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Solving Cubic Equations (JMM 2012 Short Course)
William Stein
This worksheet accompanies these slides.
Pythagorean triples
Cubic equations
The Group Law
How the figure in the slide was made:
Mark Watkins's Curve
The Rank
File: /home/wstein/sage/sage-4.8.alpha5/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_rational_field.py
Type: <type ‘instancemethod’>
Definition: E.rank(use_database=False, verbose=False, only_use_mwrank=True, algorithm=’mwrank_lib’, proof=None)
Docstring:
Return the rank of this elliptic curve, assuming no conjectures.
If we fail to provably compute the rank, raises a RuntimeError exception.
INPUT:
- use_database (bool) - (default: False), if True, try to look up the regulator in the Cremona database.
- verbose - (default: None), if specified changes the verbosity of mwrank computations. algorithm -
- - 'mwrank_shell' - call mwrank shell command
- - 'mwrank_lib' - call mwrank c library
- only_use_mwrank - (default: True) if False try using analytic rank methods first.
- proof - bool or None (default: None, see proof.elliptic_curve or sage.structure.proof). Note that results obtained from databases are considered proof = True
OUTPUT:
- rank (int) - the rank of the elliptic curve.
IMPLEMENTATION: Uses L-functions, mwrank, and databases.
EXAMPLES:
sage: EllipticCurve('11a').rank() 0 sage: EllipticCurve('37a').rank() 1 sage: EllipticCurve('389a').rank() 2 sage: EllipticCurve('5077a').rank() 3 sage: EllipticCurve([1, -1, 0, -79, 289]).rank() # This will use the default proof behavior of True 4 sage: EllipticCurve([0, 0, 1, -79, 342]).rank(proof=False) 5 sage: EllipticCurve([0, 0, 1, -79, 342]).simon_two_descent()[0] 5Examples with denominators in defining equations:
sage: E = EllipticCurve([0, 0, 0, 0, -675/4]) sage: E.rank() 0 sage: E = EllipticCurve([0, 0, 1/2, 0, -1/5]) sage: E.rank() 1 sage: E.minimal_model().rank() 1A large example where mwrank doesn’t determine the result with certainty:
sage: EllipticCurve([1,0,0,0,37455]).rank(proof=False) 0 sage: EllipticCurve([1,0,0,0,37455]).rank(proof=True) Traceback (most recent call last): ... RuntimeError: Rank not provably correct.
Try a random curve (if you try a different one it could take a long time -- press "escape" with the cursor in the box to interrupt):
A family
Exercise: Find the first such that has rank .
Elkies Curve of Rank (at least) 28
That the first few good are negative is evidence that has high rank:
Exercise: What is the smallest good prime such that ?
The following takes about 60 seconds (on my laptop), and shows that the 28 points are independent:
Primes
It takes a while to compute the string representation of .
Exercise: Make an interact that has a slider letting you select a prime, which plots the graph of modulo that prime.
The L-Series
Natural (analytic) continuation
The Birch and Swinnerton-Dyer Conjecture:
A Rank 1 Curve
A Rank 2 Curve
If you solve for the order of the Shafarevich-Tate group in the conjecture:
The following proves that does not divide the order of this group:
Open Problem: Prove that the Shafarevich-Tate group of the specific curve given above is finite.
A Rank 4 Curve
Open Problem: Prove that vanishes to order at for the specific curve above. (Or for any curve at all!)