IHP (Paris)/ ICTP (Trieste)1From: "Stephen S. Kudla" <[email protected]>2To: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]3Date: Sun, 21 Jan 2001 16:10:31 -0500456Dear IHP/ICTP participant!78We are currently in the process of writing up a final9report on the NSF Grant which provided support for US10participants in the program at the IHP in Paris last spring and the11program at the ICTP in Trieste in August. Could you please send us a12couple of sentences about your activities (e.g., what13lecture series you attended, who you met and talked math with, what14work you did there, or how participation in the program benefited your15work/ expanded your horizons, etc.) This information will make our task16very much easier and will hopefully encourage the NSF17to fund such activites in the future!1819Finally, it would be helpful if20you could take a moment to do this right away21(rather than sending information in a rush later), since22we are under some pressure to finish this chore this month.2324Best wishes!2526Steve Kudla27Freydoon Shahidi2829----------------3031OK, here I go:3233What I want to say:341) Went to many of the lectures in the automorphic forms workshop.352) Finished a paper with Kevin Buzzard on Artin's conjecture.363) Made significant progress towards finishing a paper with37Loic Merel on torsion points on elliptic curves384) Proved that every element of Sha is visible somewhere, following39a conversation with Ralph Greenberg.405) Formulated a tentative strategy for constructing points on41elliptic curves of even analytic rank > 0 whose Shafarevich-Tate42is finite.434445I was very productive during the month that I visited the IHP in46Paris. I was stimulated by the talks on automorphic forms, which I47attended regularly. Kevin Buzzard and I finished a joint paper on48Artin's conjecture about icosahedral Galois representation, which was49the culmination of a three-year project. Loic Merel and I had several50intense discussions about torsion points on elliptic curves, and made51progress that allowed us to nearly finish a joint paper. Building on52a dinner conversation with Ralph Greenberg, I noticed that there is a53simple proof that every element of the Shafarevich-Tate group of an54abelian variety is visible somewhere, thus improving on an earlier55observation of Johan de Jong. I also formulated a tentative strategy56for constructing points on elliptic curves of analytic rank > 0 whose57Shafarevich-Tate group is finite. I described this strategy in an58Utrecht talk in honor of Franz Oort, which I delivered immediately59after my stay in Paris.60616263