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SUMMARY:Sachi Hashimoto (Brown University)
DTSTART:20250909T203000Z
DTEND:20250909T213000Z
DTSTAMP:20260423T130711Z
UID:MITNT/118
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MITNT/118/">
 Rational points on $X_0(N)^*$ when $N$ is non-squarefree</a>\nby Sachi Has
 himoto (Brown University) as part of MIT number theory seminar\n\nLecture 
 held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nThe r
 ational points of the modular curve $X_0(N)$ classify pairs $(E\,C_N)$ of 
 elliptic curves over $\\mathbb{Q}$ together with a rational cyclic subgrou
 p of order $N$. The curve $X_0(N)^*$ is the quotient of $X_0(N)$ by the fu
 ll group of Atkin-Lehner involutions. Elkies showed that the rational poin
 ts on this curve classify elliptic curves over the algebraic closure of $\
 \mathbb{Q}$ that are isogenous to their Galois conjugates\, and conjecture
 d that when $N$ is large enough\, the points are all CM or cuspidal. In jo
 int work with Timo Keller and Samuel Le Fourn\, we study the rational poin
 ts on the family $X_0(N)^*$ for N non-squarefree. In particular we will re
 port on some integrality results for the $j$-invariants of points on $X_0(
 N)^*$.\n
LOCATION:https://researchseminars.org/talk/MITNT/118/
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