Abstract: We present an algorithm to decide whether a elliptic curve over Q_p has a non-trivial p-torsion part (# E(Q_p)[p] \neq 1) under certain assumptions. We use this algorithm to efficiently determine E(Q)_{tors} the group of Q-rational torsion points on an elliptic curve.
Note: The above p-adic approach actually runs in (worst case) softly quadratic time. The following l-adic approach runs in softly quadratic time and a randomized flavor of the algorithm runs in softly linear expected time.
Abstract: We introduce an l-adic algorithm to efficiently determine the group of rational torsion points on an elliptic curve. We also make a conjecture about the discriminant of the m-division polynomial of an elliptic curve.
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.
Note: The two departmental tech reports have been debugged and combined into the following publication. Also the "#E(Q_p)[p] \neq 1" decisional algorithm in the first tech report (and the ANTS poster) has been extended to cover the split multiplicative case.
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.
Abstract: We conjecturally relate via a polynomial-time reduction, a subproblem of integer factoring to the problem of computing the Mordell-Weil group of an elliptic curve from a special family. This raises an interesting question about the growth of the height of the generators of the above group with respect to the discriminant of the elliptic curve. We gather numerical evidence to shed light on this behavior.
Note: Experts believe Conjecture 4 is false. Silverman's email
ACA 2003 NCSU, July 28, 2003
MNTCG Madison, October 25, 2003
MPKC 2003 Chicago, November 7, 2003
Algebra Seminar USC, Los Angeles, December 1, 2003
Seminar/Colloquia The Institute of Mathematical Sciences, Chennai, January 2, 2004
Theory group "informal" talk Indian Institute of Technology, Madras, January 7, 2004
Workshop on Computational Arithmetic Geometry PIMS, Simon Fraser University, Burnaby, Canada, July 6, 2004
ACA 2004 (Applications of Computer Algebra) Special Session: Computational aspects of algebraic curves Beaumont, Texas, July 20, 2004
Conference on Computational Aspects of Algebraic Curves University of Idaho, Moscow, Idaho, May 27, 2005
Seminar/Colloquia The Institute of Mathematical Sciences, Chennai, January 12, 2006
Number Theory Seminar University of California, Los Angeles, March 6, 2006
Graduate Student Colloquium University of Southern California, Los Angeles, April 19, 2006
SAGE Days 1
University of California, San Diego, February 5, 2006
Slides
Audio
Photo 1
Photo 2
AWS 2006 University of Arizona, Tucson, March 15, 2006
MSRI 2006: Computing with Modular Forms MSRI, Berkeley, August 11, 2006
Slides
Pics
SAGE Days 2 University of Washington, Seattle, October 7, 2006
Slides Pics
Explicit methods for rational points on curves, BIRS, Banff, Canada, February 4-9, 2007
Slides Pics
Crypto 2001 University of California, Santa Barbara, August 19-23, 2001
Contemporary Methods in Cryptography University of California, Los Angeles, January 9-13, 2002
Phase Transitions And Algorithmic Complexity University of California, Los Angeles, June 3-5, 2002
ANTS V (Algorithmic Number Theory Symposium V) University of Sydney, Australia, July 7-12, 2002
ECHIDNA - Workshop in Arithmetic Geometry and Applications University of Sydney, Australia, July 15 - July 19, 2002 (The conference title signifies: Elliptic Curves and HIgher DimeNsional Analogues; or That uniquely Australian creature which follows ANTS!)
Computational Aspects of Algebraic Curves and Cryptography University of Florida, Gainesville, March 3-7, 2003
European School on Algebraic Geometry and Information Theory CIRM, Marseilles, France, May 12-16, 2003
ACA 2003 (Applications of Computer Algebra) Special Session: Computational aspects of algebraic curves North Carolina State University, Raleigh, July 28-31, 2003
MNTCG : Midwest Number Theory Conference for Graduate Students and Recent PhDs University of Wisconsin, Madison, October 25-26, 2003
MPKC 2003 : Mathematics of Public-Key Cryptography University of Illinois, Chicago, November 7-9, 2003
Joint India-AMS Mathematics Meeting Indian Institute of Science, Bangalore, December 17-20, 2003
AWS 2004: Arizona Winter School University of Texas, Austin, March 13-17, 2004
ANTS VI: Algorithmic Number Theory Symposium University of Vermont, Burlington, June 13-18, 2004
Workshop on Computational Arithmetic Geometry PIMS, Simon Fraser University, Burnaby, Canada, July 5-9, 2004
ACA 2004 (Applications of Computer Algebra) Special Session: Computational aspects of algebraic curves Beaumont, Texas, July 21-23, 2004
Conference on Computational Aspects of Algebraic Curves University of Idaho, Moscow, Idaho, May 26-28, 2005
Summer Institute in Algebraic Geometry: Week III: Arithmetic Geometry University of Washington, Seattle, August 8-12, 2005
SAGE Days 2006 University of California, San Diego, February 4-5, 2006
AMS Special Session on Computational Arithmetic Geometry SFSU, San Francisco, April 29-30, 2006
Arizona Winter School 2006: Computational and algorithmic aspects of algebra and arithmetic University of Arizona, Tucson, March 11-15, 2006
MSRI 2006: Computing with Modular Forms MSRI, Berkeley, July 31-August 11, 2006
SciPy 2006 Conference Caltech, Pasadena, August 17, 2006
Securing Cyberspace: Tutorials UCLA, Los Angeles, September 12-15, 2006
SAGE Days 2 UW, Seattle, October 6-8, 2006
Securing Cyberspace: Workshop I: Number Theory and Cryptography - Open Problems UCLA, Los Angeles, October 9-13, 2006 Pics
Southern California Number Theory Day UCI, Irvine, Oct 14, 2006 Pics
Visit William Stein UW, Seattle, December 5-10, 2006 Pics
Southern California Number Theory Day UCSD, San Diego, Jan 27, 2007
Interactive Parallel Computation in Support of Research in Algebra, Geometry and Number Theory. Jan 29-Feb 2, 2007, Mathematical Sciences Research Institute, Berkeley, CA.
Explicit methods for rational points on curves, BIRS, Banff, Canada, February 4-9, 2007
SAGE Days 3 IPAM, UCLA, Los Angeles, February 17-21, 2007
Arizona Winter School 2007: $p$-adic Geometry University of Arizona, Tucson, March 10-14, 2007 (***Cancelled trip***)
SAGE Days 4 University of Washington, Seattle, June 12-17, 2007
L-functions, ranks of elliptic curves, and random matrix theory, BIRS, Banff, Canada, July 8-13, 2007
Summer School on Serre's Modularity Conjecture, Luminy, France, July 9-20, 2007 (***Tentative***)
Rational Points on Curves and Higher-Dimensional Varieties: Theory and Explicit Methods, International University-Bremen, Germany, July 21-28, 2007 (***Tentative***)
Workshop on the Tate conjecture, American Institute of Mathematics Research Conference Center, Palo Alto, California, July 23-27, 2007 (Deadline: March 23) (***Tentative***)
Workshop on L-functions and modular forms, American Institute of Mathematics Research Conference Center, Palo Alto, California, July 30-August 3, 2007 (Deadline: March 30) (***Tentative***)
Summer School on Iwasawa Theory, McMaster University in Hamilton, Ontario, Canada, August 9-13, 2007 (Deadline: May 1)