Thursday, November 18, 2010

Current Research Papers and Books

This post is about the papers I'm working on these days. Each of the papers below are papers that wouldn't be too difficult to finish and submit. As far as I know, most of the serious theoretical issues in all of the papers are resolved. I'm also putting links below into a snapshot of each of the papers as of this post (which might vanish or get updated in the future).

  1. Calegari-Stein: Non-eisenstein descent
    • Numerous small technical questions to answer, still
    • Most of it should now be do-able in Sage; patch up anything that isn't
  2. Stein: Computing End(Af) and minimal degree of Af^ --> Af
    • The computations that I did with Tseno Tselkov a few years ago...
    • Polish and publish
    • More data should be computable these days
    • I'm not sure the algorithm is even implemented in Sage, but it shouldn't be too hard
  3. Kamienny-Stein-Stoll: Quartic torsion points
    • Easy approach: just let it depend on Magma. Lame. Or just put the Magma part in a separate paper not by me (use a pseudonym or just Stoll).
    • Harder: determine cupsidal torsion subgroup using new ideas; at least helps
    • Hard as hell: implement Hesse's algorithm in Sage for this (this is what I want to do -- it's weak to apply Hesse to solve this problem, if I don't truly understand it, anyways.)
    • Do a better writeup of the intro based on talks I've given
  4. Stein: Kolyvagin's conjecture
    • First ever verification of it
    • Interesting theoretical results about kolyvagin derivative: highlight in intro
    • Just needs polish, at this point. Maybe improve algorithm and redo tables, better.
    • Should add more about what is in Stein-Weinstein
  5. Stein-Weinstein: Kolyvagin densities
    • Explains data from Kolyvagin conj
    • Need tex file
    • Just needs polish, I think. Jared wants to push the results too far, maybe?
  6. Stein-Wuthrich: Aplying p-adic methods to bound Shafarevich-Tate groups
    • Need to apply it to get some big interesting tables/computational results, or what is the point!?
    • There is a weird merge issue regarding reducible case
  7. Bradshaw-Stein: Provable computation of motivic L-functions
    • Put his thesis into a paper, and give it my spin and numerical data
    • Motivate with quote from Dokchitser's paper; once this paper is published, people will be able to reference it when claiming the existence of algorithms to do blah, blah, even if they don't actually use it as such.
    • Just a part of the thesis!
  8. Bardshaw-Stein: A conjectural Kolyvagin-style bound over ring
    class fields for curves with analytic rank at least 2
    • Based on Robert's calculations in his thesis
    • Very easy sell to publish this in a good journal
    • Can be viewed as a sort of follow up to a Bertolini-Darmon paper
  9. Mazur-Stein: A popular book on the Riemann-Hypothesis
    • Tricky because of theoretical issues with non-tempered disributions, which we're confronted with. Despite being a popular book, it's definitely interesting research, as is the case with anything Barry touches!
  10. Ribet-Stein: Lectures on Modular Forms and Hecke operators
    • Just needs polish and more examples.
    • Did get some work done polishing this when I taught out of it in 2003

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