Related papers: On Heegner points of large conductors

Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/Q and non-torsion point P in E(Q), there is at most one…

In this work, we extend Howard's construction of compatible families of Heegner points to the setting of towers of Gross curves and Shimura curves over totally real fields. Following the strategy of Longo and Vigni, our approach…

We describe all special curves in the parameter space of complex cubic polynomials, that is all algebraic irreducible curves containing infinitely many post-critically finite polynomials. This solves in a strong form a conjecture by Baker…

We present several new heuristic algorithms to compute class polynomials and modular polynomials modulo a prime $p$ by revisiting the idea of working with supersingular elliptic curves. The best known algorithms to this date are based on…

We study the modular curves defined by Weber functions, and associated modular polynomials, action of $\mathrm{SL}_2(\mathbb{Z})$, and parametrizations of elliptic curves with a view to the study of the isogeny graphs that they determine,…

Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large. More precisely, we consider certain infinite…

Given a rational elliptic curve E, a suitable imaginary quadratic field K and a quaternionic Hecke eigenform g of weight 2 obtained from E by level raising such that the sign in the functional equation for L_K(E,s) (respectively, L_K(g,1))…

Veech groups uniformize Teichm\"uller geodesic curves in Riemann moduli space. Recently, examples of infinitely generated Veech groups have been given. We show that these can even have infinitely many cusps and infinitely many infinite…

In the classical setting, the modular equation of level $N$ for the modular curve $X_0(1)$ is the polynomial relation satisfied by $j(\tau)$ and $j(N\tau)$, where $j(\tau)$ is the standard elliptic $j$-function. In this paper, we will…

This work is motivated by two central questions in the birational geometry of moduli spaces of curves -- Fulton's conjecture and the effective cone of $\bar M_g$. We study the algebro-geometric aspect of Teichmuller curves parameterizing…

We exhibit $88$ examples of rank zero elliptic curves over the rationals with $|\sza(E)| > 63408^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $|\sza(E)| = 1029212^2 =…

We consider a system of quasilinear elliptic equations, with indefinite super-linear nonlinearity, depending on two real parameters $\lambda,\mu$. By using the Nehari manifold and the notion of extremal parameter, we extend some results…

We implement an algorithm to compute the number of points over finite fields for the Shimura curves $X_0^D(N)$ and their Atkin--Lehner quotients. Our computations result in many examples of curves which attain the largest known point counts…

We characterize the possible reductions of $j$-invariants of elliptic curves which admit complex multiplication by an order $\mathcal{O}$ where the curve itself is defined over $\mathbb{Z}_p$. In particular, we show that the distribution of…

In this article, we study the family of elliptic curves $E/\mathbb{Q}$, having good reduction at $2$ and $3$, and whose $j$-invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set…

Let $K$ be a composite field of some real quadratic fields. We give a sufficient condition on $K$ such that all elliptic curves over $K$ is modular.

Let $E$ be an elliptic curve defined over a number field $K$. We say that a prime number $p$ is exceptional for $(E,K)$ if $E$ admits a $p$-isogeny defined over $K$. The so-called exceptional set of all such prime numbers is finite if and…

For an elliptic curve E over an abelian extension k/K with CM by K of Shimura type, the L-functions of its [k:K] Galois representations are Mellin transforms of Hecke theta functions; a modular parametrization (surjective map) from a…

We parametrize the commensurability classes of curves on Shimura surfaces that are totally geodesic, i.e., the commensurability classes of so-called $\mathbb{C}$-Fuchsian subgroups. In particular, if a Shimura surface contains one…

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. We give a complete classification of when…