[Paper Review] Local Points on Quadratic Twists of X_0(N)

Ekin Özman|ArXiv.org|Nov 24, 2009

Algebraic Geometry and Number Theory18 references3 citations

TL;DR

This paper provides necessary and sufficient conditions for the existence of $\mathbb{Q}_p$-rational points on quadratic twists $X^d(N)$ of the modular curve $X_0(N)$, resolving a question by Ellenberg on local points everywhere. It establishes that for odd primes $p$ not simultaneously ramified in $\mathbb{Q}(\sqrt{d})$ and $\mathbb{Q}(\sqrt{-N})$, local points exist precisely under specific congruence and splitting conditions on $N$ and $d$, and shows that some curves with local points everywhere violate the Hasse principle, with the obstruction explained by the Brauer-Manin obstruction.

ABSTRACT

Let X^d(N) be the quadratic twist of the modular curve X_0(N) through the Atkin-Lehner involution w_N and a quadratic extension Q(\sqrt{d})/Q. The points of X^d(N)(Q) are precisely the Q(\sqrt{d})-rational points of X_0(N) that are fixed by σcomposition w_N, where σis the generator of Gal(Q(\sqrt{d})/Q).Ellenberg asked the following question: For which d and N does X^d(N) have rational points over every completion of Q? Given (N,d,p) we give necessary and sufficient conditions for the existence of a Q_p-rational point on X^d(N), whenever p is not simultaneously ramified in Q(\sqrt{d}) and Q(\sqrt{-N}), answering Ellenberg's question for all odd primes p when (N,d)=1. The main theorem yields a population of curves which have local points everywhere but no points over Q; in several cases we show that this obstruction to the Hasse Principle is explained by the Brauer-Manin obstruction.

Motivation & Objective

• To answer Ellenberg's question on when quadratic twists $X^d(N)$ have rational points over every completion of $\mathbb{Q}$.
• To characterize the existence of $\mathbb{Q}_p$-rational points on $X^d(N)$ for all primes $p$, under the condition that no prime is simultaneously ramified in $\mathbb{Q}(\sqrt{d})$ and $\mathbb{Q}(\sqrt{-N})$.
• To identify cases where curves have local points everywhere but no global rational points, and to determine whether such failures of the Hasse principle are explained by the Brauer-Manin obstruction.
• To extend and unify previous results on local points for twists of $X_0(N)$, particularly in the context of $\mathbb{Q}$-curves and modular curves.

Proposed method

• Uses the theory of Atkin-Lehner involutions and étale descent to define the quadratic twist $X^d(N)$ of $X_0(N)$ over $\mathbb{Q}$.
• Identifies $\mathbb{Q}$-rational points on $X^d(N)$ with $\mathbb{Q}(\sqrt{d})$-rational points on $X_0(N)$ fixed by $\sigma \circ w_N$, where $\sigma$ generates $\mathrm{Gal}(\mathbb{Q}(\sqrt{d})/\mathbb{Q})$.
• Applies Hensel's Lemma to analyze $\mathbb{Q}_p$-points when $p$ divides $N$ and $p$ is inert in $\mathbb{Q}(\sqrt{d})$.
• Employs the theory of complex multiplication (CM) elliptic curves to construct $\mathbb{Q}_p$-points when $p$ is ramified in $\mathbb{Q}(\sqrt{d})$ and $p \nmid N$.
• Uses Scharaschkin's method and computations in Magma to analyze the Mordell-Weil group of the Jacobian and rule out rational points in specific cases.
• Applies the Weil formula for functional equations of $L$-functions to deduce that certain Jacobians are simple and have no nontrivial quotients with rank 0, ruling out Mazur-style methods.

Experimental results

Research questions

• RQ1 For which integers $d$ and square-free $N$ does the quadratic twist $X^d(N)$ have a $\mathbb{Q}_p$-rational point for every prime $p$?
• RQ2 Under what conditions on $d$ and $N$ does $X^d(N)$ have local points everywhere but no global rational points over $\mathbb{Q}$?
• RQ3 Can the failure of the Hasse principle for such curves be explained by the Brauer-Manin obstruction?
• RQ4 How do the ramification behavior of $p$ in $\mathbb{Q}(\sqrt{d})$ and $\mathbb{Q}(\sqrt{-N})$ affect the existence of local points?
• RQ5 What is the structure of the Mordell-Weil group of the Jacobian of $X^d(N)$, and how does it influence the existence of rational points?

Key findings

• For odd primes $p$ inert in $\mathbb{Q}(\sqrt{d})$ and dividing $N$, $X^d(N)$ has a $\mathbb{Q}_p$-point if and only if $N = p \prod q_i$ with $p \equiv 3 \pmod{4}$ and all $q_i \equiv 1 \pmod{4}$, or $N = 2p \prod q_i$ under the same congruence conditions.
• When $2$ is inert in $\mathbb{Q}(\sqrt{d})$ and divides $N$, a $\mathbb{Q}_2$-point exists if and only if $N = 2 \prod q_i$ with all $q_i \equiv 1 \pmod{4}$.
• For odd primes $p$ ramified in $\mathbb{Q}(\sqrt{d})$ and not dividing $N$, a $\mathbb{Q}_p$-point exists if and only if $p$ lies in a specific finite set $S$ defined in Proposition 5.5.
• The curve $X^{17}(23)$ has local points everywhere but no $\mathbb{Q}$-rational points, and this failure of the Hasse principle is explained by the Brauer-Manin obstruction.
• For $d \in \{-223, -211, -59, 101, 173\}$, the Jacobian of $X^d(23)$ has trivial torsion and rank 0, implying $X^d(23)(\mathbb{Q}) = \emptyset$.
• The Jacobian of $X^{17}(23)$ is simple and has rank 2, with $q$-expansion coefficients in $\mathbb{Q}(\sqrt{-5})$, and the associated newform is self-conjugate, confirming the absence of nontrivial isogeny factors.