Quadratic points on modular curves

Ozman, Ekin; Siksek, Samir

doi:10.1090/mcom/3407

Quadratic points on modular curves
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Math. Comp. 88 (2019), 2461-2484 Request permission

Abstract:

In this paper we determine the quadratic points on the modular curves $X_0(N)$, where the curve is non-hyperelliptic, the genus is $3$, $4$, or $5$, and the Mordell–Weil group of $J_0(N)$ is finite. The values of $N$ are $34$, $38$, $42$, $44$, $45$, $51$, $52$, $54$, $55$, $56$, $63$, $64$, $72$, $75$, $81$.

As well as determining the non-cuspidal quadratic points, we give the $j$-invariants of the elliptic curves parametrized by those points, and determine if they have complex multiplication or are quadratic $\mathbb {Q}$-curves.

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