Introduction to Galois cohomology

Jeremy Rouse

October 2, 4-5 pm. Manchester 124.

Let $E$ be an elliptic curve and $\alpha$ be a rational point on $E$. For each prime number $p$, we can reduce $\alpha \bmod p$ and obtain a point in $E(\mathbb{F}_{p})$, which is a finite group. How often is the order of this point odd? There are many tools needed to study this question, and this talk will be an introduction to those tools, chief among them being the cohomology group $H^{1}(G,M)$. In addition to the motivating question above, these tools can be used to classify twists of an elliptic curve (given an elliptic curve $E/\mathbb{Q}$, what are elliptic curves $E'/\mathbb{Q}$ that are not isomorphic to $E$ over $\mathbb{Q}$, but become isomorphic over some extension $K/\mathbb{Q}$?), and to describe the theory of descent (which is used to compute the rank of an elliptic curve).

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Last modified: Tuesday, 24-Sep-2019 10:36:19 EDT