Modular curves and isolated points
Linfoot Number Theory Seminar
19th February 2025, 11:00 am – 12:00 pm
Fry Building, 2.04
Modular curves are objects of central importance in arithmetic geometry, parametrizing elliptic curves with particular Galois representations. They form a key part of the proof of results such as Fermat's Last Theorem and Mazur's torsion theorem. On the other hand, isolated points are "exceptional" low-degree points on curves, which lie outside the infinite families of low-degree points effected by the geometry of the curve. In this talk, I will aim to give a gentle introduction to these two concepts, the former via a stroll through the world of moduli spaces, the latter via an examination of Faltings's proof of the Mordell conjecture. In particular, little to no prior knowledge will be assumed. Time permitting, I will conclude with some recent advances on the intersection of these two notions.
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