| We will study modular Abelian varieties with odd congruence numbers, by studying the cuspidal subgroup of J0( N). We show the conductor of such Abelian varieties must be of a special type, for example if N is odd then N = palpha or N = pq for some prime p and q. We then focus our attention to modular elliptic curves, and using result of Agashe, Ribet, and Stein [1], we try to classify all elliptic curves of odd modular degree. Our studies prove many cases of the Stein and Watkins's conjecture on elliptic curves with odd modular degree. |
