We study curves of order 2 from the viewpoint of submanifold theory. We give a characterization of the parallel Kähler embeddings of a complex projective space into an ambient complex projective space from this point of view. This characterization is an improvement of the results in [N, PS].

A one-dimensional free boundary problem of hyperbolic type will be analyzed. This problem arises from the physical model ``Peel a thin film from a domain''. The local solutions have been already given for our problem. However, it is not sure that the solutions can be extended globally. In this article, by applying an iteration method, it will be shown that time-global solutions exist.

Let $G = \Gamma_0(N)$ and $g$ be the group generated by the involution $z \mapsto -1/Nz$ of the upper half plane. We determine the cohomology set $H^1(g,G)$ in terms of the class numbers $h(-N)$ and $h(-4N)$ of quadratic forms.

The present note reports on an explicit spectral formula for the fourth moment of the Dedekind zeta function $\zeta_{\mathrm{F}}$ of the Gaussian number field $\mathrm{F} = \mathbf{Q}(i)$, and on a new version of the sum formula of Kuznetsov type for $\mathrm{PSL}_2(\mathbf{Z}[i])\backslash \mathrm{PSL}_2(\mathbf{C})$. Our explicit formula (Theorem 5, below) for $\zeta_{\mathrm{F}}$ gives rise to a solution to a problem that has been posed on p. 183 of [M3] and, more explicitly, in [M4]. Also, our sum formula (Theorem 4, below) is an answer to a problem raised in [M4] concerning the inversion of a spectral sum formula over the Picard group $\mathrm{PSL}_2(\mathbf{Z}[i])$ acting on the three dimensional hyperbolic space (the $K$-trivial situation). To solve this problem, it was necessary to include the $K$-nontrivial situation into consideration, which is analogous to what has been experienced in the modular case.

Following the conceptual analogies between knots and primes, 3-manifolds and number fields, we discuss an analogue in knot theory after the model of the arithmetical theory of genera initiated by Gauss. We present an analog for cyclic coverings of links following along the line of Iyanaga-Tamagawa's genus theory for cyclic extentions over the rational number field. We also give examples of $\mathbf{Z} / 2\mathbf{Z} \times \mathbf{Z} / 2\mathbf{Z}$-coverings of links for which the principal genus theorem does not hold.

We rewrite the defining relations [5] of the simply-laced elliptic Lie algebras in terms of the extended elliptic Cartan matrix by considering the extended elliptic diagram.

We present an explicit formula for the Selmer rank of the elliptic curve $y^2 = x^3 + Dx$. Using this formula, we give some results analogous to Iskra's theorem.

We give an upper bound of the level $\mathfrak{n}$ by means of $d$-gonality of the Drinfel'd modular curves $X_0(\mathfrak{n})$ mod $\mathfrak{p}$ for $\mathfrak{p} \nmid \mathfrak{n}$. As a corollary of the result, we obtain an estimation in the strong Uniform Boundedness Conjecture for Drinfel'd modules of rank 2. We also discuss some asymptotically (and practically) good bound in this connection.

We obtain a lower bound for the error term of the prime geodesic theorem for hyperbolic 3-manifolds. Our result is $\Omega_{\pm}(x(\log\log x)^{1/3} / \log x)$. We also generalize Sarnak's upper bound $O(x^{(5/3) + \varepsilon})$ to compact manifolds.

Let $0 < \theta < 1$ be irrational and $T_{\theta} x = x + \theta \bmod 1$ on $[0,1)$. Consider the partition $\mathcal{Q}_n = \{[(i - 1) / 2^n, i/2^n) : 1 \leq i \leq 2^n\}$ and let $Q_n(x)$ denote the interval in $\mathcal{Q}_n$ containing $x$. Define two versions of the first return time: $J_n(x) = \min\{ j \geq 1 : \| x - {T_{\theta}}^j x \| = \| j \cdot \theta \| < 1/2^n \}$ where $\| t \| = \min_{n \in \mathbf{Z}} |t - n|$, and $K_n(x) = \min\{ j \geq 1 : {T_\theta}^j x \in Q_n(x) \}$. We show that $\log J_n / n \to 1$ and $\log K_n(x) / n \to 1$ a.e. as $n \to \infty$ for a.e. $\theta$.