A new Euclidean tight 6-design

We give a new example of Euclidean tight 6-design in $\mathbb R^{22}$.Comment: 9 page

On relative tt-designs in polynomial association schemes

Motivated by the similarities between the theory of spherical $t$-designs and that of $t$-designs in $Q$-polynomial association schemes, we study two versions of relative $t$-designs, the counterparts of Euclidean $t$-designs for $P$- and/or $Q$-polynomial association schemes. We develop the theory based on the Terwilliger algebra, which is a noncommutative associative semisimple $\mathbb{C}$-algebra associated with each vertex of an association scheme. We compute explicitly the Fisher type lower bounds on the sizes of relative $t$-designs, assuming that certain irreducible modules behave nicely. The two versions of relative $t$-designs turn out to be equivalent in the case of the Hamming schemes. From this point of view, we establish a new algebraic characterization of the Hamming schemes.Comment: 17 page

Note on cubature formulae and designs obtained from group orbits

In 1960, Sobolev proved that for a finite reflection group G, a G-invariant cubature formula is of degree t if and only if it is exact for all G-invariant polynomials of degree at most t. In this paper, we find some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and moreover gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007) which classifies tight Euclidean designs invariant under the Weyl group of type B to other finite reflection groups.Comment: 18 pages, no figur

Nonexistence of exceptional imprimitive Q-polynomial association schemes with six classes

Suzuki (1998) showed that an imprimitive Q-polynomial association scheme with first multiplicity at least three is either Q-bipartite, Q-antipodal, or with four or six classes. The exceptional case with four classes has recently been ruled out by Cerzo and Suzuki (2009). In this paper, we show the nonexistence of the last case with six classes. Hence Suzuki's theorem now exactly mirrors its well-known counterpart for imprimitive distance-regular graphs.Comment: 7 page

The Kronecker limit formulas via the distribution relation

In this paper, we give a proof of the classical Kronecker limit formulas using the distribution relation of the Eisenstein-Kronecker series. Using a similar idea, we then prove $p$-adic analogues of the Kronecker limit formulas for the $p$-adic Eisenstein-Kronecker functions defined in our previous paper

On a property of 2-dimensional integral Euclidean lattices

Let $L$ be any integral lattice in the 2-dimensional Euclidean space. Generalizing the earlier works of Hiroshi Maehara and others, we prove that for every integer $n>0$, there is a circle in the plane $\mathbb{R}^{2}$ that passes through exactly $n$ points of $L$.Comment: 9 page

p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure

The specializations of the motivic elliptic polylog are called motivic Eisenstein classes. For applications to special values of L-Functions, it is important to compute the realizations of these classes. In this paper, we prove that the syntomic realization of the motivic Eisenstein classes, restricted to the ordinary locus of the modular curve, may be expressed using p-adic Eisenstein-Kronecker series. These p-adic modular forms are defined using the two-variable p-adic measure with values in p-adic modular forms constructed by Katz.Comment: 40 page