Given any prime number l which is a primitive root modulo 529 (=23²), we shall prove that the l-class group of the Z₂₃-extension over the rational field is trivial.

Let p be an odd prime number, and F a number field. We show that when F/Q is unramified at p, any tame cyclic extension N/F of degree p has a normal integral basis if the pushed up extension $N(\zeta_p)/F(\zeta_p)$ has a normal integral basis.

From a finite number of Riemann surfaces $W_j\ (j\in J:=\{ 1,2,\cdots ,m\} )$ we form two kinds of Riemann surfaces, one of which is a united surface ${\bigcup {\times}}_{j \in J}W_j$ and the other is simply a bunched surface $\bigcup _{j \in J} W_j$. We compare the space $H( {\bigcup {\times}} _{j \in J}W_j)$ of harmonic functions on $ {\bigcup {\times}}_{j \in J}W_j$ and the space $H(\bigcup _{j \in J}W_j)$ of harmonic functions on $\bigcup _{j \in J}W_j$ and show that these are canonically isomorphic, i.e. $$ H\Bigl ({\bigcup {\times}}_{j \in J}W_j\Bigr )\cong H\Bigl ({\bigcup }_{j \in J}W_j\Bigr ) $$ in the sense that there is a bijective mapping $t$ of the former space onto the latter space such that $t$ is linearly isomorphic, $t$ preserves orders, i.e. $tu\geqq 0$ if and only if $u\geqq 0$, and $t$ fixes the real number field ${\bf R}$, i.e. $t\lambda =\lambda $ for every $\lambda \in {\bf R}$, under the standing assumption that all the $W_j$ are hyperbolic. The result is then applied to give a sufficient condition better than our former one for an afforested surface to belong to the class ${\cal O}_s$ of hyperbolic Riemann surfaces carrying no nonzero singular harmonic functions when its plantation and trees on it are all in ${\cal O}_s$.

One-parameter mirror families to weighted hypersurfaces are already constructed and well studied. A generic Torelli theorem for quintic-mirror family is proved by Sampei Usui. In this article, we give the proof of a generic Torelli theorem for the other families after Usui's proof for quintic-mirror family.

We study Humbert's modular equation which characterizes curves of genus two having real multiplication by the quadratic order of discriminant 5. We give it a simple, but general expression as a polynomial in $x_1,\ldots ,x_6$ the coordinate of the Weierstrass points, and show that it is invariant under a transitive permutation group of degree 6 isomorphic to $\frak S_5$. We also prove the rationality of the hypersurface in P⁵ defined by the generalized modular equation.