Number Theory Seminar HS 2020
Because of the Coronavarius pandemic all talks after March 13 must be cancelled or moved online.
|30.01.2020||Primes dividing singular moduli||A remarkable property of singular invariants of CM elliptic curves (singular moduli) is that they are always algebraic integers. Hence it makes sense to ask whether, for a fixed set of rational primes S, there exist singular moduli that are S-units. When the set S is empty, Yu. Bilu, P. Habegger and L. Kühne have given a negative answer to the question, proving that singular units do not exist. In this talk we show how the same result holds for the infinite set S of primes congruent to 1 modulo 3 and discuss the more general problem of understanding the prime factorization of singular moduli.|
|20.02.2020||Isogenies of elliptic curves over function fields||This talk will focus on elliptic curves over function fields and their isogenies. In recent work (in progress) with Fabien Pazuki, we aim at proving, in the function field setting, analogues of two famous theorems concerning elliptic curves over number fields. The first of these describes the effect of an isogeny on the Weil height of the j-invariant of an elliptic curve. The second one is an ``isogeny estimate’’ in the spirit of theorems by Masser—Wüstholz and by Gaudron—Rémond. I will state our results and sketch their proof. I will try to highlight the similarities and differences between the number field and the function field cases.|