is a spherical 5-design. This note considers the classification of integral lattices whose shells of norm 3 are 5-designs.Comment: 10 pages, http://www2.math.kyushu-u.ac.jp/~j.shigezumi

Cubature formulas, geometrical designs, reproducing kernels, and Markov operators

Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to large classes of finite measure spaces $(\Omega,\sigma)$ and appropriate spaces of functions inside $L^2(\Omega,\sigma)$. The last section points out how spherical designs are related to a class of reflection groups which are (in general dense) subgroups of orthogonal groups

Spherical designs and modular forms of the D4D_4 lattice

In this paper, we study shells of the $D_4$ lattice with a slightly general concept of spherical $t$-designs due to Delsarte-Goethals-Seidel, namely, the spherical design of harmonic index $T$ (spherical $T$-design for short) introduced by Delsarte-Seidel. We first observe that the $2m$-shell of $D_4$ is an antipodal spherical $\{10,4,2\}$-design on the three dimensional sphere. We then prove that the $2$-shell, which is the $D_4$ root system, is tight $\{10,4,2\}$-design, using the linear programming method. The uniqueness of the $D_4$ root system as an antipodal spherical $\{10,4,2\}$-design with 24 points is shown. We give two applications of the uniqueness: a decomposition of the shells of the $D_4$ lattice in terms of orthogonal transformations of the $D_4$ root system: and the uniqueness of the $D_4$ lattice as an even integral lattice of level 2 in the four dimensional Euclidean space. We also reveal a connection between the harmonic strength of the shells of the $D_4$ lattice and non-vanishing of the Fourier coefficient of a certain newforms of level 2. Motivated by this, congruence relations for the Fourier coefficients are discussed

Some Cases of Kudla’s Modularity Conjecture for Unitary Shimura Varieties

A common theme of the thesis is the interplay of symmetry and rigidity, which is a general phenomenon in mathematics. Symmetry is a notion related to the degree to which an object remains unchanged under transformations, and rigidity is a notion that in terms of physics can be thought of as a lack of freedom, which leads to stronger properties of an object than we normally expect. An object of higher symmetry often also exhibits a higher extent of rigidity, and vice versa. In the introduction of the thesis, we provide some background on modular forms, number theory, and geometry in a way that does not require familiarity with these subjects. The contributions of this thesis are presented in three articles.In Article I, we establish the existence of rational geometric designs for rational polytopes via the circle method and convex geometry, and discuss the existence of rational spherical designs which relates to Lehmer\u27s conjecture on the Ramanujan tau function. In Article II, we break the barrier of expressing weight-2 modular forms of higher level whose central L-values vanish by products of at most two Eisenstein series. This work shows the power of Rankin--Selberg method and also contributes to the computation of elliptic modular forms.In Preprint III, we prove unconditionally some cases of Kudla\u27s conjecture on the modularity of generating functions of special cycles on unitary Shimura varieties, for norm-Euclidean imaginary quadratic fields. Our method is based on a result of Liu and work of Bruinier--Raum, who confirmed the orthogonal Kudla conjecture over Q