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The complexity of the elliptic curve method of factorization (ECM) is proven under the celebrated conjecture of existence of smooth numbers in short intervals. In this work we tackle a different version of ECM which is actually much more…

Let $E$ be an elliptic curve defined over $\mathbf{Q}$ without complex multiplication. For each prime $\ell$, there is a representation $\rho_{E,\ell}\colon \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \text{GL}_2(\mathbf{F}_{\ell})$…

We prove that integral points can be effectively determined on all but finitely many modular curves, and on all but one modular curve of prime power level.

We show that the shape of a complex cubic field lies on the geodesic of the modular surface defined by the field's trace-zero form. We also prove a general such statement for all orders in \'etale Q-algebras. Applying a method of Manjul…

We consider the family of dynamical modular curves associated to quadratic polynomial maps and determine precisely which of these curves have infinitely many cubic points. We use this to prove a classification statement on preperiodic…

From the Modularity Theorem proven by Wiles, Taylor, et al, we know that all elliptic curves are modular. It has been shown by Martin and Ono exactly which are represented by eta-quotients, and some examples of elliptic curves represented…

Given a smooth cubic hypersurface $X$ over a finite field of characteristic greater than 3 and two generic points on $X$, we use a function field analogue of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number…

In this study, we determine all modular curves $X_0(N)$ that admit infinitely many cubic points.

Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and…

Given a lattice polygon, we study the moduli space of all tropical plane curves with that Newton polygon. We determine a formula for the dimension of this space in terms of combinatorial properties of that polygon. We prove that if this…

Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j_{K,f})$,…

In this paper, we study the finiteness problem of torsion points on an elliptic curve whose coordinates satisfy some multiplicative dependence relations. In particular, we prove that on an elliptic curve defined over a number field there…

This paper presents the theory of holomorphic vector valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are…

Following the work of Mestre, we use Weil's explicit formulas to compute explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields. Moreover, we obtain bounds for the conductor of elliptic curves…

We prove that any convex domain of C^2 carries properly embedded complete complex curves. In particular, we exhibit the first examples of complete bounded embedded complex curves in C^2

Baba and Granath generalize Elkies' theorem on infinitude of supersingular primes for elliptic curves to abelian surfaces with quaternionic multiplication of discriminant $6$, whose field of moduli is $\mathbb{Q}$ and which is a Jacobian in…

We consider the class of curves of finite total curvature, as introduced by Milnor. This is a natural class for variational problems and geometric knot theory, and since it includes both smooth and polygonal curves, its study shows us…

We study the arithmetic (geometric) progressions in the $x$-coordinates of quadratic points on smooth projective planar curves defined over a number field $k$. Unless the curve is hyperelliptic, we prove that these progressions must be…

We show how the modular symmetries that have been found to be consistent with most available scaling data from quantum Hall systems, derive from a rigid family of algebraic curves of the elliptic type. The complicated special functions…

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times…