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SUMMARY:Rational points on $X_0(N)^*$ and local heights in quadratic Chaba
 uty [MPIM]
DTSTART:20261104T133000Z
DTEND:20261104T143000Z
DTSTAMP:20261030T013300Z
UID:indico-event-1715@math-events.uni-bonn.de
DESCRIPTION:Speakers: Nikola Adžaga (University of Zagreb/z. Z. MPIM)\n\n
 Number theory lunch seminar\nRational points on the full Atkin--Lehner quo
 tient $X_0(N)^*$ correspond to $\\mathbb{Q}$-curves\, and Elkies conjectur
 ed that $X_0(N)^*$ has exceptional (non-cuspidal\, non-CM) rational points
  for only finitely many levels $N$. Quadratic Chabauty has become the main
  tool for provably determining the rational points on these curves. After 
 briefly surveying joint work on the hyperelliptic curves $X_0(N)^*$ and on
  $X_0^+(p)$ of genus 4--6\, I will focus on recent joint work with Maarten
  Derickx and Timo Keller. That work removes the main obstacle in these com
 putations\, namely the determination of local heights at primes of bad red
 uction. For squarefree $N$\, we describe the dual graphs of semi-stable re
 gular models of $X_0(N)^*$. Their long cycles impose linear conditions on 
 the correspondence\, and we bound the number of these conditions using emb
 eddings of quadratic orders into quaternion algebras. As a result\, for al
 l but finitely many squarefree $N$\, we can choose a correspondence for wh
 ich the local heights at all primes of bad reduction vanish. As an applica
 tion\, we determine the rational points on several new curves $X_0(N)^*$.\
 n\nhttps://math-events.uni-bonn.de/event/1715/
LOCATION:MPIM\, Vivatsgasse\,  7 - Lecture Hall (Max Planck Institute for 
 Mathematics)
URL:https://math-events.uni-bonn.de/event/1715/
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