We prove that theta functions of binary quadratic forms with congruence conditions coincide with theta functions attached to ray classes of quadratic fields. By Hecke's Theorem, we can construct Hecke eigenforms by forming linear combinations of them.

The aim of this work is to study regularity of solutions of the initial value problem for the Schrӧdinger equations with sub-quadratic potential. More precisely, we determine the $H^s$ wave front sets of solutions from the behavior at infinity of the initial data by using the characterization of the $H^s$ wave front set via wave packet transform.

Ishioka-Kunugi [9] gives an equivalent condition for Scott modules to be Brauer indecomposable. This paper generalizes the equivalent condition to that for Brauer-friendly modules to be slash indecomposable.

Zeta functions for linear codes were defined by I. Duursma in 1999. In the cases of genera less than three, S. Nishimura gave equivalent conditions for their Riemann hypothesis. In this paper, using a new method, we give similar equivalent conditions for the cases of genera three and four. Our method can be applied to smaller genera and leads to an alternative simple proofs of Nishimura’s theorems. Using these results, we examine the Riemann hypothesis of some invariant polynomials. We also discuss the cases of genera greater than four and propose some new problems.

Let $(X_0,X_1)$ be a compatible couple of Banach spaces and $T_j$ a bounded operator from $X_j$ into itself $(j = 0, 1)$ satisfying $T_0x = T_1x$ for all $x \in X_0 \cap X_1$. On an additional assumption concerning a boundedness of $T_0|_{X_0\cap X_1} (=T_1|_{X_0\cap X_1})$, the next relations of spectra are proved: $$\sigma(T_\theta) \subset \sigma(T_0)\cup \sigma (T_1) = \sigma(T\Delta)\cup \sigma (T_\Sigma)\quad (\theta \in (0,1)),$$ where $T_\theta $, $T_\Delta$ and $T_\Sigma$ are the bounded operators induced by $T_0$ and $T_1$ on the complex interpolation space $(X_0,X_1)_{[\theta ]}$, the intersection $X_0 \cap X_1$ and the sum $X_0 + X_1$, respectively.