Narrow Results

The theory of elliptic curves and modular forms provides a precise relationship between the supersingular j-invariants and the congruences between modular forms. Kaneko and Zagier discuss a surprising generalization of these results in their paper on Atkin orthogonal polynomials. In this paper, we define an analog of the Atkin orthogonal polynomials for rank two Drinfel'd modules.

We obtain a new proof of an asymptotic formula for the coefficients of the j-invariant of elliptic curves. Our proof does not use the circle method. We use Laplace's method of steepest descent and the Hardy–Ramanujan asymptotic formula for the partition function. (The latter asymptotic formula can be derived without the circle method.)

In this paper, we obtain two effective bounds for the j-invariant of integral points on certain modular curves which have positive genus and less than three cusps.

For each algebraic number $\alpha \in \bar{\mathbb{Q}}$, a result of Habegger [P. Habegger, Singular moduli that are algebraic units, Algebra Number Theory 9(7) (2015) 1515–1524] shows that there are only finitely many singular moduli $j$ such that $j-\alpha$ is an algebraic unit. His result uses Duke’s Equidistribution Theorem and is thus not effective. In this paper, we give an effective proof of Habegger’s result assuming that $\alpha$ is not a singular modulus itself. We give an explicit bound, which depends only on $\alpha$, on the discriminant $\mathrm{\Delta }$ associated with a singular modulus $j$ such that $j-\alpha$ is a unit. This implies explicit bounds on the number of these singular moduli.