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Archivio digitale delle tesi discusse presso l’Università di Pisa

Tesi etd-10022021-165356


Tipo di tesi
Tesi di laurea magistrale
Autore
FURIO, LORENZO
URN
etd-10022021-165356
Titolo
Finiteness of Multiplicatively Dependent n-tuples of Singular Moduli
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
relatore Prof. Bilu, Yuri
correlatore Dott. Lombardo, Davide
Parole chiave
  • algebraic number theory
  • diophantine geometry
  • elliptic curves
  • number theory
  • Pila-Wilkie
  • singular moduli
  • unlikely intersections
Data inizio appello
29/10/2021
Consultabilità
Tesi non consultabile
Riassunto
The aim of the thesis is to explain a result of Pila and Tsimerman stating that for every positive integer n there exist finitely many multiplicatively dependent n-tuples of singular moduli, excluding trivial cases with dependent subsets. The main ingredient of the proof is the theorem of Pila-Wilkie. In particular, one defines a particular set X of quadratic points, corresponding to the dependent n-tuples of singular moduli, then with some arithmetic estimates finds a lower bound for the number of points with bounded height, depending on the height. The theorem of Pila-Wilkie shows that the number of quadratic points in the transcendental part of X grows less than the lower bound, then they must be contained in the algebraic part of X. With some mixed Ax-Schanuel results we show that an algebraic subvariety of X containing a singular-dependent n-tuple must be atypical and with some other work we show that such a subvariety cannot exist. This concludes the proof.
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