Search

For a modular elliptic curve $E/\mathbb{Q}$, we show a number of links between the primes $\ell $ for which the mod $\ell $ representation of $E/\mathbb{Q}$ has projective dihedral image and congruence primes for the newform associated to $E/\mathbb{Q}$.

Let $f(x, y)$ be a binary cubic form with integral rational coefficients, and suppose that the polynomial $f(x, y)$ is irreducible in $\mathbb{Q}[x, y]$ and no prime divides all the coefficients of $f$. We prove that the set $f(\mathbb{Z}^{2})$ contains infinitely many primes unless $f(a, b)$ is even for each $(a, b)$ in $\mathbb{Z}^{2}$, in which case the set $\frac{1}{2}f(\mathbb{Z}^{2})$ contains infinitely many primes.

2000 Mathematical Subject Classification: primary 11N32; secondary 11N36, 11R44.

Search Results

Refine search

Refine search

Actions for selected content:

2 results

Surjectivity of mod ℓ Representations Attached to Elliptic Curves and Congruence Primes

PRIMES REPRESENTED BY BINARY CUBIC FORMS

Search Results

Refine search

Refine search

Actions for selected content:

Save Search

2 results

Surjectivity of mod ℓ Representations Attached to Elliptic Curves and Congruence Primes

PRIMES REPRESENTED BY BINARY CUBIC FORMS