Research

Publications and Preprints


Abstract: Let $\mathcal{O}_K$ be a Henselian discrete valuation domain with field of fractions $K$. Assume that $\mathcal{O}_K$ has algebraically closed residue field $k$. Let $E/K$ be an elliptic curve with additive reduction. The semi-stable reduction theorem asserts that there exists a minimal extension $L/K$ such that the base change $E_L/L$ has semi-stable reduction.

It is natural to wonder whether specific properties of the semi-stable reduction and of the extension $L/K$ impose restrictions on what types of Kodaira type the special fiber of $E/K$ may have. In this paper we study the restrictions imposed on the reduction type when the extension $L/K$ is wildly ramified of degree $2$, and the curve $E/K$ has potentially good supersingular reduction. We will also analyze the question of possible reduction types of two isogenous elliptic curves with these properties.

Abstract: The classical modular polynomials are used to define the modular curves $X_0(N)/\mathbb{Q}$ and have been extensively studied. Motivated by computations, this article introduces some conjectures on congruences modulo powers of the primes $2,3$ and $5$ satisfied by the coefficients of these polynomials. As a step towards a possible proof of the conjectures, we provide closed formulas for $\ell$ nontrivial coefficients of the classical modular polynomials in terms of the Fourier coefficients of the modular invariant function $j(z)$. We deduce congruences from these formulas supporting the conjectures.