Patrick Ingram

This is roughly what I look like (photo by Elissa Ross)

Patrick Ingram


NSERC postdoctoral fellow
University of Toronto

























Research

My research is in number theory, and in particular diophantine equations. More specifically, I'm interested in diophantine approximation on elliptic curves or more general settings, with a bias towards "dynamical" problems.

As one particular example, let W(n) be a sequence of integers satisfying

W(m+n)W(m-n)=W(m+1)W(m-1)W(n)2-W(n+1)W(n-1)W(m)2.

Such a sequence corresponds, in a natural way, to a point on an elliptic curve, and the sequence of values n-2log|W(n)| converges to the global canonical height h(P) of that point. But how quickly? Can any broad, uniform results be shown about the rate of convergence, either effectively or ineffectively? If the limit of n-2log|W(n)| is non-zero, how small can it be? Analogous questions arise for sequences related to iteration of rational maps on a number field, say, and applications are found in questions about integral points on elliptic curves, or primitive divisors in sequence that satisfy the above relation.

Or, suppose that E(n) is a sequence of elliptic curves (over a number field K) whose j-invariants converge very rapidly to some K-rational number. What can one say about the structure of the groups E(n)(K)? Surprisingly, the torsion subgroups of these curves will be quite restricted, a consequence of diophantine approximation on some appropriate modular curves.

If you'd like a visual interpretation of my subject, click here.

Some Publications (in print/accepted)

  1. 'Torsion subgroups of elliptic curves in short Weierstrass form', Transactions of the American Mathematical Society 357 no. 8 (2005), p. 3325-3337 (joint paper with M. A. Bennett)
  2. 'On k-th power numerical centres', Comptes rendus mathematiques de l'Academie des sciences (2006)
  3. 'Diophantine analysis and torsion on elliptic curves', Proceedings of the London Mathematical Society 94 no. 1 (2007), p. 137-154
  4. 'Elliptic divisibility sequences over certain curves', Journal of Number Theory 123 issue 2 (2007), p. 473-486
  5. 'Approximating algebraic numbers by j-invariants of elliptic curves', Acta Arithmetica 131 (2008), p. 57-68
  6. 'Uniform estimates for primitive divisors in elliptic divisibility sequences' (joint paper with J. H. Silverman), to appear in a forthcoming memorial volume for Serge Lang, published by Springer-Verlag
  7. 'Primitive divisors in arithmetic dynamics', to appear in Mathematical Proceedings of the Cambridge Philosophical Society (joint paper with J. H. Silverman)
  8. 'Lower bounds on the canonical height associated to the morphism \phi(z)= z^d+c', to appear in Monatshefte für Mathematik
  9. 'The uniform primality conjecture for elliptic curves', to appear in Acta Arithmetica (joint paper with G. Everest, V. Mahé, and S. Stevens

Works in Progress (submitted)

  1. 'Multiples of integral points on elliptic curves'
  2. 'Primitive divisors on twists of the Fermat cubic' (joint paper with G. Everest and S. Stevens)
  3. 'A quantitative primitive divisor result for elliptic divisibility sequences'
  4. 'Uniform bounds on pre-images under quadratic dynamical systems' (joint paper with X. W. C. Faber, B. Hutz, R. Jones, M. Manes, T. J. Tucker, and M. E. Zieve)

Teaching

I am currently teaching Math 137 (Calculus), and supervising a senior thesis. (All material for Math 137, including my office hours, is available on the BlackBoard system, accessible throught the U of T Portal.)

Personal

When I'm not making math, or distributing it to students, I'm often found on the streets of Toronto photographing things, in the dingy darkroom at Hart House developing said photos, or eating Portuguese custard tarts near my home in Toronto with Elissa and our slightly arthritic cat Roxy.

My people

For reasons of Google-rank, I shall mention Elissa Ross, Liam Watson, and Adam Clay.