Related papers: Modular curves and bad reduction

Let $C$ be a genus $2$ curve with Jacobian isomorphic to the square of an elliptic curve with complex multiplication by a maximal order in an imaginary quadratic field of discriminant $-d<0$. We show that if the stable model of $C$ has bad…

We prove the non-existence of elliptic curves having good reduction everywhere over some real quadratic fields.

We compute the Mordell-Weil groups of the modular Jacobian varieties of hyperelliptic modular curves $X_1(M, MN)$ over every number field which is the composition of quadratic fields. Also we prove criteria for the existence of elliptic…

Let E be an elliptic curve with complex multiplication by R, where R is an order of discriminant D<-4 of an imaginary quadratic field K . If a prime number p is decomposed completely in the ring class field associated with R, then E has…

We study negative curves on surfaces obtained by blowing up special configurations of points in the complex projective palne. Our main results concern the following configurations: very general points on a cubic, 3-torsion points on an…

We complete the classification of torsion subgroups $E(K)_{\text{tors}}$ that can occur for an elliptic curve $E/\mathbb{Q}$ over a sextic number field $K$. Previous work determined the complete set of these groups, leaving the existence of…

For certain elliptic curves $E/\mathbb{Q}$ with $E(\mathbb{Q})[2]=\mathbb{Z}/2 \mathbb{Z}$, we prove a criterion for prime twists of $E$ to have analytic rank 0 or 1, based on a mod 4 congruence of 2-adic logarithms of Heegner points. As an…

Given an elliptic curve E/Q and a prime p at which E has good reduction, let e_p be the exponent of the group E_p(F_p) of F_p-rational points on the reduction of E modulo p. Under the Generalized Riemann Hypothesis (GRH) for the Dedekind…

Merel has shown that the order of torsion subgroup of an elliptic curve over a number field can be bounded in terms of only the degree of the number field. The purpose of this note is to investigate what could be the `right bound'. In this…

In this work, we consider the rational points on elliptic curves over finite fields F_{p}. We give results concerning the number of points on the elliptic curve y^2{\equiv}x^3+a^3(mod p)where p is a prime congruent to 1 modulo 6. Also some…

Given an elliptic curve defined over the field of rational numbers, it is known how its torsion subgroup may grow when we make a base change to a quadratic number field. In this paper we consider the inverse question: if we have the…

For E/k an elliptic curve with CM by O, we determine a formula for (a generalization of) the arithmetic local constant of [4] at almost all primes of good reduction. We apply this formula to the CM curves defined over Q and are able to…

For a superelliptic curve $\mathcal X$, defined over $\mathbb Q$, let $\mathfrak p$ denote the corresponding moduli point in the weighted moduli space. We describe a method how to determine a minimal integral model of $\mathcal X$ such…

For a quadratic field $\mathcal{K}$ without rationally defined CM, we prove that there exists of a prime $p_{\mathcal{K}}$ depending only on $\mathcal{K}$ such that if $d$ is a positive integer whose minimal prime divisor is greater than…

In 2002 Watkins conjectured that given an elliptic curve defined over $\mathbb{Q}$, its Mordell-Weil rank is at most the $2$-adic valuation of its modular degree. We consider the analogous problem over function fields of positive…

It is a classical fact going back to F. Klein that an elliptic curve $E$ over $\bar{\mathbb{Q}}$ is defined by a homogeneous polynomial in $3$ variables with coefficients in $\mathbb{Q}(j_{E})$, where $j_{E}$ is the $j$-invariant of $E$,…

We give a classification of the cuspidal automorphic representations attached to rational elliptic curves with a non-trivial torsion point of odd order. Such elliptic curves are parameterizable, and in this paper, we find the necessary and…

We prove asymptotic formulas for cyclicity of reductions of elliptic curves over the rationals in a family of curves having specified torsion. These results agree with established conditional results and with average results taken over…

By focusing on the family $E:y^2=x^3+a$, we present strategies for determining the structure of the torsion subgroup of the Mordell-Weil group of an elliptic curve, $E(K)$, over quadratic field $K$. Generalizations of the Nagell-Lutz…

Let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be a prime of good supersingular reduction. Attached to $E$ are pairs of Iwasawa invariants $\mu_p^\pm$ and $\lambda_p^\pm$ which encode arithmetic properties of $E$ along the cyclotomic…