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Deciding whether the p-torsion group of the Q_p-rational points of an elliptic curve is non-trivial. |
1. Deciding whether the p-torsion group of the Q_p-rational points of an elliptic curve is non-trivial. |
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2. Elliptic curve torsion points and division polynomials 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. http://sage.math.washington.edu/home/burhanud/papers/burhanuddin.elliptic.curve.rational.torsion.pdf 3. Some computational problems motivated by the Birch and Swinnerton-Dyer conjecture, Ph.D. dissertation, University of Southern California, 2007. Abstract: http://sage.math.washington.edu/home/burhanud/papers/burhanuddin.dissertation.pdf 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: http://sage.math.washington.edu/home/burhanud/papers/burhanuddin.mod.7.galois.representations.pdf 5. Elliptic curves with large Shafarevich-Tate group 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. http://sage.math.washington.edu/home/burhanud/papers/burhanuddin.elliptic.curve.large.sha.pdf |
I was a Program in Computing Assistant Adjunct Professor in the Department of Mathematics at UCLA (2007-2011) and am a SAGEvangelist (2005-).
Papers
http://sage.math.washington.edu/home/burhanud/papers
1. Deciding whether the p-torsion group of the Q_p-rational points of an elliptic curve is non-trivial. 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.
http://sage.math.washington.edu/home/burhanud/papers/burhanuddin.elliptic.curve.p-adic.torsion.pdf
2. Elliptic curve torsion points and division polynomials 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.
http://sage.math.washington.edu/home/burhanud/papers/burhanuddin.elliptic.curve.rational.torsion.pdf
3. Some computational problems motivated by the Birch and Swinnerton-Dyer conjecture, Ph.D. dissertation, University of Southern California, 2007.
Abstract:
http://sage.math.washington.edu/home/burhanud/papers/burhanuddin.dissertation.pdf
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:
http://sage.math.washington.edu/home/burhanud/papers/burhanuddin.mod.7.galois.representations.pdf
5. Elliptic curves with large Shafarevich-Tate group
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.
http://sage.math.washington.edu/home/burhanud/papers/burhanuddin.elliptic.curve.large.sha.pdf
Stuff I have worked/am working/plan to work on wrt to SAGE
* Doctoral dissertation http://www.sagemath.org/files/thesis/burhanuddin-thesis-2007.pdf
* On the reducibility of Hecke polynomials over ZZ http://sage.math.washington.edu/home/burhanud/heckered/heckered.pdf
* Mestre's method of graphs project which started at the MSRI Computing with Modular Forms workshop.
- Check out:
* Implementing asymptotically fast elliptic curve rational torsion computation algorithms.
kurrently kludgey kode http://sage.math.washington.edu/home/burhanud/tor/tor.py.txt
algorithms http://modular.math.washington.edu/home/burhanud/volume.pdf
* My research is pretty SAGEy
Research Statement http://sage.math.washington.edu/home/burhanud/app11/restat11/iftikhar.burhanuddin.restat11.pdf
Old Stuff
* Make wikipage about Talks related to SAGE (plan)
* Editing the SAGE programming guide in time for the release of sage-2.0
* Editing the SAGE reference manual (and build process?) in time for the release of sage-2.0 (plan)
* Wrapping Denis Simon's 2-descent (plan)
* Dekinking some SAGE tab completion kinks (plan)
* SAGE + Parallel, The Problem Book (plan)