I am a Computer Scientist at Adobe Research and a SAGEvangelist (2005 - ).
- http://www.adobe.com/technology/people/bangalore/iftikhar-ahamath-burhanuddin.html
- https://www.linkedin.com/pub/iftikhar-burhanuddin/3/681/24a
Papers
1. Deciding whether the p-torsion group of the Q_p-rational points of an elliptic curve is non-trivial, with Ming-Deh Huang. ANTS VI Poster Abstracts. SIGSAM Bulletin, Volume 38, Number 3 September 2004 Issue 149.
Abstract: This note describes an algorithm to decide whether an elliptic curve over Q_p has a non-trivial p-torsion part (# E(Q_p)[p] is not equal to 1) under certain assumptions.
2. Elliptic curve torsion points and division polynomials, with Ming-Deh Huang. Computational Aspects of Algebraic Curves, T. Shaska (Ed.), Lecture Notes Series on Computing, 13 (2005), 13--37, World Scientific.
Abstract: We present two algorithms - p-adic and l-adic - to determine E(Q)_{tors} the group of rational torsion points on an elliptic curve. Another algorithm we introduce is one which decides whether an elliptic curve over Q_p has a non-trivial p-torsion part and this comes into play in the p-adic torsion computation procedure. We also make some remarks about the discriminant of the m-division polynomial of an elliptic curve and the information it reveals about torsion points.
3. Some computational problems motivated by the Birch and Swinnerton-Dyer conjecture. Ph.D. dissertation, University of Southern California, 2007.
Abstract: This dissertation revolves around the BSD (Birch and Swinnerton-Dyer) conjecture for elliptic curves defined over the rational numbers, a famous problem that has been open for over forty years and one of the seven Millennium Prize problems. The BSD conjecture is considered to be the first nontrivial number theoretic problem put forth as a result of explicit machine computation --- in the late '50s at Cambridge University. The BSD conjecture relates the rank of the Mordell-Weil group, the group of rational points of an elliptic curve, a quantity which seems to be difficult to pin down, to the order of vanishing of the L-function of the elliptic curve at its central point.
We make algorithmic and theoretical advances with regards to some of the terms appearing in the BSD formula, namely the sizes of the torsion subgroup and the Shafarevich-Tate group.
Firstly, we introduce an algorithm to compute elliptic curve torsion subgroup. The randomized version of this procedure runs in expected time which is essentially linear in the number of bits required to write down the equation of the elliptic curve.
Next, we discuss a conjecture of Hindry, who proposed a Brauer-Siegel type formula for elliptic curves. Driven by a suggestion of Hindry, we prove assuming standard conjectures that there are infinitely many elliptic curves with Shafarevich-Tate group of size about as large as the square root of the minimal discriminant of the curve. This improves on a result of de Weger.
Thirdly, we consider certain quartic twists of an elliptic curve. We establish a reduction between the problem of factoring integers of a certain form and the problem of computing rational points on these twists. We illustrate that the size of Shafarevich-Tate group of these curves will make it computationally expensive to factor integers by computing rational points via the Heegner point method.
Finally, we sketch existing algorithms that compute the quantities appearing in the BSD formula and introduce strategies to parallelize them.
4. On projectively rational lifts of mod 7 Galois representations, with Luis Dieulefait. JP Journal of Algebra, Number Theory and Applications, Volume 20, Issue 1, 109 -- 119, February 2011.
Abstract: We consider the problem of constructing projectively rational lifts of odd, two-dimensional Galois representations with values in F_7. Using modular forms, in particular the theory of congruences, we compute such lifts for many examples of mod 7 representations thus giving evidence that suggests that such lifts may always exist. We also consider the invariance after twist (weight change) of the existence of such lifts.
5. Elliptic curves with large Shafarevich-Tate group, with Ming-Deh Huang. Journal of Number Theory, Volume 133, Issue 2, February 2013, Pages 369 -- 374.
Abstract: We show that there exist infinitely many elliptic curves with Shafarevich-Tate group of order essentially as large as the the square root of the minimal discriminant assuming certain conjectures. This improves on a result of de Weger.
6. On the Equation y^2 = x^3 - pqx, with Ming-Deh Huang. Journal of Numbers, Volume 2014 (2014), Article ID 825634, 5 pages.
Abtsract: We consider certain quartic twists of an elliptic curve. We establish the rank of these curves under the Birch and Swinnerton-Dyer conjecture and obtain bounds on the size of Shafarevich-Tate group of these curves. We also establish a reduction between the problem of factoring integers of a certain form and the problem of computing rational points on these twists.