arXiv subjects
Publications and source records attributed to Noburo Ishii.
In our preceding article, we defined a generalized lambda function and showed that the genaralized lambda function and the modular invariant function generate the modular function field with respect to a principal congruence subgroup. In this article we shall study a minimal equation and values of the genaralized lambda function.
We define a modular function which is a generalization of the elliptic modular lambda function. We show this function and the modular invariant function generate the modular function field with respect to the principal congruence subgroup. Further we study its values at imaginary quadratic points.
We study a modular function $Λ_{k,\ell}$ which is one of generalized $λ$ functions. We show $Λ_{k,\ell}$ and the modular invariant function $j$ generate the modular function field with respect to the modular subgroup $Γ_1(N)$. Further we prove that $Λ_{k,\ell}$ is integral over $\mathbf Z[j]$. From these results, we obtain that the value of $Λ_{k,\ell}$ at an imaginary quadratic point is an algebraic integer and generates a ray class field over the Hilbert class field.
We study special values of a modular function $Λ$ which is one of generalized $λ$ functions. We show special values of $Λ$ at imaginary quadratic points are algebraic integers. Further we prove that $Λ$ and the modular invariant function generate the modular function field with respect to the modular subgroup $Γ_1(N)$.
We study the properties of special values of the modular functions obtained from Weierstrass P-function at imaginary quadratic points.
Enge and Schertz gave the method of using the double eta-quotient for the construction of elliptic curves over finite fields. In their method, it is necessary to count the number of rational points of elliptic curves corresponding to solutions of the modular equation over a finite field, because in advance we can not know which solution of the modular equation is that corresponding to the modular invariant. We give a condition that the modular invariant is a multiple root of the modular polynomial. Consequently, we give a method to reduce the amount of computation in the process of counting the number of rational points.
We give a method of representing the modular invariant function by generators of a modular function field.