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SUMMARY:Levent Alpoge (Columbia University)
DTSTART:20210318T210000Z
DTEND:20210318T220000Z
DTSTAMP:20260423T024018Z
UID:JNTS/17
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/JNTS/17/">Ef
 fective height bounds for odd-degree totally real points on some curves</a
 >\nby Levent Alpoge (Columbia University) as part of Columbia CUNY NYU num
 ber theory seminar\n\n\nAbstract\nLet $\\mathcal O$ be an order in a total
 ly real field $F.$\n Let $K$ be an odd-degree totally real field. Let $S$ 
 be a finite set of places of $K.$ We study $S$-integral $K$-points on inte
 gral models $H_\\mathcal O$ of Hilbert modular varieties because not only 
 do said varieties admit complete curves (thus reducing questions about suc
 h curves' $K$-rational points to questions about $S$-integral $K$-points o
 n these integral models)\, they also have their $S$-integral $K$-points co
 ntrolled by known cases of modularity\, in the following way. First assume
  for clarity modularity of all $\\text{\\rm GL}_2$-type abelian varieties 
 over $K$ --- then all $S$-integral $K$-points on $H_{\\mathcal O}$ \n aris
 e from K-isogeny factors of the \n $[F:\\mathbb Q]$-th power of the Jacobi
 an of a single Shimura curve with level structure (by Jacquet-Langlands tr
 ansfer). By a generalization of an argument of von Kanel\, isogeny estimat
 es of Raynaud/Masser-Wustholz and Bost's lower bound on the Faltings heigh
 t suffice to then bound the heights of all points in $H_{\\mathcal O}(\\ma
 thcal O_{K\,S}).$ \n As for the assumption\, though modularity is of cours
 e not known in this generality\, by following Taylor's (sufficiently expli
 cit for us) proof of his potential modularity theorem we are able to make 
 the above unconditional.\n\n\n \nFinally we use the hypergeometric abelian
  varieties associated to the arithmetic triangle group $\\Delta(3\,6\,6)$ 
 to give explicit examples of curves to which the above height bounds apply
 . Specifically\, we prove that\, for $a\\in \\overline{\\Bbb Q}^\\times$\n
 totally real of odd degree (e.g. $a = 1$)\, for all $L/\\Bbb Q(a)$ totally
  real of odd degree and $S$ a finite set of places of $L\,$ there is an ef
 fectively computable $c = c_{a\,{\\scriptscriptstyle L}\,{\\scriptscriptst
 yle S}}\\in \\Bbb Z^+$  such that all $x\,y\\in L$ satisfying $x^6 + 4y^3 
 = a^2 $ satisfy $h(x) < c.$ Note that this gives infinitely many curves fo
 r each of which Faltings' theorem is now effective over infinitely many nu
 mber fields.\n
LOCATION:https://researchseminars.org/talk/JNTS/17/
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