Abstract: 
For a prime number l, the classical modular polynomial Phi_l (sometimes called the modular equation of level l) is a polynomial with integer coefficients such that Phi_l(j(t),j(lt))=0 where j(t) is the jinvariant well known from the theory of elliptic functions.
While it is difficult to compute Phi_l over the integers due to the rapid growth of its coefficients, there is a beautiful structure emerging if one considers Phi_l over GF(2), where it becomes a sparse polynomial. Even though the modular polynomials considered over GF(2) have important computational applications in elliptic curves cryptography, it appears that the amazing structure of their nonvanishing terms has not been considered in the literature.
By considering the power series expansion of j(q), we prove some necessary conditions that a monomial of Phi_l must satisfy in order to have coefficient 1, and we conjecture many more such conditions based on results from our computational investigations.
