arXiv subjects
Publications and source records attributed to Yukitaka Abe.
Let $\wp $ be a Weierstrass $\wp $-function with algebraic $g_2$ and $g_3$, which has a non-zero real period $ω$. We show that at least one of $πr$ and $\wp(ωr)$ is transcendental for any non-zero real number $r$.
Let $X$ be a compact Riemann surface of genus $g$. Jacobi's inversion theorem states that the Abel-Jacobi map $φ: X^{(g)} \longrightarrow J(X)$ is surjective, where $X^{(g)}$ is the symmetric product of $X$ of degree $g$ and $J(X)$ is the Jacobi variety of $X$. Riemann obtained the explicit solution of the Jacobi inversion problem introducing Riemann's theta functions. We study such a problem for singular curves. We define a generalization of Riemann's theta functions and Riemann's constants. We obtain similar results for singular curves.
Originally, an abelian function field is the field of meromorphic functions on the Jacobi variety J(X) of a compact Riemann surface X. It is generated by the fundamental abelian functions belonging to the meromorphic function field on X. We study this relation for singular curves.
We study singular curves from analytic point of view. We give completely analytic proofs for the Serre duality and a generalized Abel's theorem. We also reconsider Picard varieties, Albanese varieties and generalized Jacobi varieties of singular curves analytically. We call an Albanese variety considered as a complex Lie group an analytic Albanese variety. We investigate them in detail. For a non-singular curve (a compact Riemann surface) $X$, there is the relation between the meromorphic function fields on $X$ and on its Jacobi variety $J(X)$. We try to extend this relation to the case of singular curves.
We characterize meromorphic function fields closed by partial derivatives in n variables.
We define the geometric simpleness for toroidal groups. We give an example of quasi-abelian variety which is geometrically simple, but not simple. We show that any quasi-abelian variety is isogenous to a product of geometrically simple quasi-abelian varieties. We also show that the ${\mathbb Q}$-extension of the ring of all endomorphisms of a geometrically simple quasi-abelian variety is a division algebra over ${\mathbb Q}$.
We completely determine cohomology groups of sections of homogeneous line bundles over a toroidal group.