Study on arithmetic properties of modular function fields and elliptic curves by constructive methods.

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In this research project, we studied following three subjects.(1)We studied the group structure of rational points of an elliptic curve E over finite fields in understanding arithmetic properties of solutions of a defining equation of a modular function field. In the case the elliptic curve E is a reduction of an elliptic curve with complex multiplication O, the group structure of rational points of E is determined by the trace α of the Frobenius endomorphism. Since the absolute value of α is easily known, the problem is to determine the sign of α. We showed a method to determine the sign and determined the sign of the trace in the case O is an orders of discriminant divided by 2, 3 or 5 and of class number 2 or 3.(2)Let p=5, 7, 11. We studied the representation of the modular invariant function J as a polynomial of degree p by a generator of a modular function field associated with the subgroup of SL_2(Z) of index p. We applied this representation to construct a family of elliptic curves with cyclic rational points groups over a finite field and to determine the Galois representation on the group of p-division points of elliptic curves.(3)Each solution of the defining equation of a modular function field corresponds to an elliptic curve. To determine this correspondence, we studied the representation of modular invariant function J by generators of the modular function field. Let g be the genus of the modular function field. We have obtained an algorithm to calculate the defining equation and the representation of J from g+1 modular functions fj which are regular except one cuspidal non-Weierstrass point. The essential part in practising the algorithm is to construct g+1 modular functions fj. In the case of Hecke group of level N, we constructed the modular functions fj for every N<53.