In this note we extend Hurwitz-type multiplication of quadratic forms. For a regular quadratic space $(K^n, q)$, we restrict the domain of $q$ to an algebraic variety $V \subsetneq K^n$ and require a Hurwitz-type ``bilinear condition'' on $V$. This means the existence of a bilinear map $\varphi\colon K^n \times K^n \rightarrow K^n$ such that $\varphi(V \times V) \subset V$ and $q(\mathbf{X}) q(\mathbf{Y}) = q(\varphi(\mathbf{X}, \mathbf{Y}))$ for any $\mathbf{X}, \mathbf{Y} \in V$. We show that the $m$-fold Pfister form is multiplicative on certain proper subvariety in $K^{2^m}$ for any $m$. We also show the existence of multiplicative quadratic forms which are different from Pfister forms on certain algebraic varieties for $n = 4, 6$. Especially for $n = 4$ we give a certain family of them.

We introduced an infinite sequence of $L^2$-torsion invariants for surface bundles over the circle in [4]. In this note, we investigate in detail the first two terms for a torus bundle case. In particular, we show that the first invariant can be described by the asymptotic behavior of the order of the first homology group of a cyclic covering.

The elliptic Hecke algebras associated to the 1-codimensional elliptic root systems have been defined by H. Yamada [10], which are subalgebras of Cherednik's double affine Hecke algebras [2, 3]. The elliptic Hecke algebras associated to the elliptic root systems of type $X^{(1,1)}$ have been defined similarly by the author [11] in terms of generators and relations associated to the completed elliptic diagram. On the other hand, M. Kapranov [6] has defined modified Cherednik algebras associated to the double coset decomposition of the group schemes over 2-dimensional local field. In this paper, we see that modified Cherednik algebras are isomorphic to elliptic Hecke algebras of type $X^{(1,1)}$.

Let $S(k + 1/2, N, \chi)$ denote the space of cusp forms of weight $k+1/2$, level $N$, and character $\chi$. Let $R_{\psi}$ be a twisting operator for a quadratic primitive character $\psi$ of even conductor and $\tilde{T}(n^2)$ the $n^2$-th Hecke operator. We give an explicit trace formula of $R_{\psi} \tilde{T}(n^2)$ on $S(k + 1/2, N, \chi)$.

We prove a general optimal inequality for warped products in complex projective spaces and determine warped products which satisfy the equality case of the inequality. Two non-immersion theorems are obtained as immediate applications.

A simple and beautiful idea of Poincaré on Poincaré series in automorphic functions can be applied to an arbitrary ring $R$ acted by a group $G$. When $G$ is finite, the key is to look at the 0-dimensional Tate cohomology of $(G, R)$ twisted by the 1-cohomology class of the group of units of $R$. As a simplest case, we examine when $R$ is the ring of integers of a quadratic field.