Research
Publications and Preprints
On the Kodaira types of Elliptic curves with potentially good supersingular reduction, preprint, (2024) (manuscript available upon request).
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.
Congruence properties of the coefficients of the classical modular polynomials, preprint, (2024) (manuscript available upon request).
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.
Rational spherical triangles, preprint, (2023). pdf
Integer dynamics, with D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, Int. J. Number Theory 18 (2022), no. 2, 397–415. pdf
Orthogonal multiplications of type [3, 4, p], p \leq 12, with Q. S. Chi, Beitraege zur Algebra und Geometrie/Contributions to Algebra and Geometry 59 (2018), 167-197. pdf