- I was contacted by Giancarlo Urzúa with an intriguing question about configurations of rational curves on elliptic surfaces. We eventually resolved it and found interesting applications: A result on square-freeness of elliptic divisibility sequences over function fields which seems to be much stronger than anything proven before, and an application to geography of surfaces, where we showed the existence of mildly singular surfaces with arbitrary geometric genus, ample canonical class, and with K2 arbitrarily large. (For smooth surfaces, Noether's formula bounds K2 in terms of the geometric genus.) You can read all about in our preprint on the arxiv.
- More recently, we found a strong upper bound on the number of tangencies considered in the paper mentioned just above. See our preprint for details.
- Richard Griffon and I studied the arithmetic of a family of twisted-constant elliptic curves over function fields, showing among other things that the Tate-Shafarevich group exhibits interesting behavior: In all cases we study it is large (as measured by the Brauer-Siegel ratio), but it may be a p-group (p = the characteristic) or it may have order prime to p. Read the preprint for more details.
- I led a group of eight participants at a recent AIM workshop on a project to write down higher genus analogues of the Legendre curve and prove results similar to those in my papers 2014a, 2014c, and 2014d. We've completed a long paper (131 pages!) which is in press at the Memoirs of the AMS.
I have projects for students at several levels. For the last few years, I have enjoyed working with advanced undergraduates on semester-length projects in number theory and related areas. I'd also be happy to talk with graduate students about projects in number theory and algebraic geometry ranging from a quick MS with good prospects for a paper to more substantial PhD projects.
The fastest and most reliable way to reach me is via e-mail at firstname.lastname@example.org. To make an appointment, please contact my colleagues Alejandra (Ali) Gaona (520-621-2868, email@example.com) or Aubrey Mouradian (520-621-2713, firstname.lastname@example.org).
My short CV.
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