Arithmetic of modular forms and its application

Abstract

The theory of modular forms plays important role in many fields of number theory. In this thesis we study various aspects of arithmetic properties of modular forms. The first theme of this thesis is self-recurrences of Hauptmodul $N(j_{1,3})$. As a $q$-series the coefficients of Hauptmodul satisfy nontrivial recurrence relations. ([3], [24],[28]) We find new recurrences which are not covered by the works of many other people and give some new results. The second theme is divisibility of traces of singular moduli. Singular moduli are algebraic numbers which is defined by a value of modular form $j(z)$ at the Heegner points. ([1], [35]) We study mod $2^n$ divisibility of the traces of these algebraic numbers. In the last we study an affine model of modular curve $X(p)$. For the arithmetic purpose it is an interesting problem to find a model of modular curve. Farkas, Kra and Kopeliovich ([7]) showed that the quotients $F_1$ and $F_2$ of modified theta functions generate the function field $\mathcal{K}(X(p))$ of the modular curve $X(p)$ for a principal congruence subgroup $\Gamma(p)$ with prime $p \geq 7$. For such primes $p$ we first find affine models of $X(p)$ over $\bbQ$ represented by $\Phi_p(X, Y)=0$, from which we are able to obtain the algebraic relations $\Psi_p(X, Y) = 0$ of $F_1$ and $F_2$ raised by Farkas et al. As its application we construct the ray class field $K_{(p)}$ modulo $p$ over an imaginary quadratic field $K$ and then explicitly calculate its class polynomial by using the Shimura reciprocity law.