The Eisenstein series of heigher weight has at least one transcendental zero point on upper half plane.

Explicit form of Fourier expansion of automorphic forms plays an important role in the theory. Here we investigate the case of $SU(2,1)$ and give an explicit formula of generalized Whittaker functions for the standard representations of the group. Togather with a result of [2], we obtain a form of fully developed Fourier expansion of automorphic forms belonging to arbitrary standard representations.

In this paper, we study length spectrum of circles on a complex projective space and on a complex hyperbolic space. In particular, we focus ourselves on the asymptotic behaviour of the number of congruency classes of circles with length $\lambda$ and on the asymptotic behaviour of the number of congruency classes of circles of prescribed geodesic curvature with length not greater than $\lambda$.

A rational Diophantine $m$-tuple is a set of $m$ nonzero rationals such that the product of any two of them is one less than a perfect square. In this paper we characterize the notions of regular Diophantine quadruples and quintuples, introduced by Gibbs, by means of elliptic curves. Motivated by these characterizations, we find examples of elliptic curves over $\mathbf{Q}$ with torsion group $\mathbf{Z}/2\mathbf{Z} \times \mathbf{Z}/2\mathbf{Z}$ and with rank equal 8.

We study the Mordell-Weil lattices ([9]) of the elliptic $K3$ surfaces which have been introduced by Inose [4] and Kuwata [6]. The point is that the lattices (of rank up to 18) arising this way can be of some interest in terms of sphere packings. In this note, we treat the case of rank 16, 17, or 18, and show that these Morderll-Weil lattices are potentially comparable with the record lattices in these dimensions ([2]). The detailed accont is in preparation, which will include the corresponding results for other dimensions as well.

We give an effective bound of the Gorenstein index of weak $\mathbf{Q}$-Fano 3-folds, and prove the boundedness of the terminal $\mathbf{Q}$-Fano 3-folds. Combined with [Bor99], this result implies furthermore the boundedness of the canonical $\mathbf{Q}$-Fano 3-folds.

We construct for each integer $n$ ($\geq 3$), infinitely many number fields of degree $n$ each of which has an unramified quadratic extension with a power integral basis but no normal integral basis.