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On Thursday Shou-wu Zhang concluded his lecture series by talking about the higher genus case , and in particular focusing on some recent work of his which is related to the effective Mordell conjecture and the abc conjecture. The higher genus case is substantially more difficult than the genus 0 or genus 1 cases, and one often needs to use techniques from many different areas of mathematics (together with one or two unproven conjectures) to get somewhere.
This is perhaps the most technical of all the talks, but also the closest to recent developments, in particular the modern attacks on the abc conjecture. (Shou-wu made the point that one sometimes needs to move away from naive formulations of problems to obtain deeper formulations which are more difficult to understand, but can be easier to prove due to the availability of tools, structures, and intuition that were difficult to access in a naive setting, as well as the ability to precisely formulate and quantify what would otherwise be very fuzzy analogies.)
On Wednesday, Shou-wu Zhang continued his lecture series. Whereas the first lecture was a general overview of the rational points on curves problem, the second talk focused entirely on the genus 1 case – i.e. the problem of finding rational points on elliptic curves. This is already a very deep and important problem in number theory – for instance, this theory is decisive in Wiles’ proof of Fermat’s last theorem. It was also somewhat more technical than the previous talk, and I had more difficulty following all the details, but in any case here is my attempt to reconstruct the talk from my notes. Once again, the inevitable inaccuracies here are my fault and not Shou-wu’s, and corrections or comments are greatly appreciated.
NB: the talk here seems to be loosely based in part on Shou-wu’s “Current developments in Mathematics” article from 2001.
[This lecture is also doubling as this week's "open problem of the week", as it discusses the Birch and Swinnerton-Dyer conjecture and the effective Mordell conjecture.]
Like many other maths departments, UCLA has a distinguished lecture series for eminent mathematicians to present recent developments in a field of mathematics, both to a broad audience and to specialists. Unlike most departments, though, our lecture series goes by the descriptive (but unimaginative) name of “Distinguished Lecture Series“, supported by the Gill Foundation. This week the lecture series is given by Shou-wu Zhang from Columbia, and revolves around the topic of rational points on curves, a key subject of interest in arithmetic geometry and number theory. The first of three talks, which was on Tuesday, was a very accessible and enjoyable overview talk, which I am reproducing here (to use this opportunity to learn this stuff myself, and also to continue the diversification of subject matter here on this blog). As before, I do not vouch for 100% accuracy, and all errors are my responsibility rather than Shou-wu’s.