“The pursuit of mathematics is a divine madness of the human spirit.”

Alfred North Whitehead

In this post, we prove the uniformization theorem for elliptic curves. The theorem states that every elliptic curve over the complex numbers arose from the complex plane modulo a lattice, and vice versa. In fact there is an isomorphism between them which is both complex analytic and algebraic. This shows that elliptic curves over complex numbers is a torus.

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It is a well known fact that there is no general formula for quintic equations or algebraic equations of any higher degree, and a typical proof of this fact uses heavy machinery from Galois theory. However, there is a far more elementary but much less well known proof by V.I. Arnold using nothing more than basic knowledge of complex numbers and topology. Last week, I gave a talk on the Short Attention Span Math Seminars organized by the Pure Math Club at University of Waterloo. In my talk, I explained the main idea of his proof: moving the coefficients along loops to induce permutations of the roots. I also talked about how it could be turned into a rigorous proof using Riemann surfaces of algebraic functions and their monodromy groups, as well as its connections to Galois theory. In the end, I also discussed briefly some ideas related to this proof. Here are my slides.

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“In mathematics the art of proposing a question must be held of higher value than solving it.”

George Cantor

This week I finally finished my final project for my PMATH 499 reading course in arithmetic geometry. My project is called “Galois cohomology and weak Mordell–Weil theorem”, which is available here. Galois cohomology is the application of group cohomology to Galois groups. In particular, we know that for a perfect field \(k\) with an algebraic closure \(K\), the Galois group \(\mathrm{Gal}(K/k)\) is isomorphic to an inverse limit of topological Galois groups \(\mathrm{Gal}(L/k)\) ranging over the finite Galois extensions \(L/k\) with the natural projections. This fact makes \(G\) a profinite group, and the Krull topology, the topology of \(G\) endowed by the inverse limit process by viewing each finite Galois group as a topological group with the discrete topology, is a topology where a basis for the neighborhood at the identity is the collection of normal subgroups having finite index in \(G\). This is a motivation to use homological algebra in such situation.

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