Quadratic Points on Non-Split Cartan Modular Curves

Abstract

In this paper we study quadratic points on the non-split Cartan modular curves $X_{ns}(p)$, for $p=7,11,$ and $13$. Recently, Siksek proved that all quadratic points on $X_{ns}(7)$ arise as pullbacks of rational points on $X_{ns}^{+}(7)$. Using similar techniques for $p=11$, and employing a version of Chabauty for symmetric powers of curves for $p=13$, we show that the same holds for $X_{ns}(11)$ and $X_{ns}(13)$. As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular.

arXiv comment18 pages. To appear in International Journal of Number Theory. Minor corrections and rewriting. Slightly improved sieving step and more computation details included

References in corpus (5)

• The Mordell-Weil sieve: Proving non-existence of rational points on curves
• Hyperelliptic modular curves $X_0(n)$ and isogenies of elliptic curves over quadratic fields
• The automorphism group of the non-split Cartan modular curve of level 11
• On the automorphisms of the non-split Cartan modular curves of prime level
• Double Covers of Cartan Modular Curves

Cited by in corpus (2)

• Computing quadratic points on modular curves $X_0(N)$
• Modular Curves with many Points over Finite Fields