  For quadratic curves over $Q$ in $F_p$, the number of solutions, which is
governed by an analogue of the Mordell-Weil group, is expressed with the
Legendre symbol of a coefficient of quadratic curves. Focusing on the number of
solutions, a quadratic curve analogue of the modular form in the
Taniyama-Shimura conjecture is proposed. This modular form yields the Gaussian
sum and also possesses some modular transformation structure.
