If is an elliptic curve over a global field , the Shafarevich-Tate group is the subgroup of elements in which map to zero under every global-to-local restriction map , one for each place of . It is known to be a torsion group and the -torsion is finite for all . It is conjectured to be finite.
AUTHORS:
Bases: sage.structure.sage_object.SageObject
The Shafarevich-Tate group associated to an elliptic curve.
If is an elliptic curve over a global field , the Shafarevich-Tate group is the subgroup of elements in which map to zero under every global-to-local restriction map , one for each place of .
EXAMPLES:
sage: E = EllipticCurve('389a')
sage: E.sha()
Shafarevich-Tate group for the Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
Returns the Birch and Swinnerton-Dyer conjectural order of Sha as a provably correct integer, unless the analytic rank is > 1, in which case this function returns a numerical value.
INPUT:
- use_database – bool (default: False); if True, try to use any databases installed to lookup the analytic order of Sha, if possible. The order of Sha is computed if it can’t be looked up.
- descent_second_limit – int (default: 12); limit to use on point searching for the quartic twist in the hard case
This result is proved correct if the order of vanishing is 0 and the Manin constant is <= 2.
If the optional parameter use_database is True (default: False), this function returns the analytic order of Sha as listed in Cremona’s tables, if this curve appears in Cremona’s tables.
NOTE:
If you come across the following error:
sage: E = EllipticCurve([0, 0, 1, -34874, -2506691])
sage: E.sha().an()
...
RuntimeError: Unable to compute the rank, hence generators, with certainty (lower bound=0, generators found=[]). This could be because Sha(E/Q)[2] is nontrivial.
Try increasing descent_second_limit then trying this command again.
You can increase the (in the above example, set to the default, 12) option to try again:
sage: E.sha().an(descent_second_limit=16)
1
EXAMPLES:
sage: E = EllipticCurve([0, -1, 1, -10, -20]) # 11A = X_0(11)
sage: E.sha().an()
1
sage: E = EllipticCurve([0, -1, 1, 0, 0]) # X_1(11)
sage: E.sha().an()
1
sage: EllipticCurve('14a4').sha().an()
1
sage: EllipticCurve('14a4').sha().an(use_database=True) # will be faster if you have large Cremona database installed
1
The smallest conductor curve with nontrivial Sha:
sage: E = EllipticCurve([1,1,1,-352,-2689]) # 66b3
sage: E.sha().an()
4
The four optimal quotients with nontrivial Sha and conductor <= 1000:
sage: E = EllipticCurve([0, -1, 1, -929, -10595]) # 571A
sage: E.sha().an()
4
sage: E = EllipticCurve([1, 1, 0, -1154, -15345]) # 681B
sage: E.sha().an()
9
sage: E = EllipticCurve([0, -1, 0, -900, -10098]) # 960D
sage: E.sha().an()
4
sage: E = EllipticCurve([0, 1, 0, -20, -42]) # 960N
sage: E.sha().an()
4
The smallest conductor curve of rank > 1:
sage: E = EllipticCurve([0, 1, 1, -2, 0]) # 389A (rank 2)
sage: E.sha().an()
1.00000000000000
The following are examples that require computation of the Mordell-Weil group and regulator:
sage: E = EllipticCurve([0, 0, 1, -1, 0]) # 37A (rank 1)
sage: E.sha().an()
1
sage: E = EllipticCurve("1610f3")
sage: E.sha().an()
4
In this case the input curve is not minimal, and if this function didn’t transform it to be minimal, it would give nonsense:
sage: E = EllipticCurve([0,-432*6^2])
sage: E.sha().an()
1
Return the numerical analytic order of Sha, which is a floating point number in all cases.
INPUT:
Note
See also the an() command, which will return a provably correct integer when the rank is 0 or 1.
Warning
If the curve’s generators are not known, computing them may be very time-consuming. Also, computation of the L-series derivative will be time-consuming for large rank and large conductor, and the computation time for this may increase substantially at greater precision. However, use of very low precision less than about 10 can cause the underlying pari library functions to fail.
EXAMPLES:
sage: EllipticCurve('11a').sha().an_numerical()
1.00000000000000
sage: EllipticCurve('37a').sha().an_numerical() # long time
1.00000000000000
sage: EllipticCurve('389a').sha().an_numerical() # long time
1.00000000000000
sage: EllipticCurve('66b3').sha().an_numerical()
4.00000000000000
sage: EllipticCurve('5077a').sha().an_numerical() # long time
1.00000000000000
A rank 4 curve:
sage: EllipticCurve([1, -1, 0, -79, 289]).sha().an_numerical() # long time
1.00000000000000
A rank 5 curve:
sage: EllipticCurve([0, 0, 1, -79, 342]).sha().an_numerical(prec=10, proof=False) # long time -- about 30 seconds.
1.0
# See trac #1115
sage: sha=EllipticCurve('37a1').sha()
sage: [sha.an_numerical(prec) for prec in xrange(40,100,10)] # long time
[1.0000000000,
1.0000000000000,
1.0000000000000000,
1.0000000000000000000,
1.0000000000000000000000,
1.0000000000000000000000000]
Returns the conjectural order of Sha(E), according to the -adic analogue of the Birch and Swinnerton-Dyer conjecture as formulated in [MTT] and [BP].
REFERENCES:
INPUT:
OUTPUT: -adic number - that conjecturally equals .
If prec is set to zero (default) then the precision is set so that at least the first -adic digit of conjectural is determined.
EXAMPLES: Good ordinary examples:
sage: EllipticCurve('11a1').sha().an_padic(5) #rank 0
1 + O(5^2)
sage: EllipticCurve('43a1').sha().an_padic(5) #rank 1
1 + O(5)
sage: EllipticCurve('389a1').sha().an_padic(5,4) #rank 2 (long time)
1 + O(5^3)
sage: EllipticCurve('858k2').sha().an_padic(7) #rank 0, non trivial sha (long time)
7^2 + O(7^6)
sage: EllipticCurve('300b2').sha().an_padic(3) # an example with 9 elements in sha
3^2 + O(3^6)
sage: EllipticCurve('300b2').sha().an_padic(7, prec=6)
2 + 7 + O(7^8)
Exceptional cases:
sage: EllipticCurve('11a1').sha().an_padic(11) #rank 0
1 + O(11^2)
sage: EllipticCurve('130a1').sha().an_padic(5) #rank 1
1 + O(5)
Non-split, but rank 0 case (trac #7331):
sage: EllipticCurve('270b1').sha().an_padic(5) #rank 0
1 + O(5^2)
The output has the correct sign
sage: EllipticCurve('123a1').sha().an_padic(41) #rank 1 (long time)
1 + O(41)
Supersingular cases:
sage: EllipticCurve('34a1').sha().an_padic(5) # rank 0
1 + O(5^2)
sage: EllipticCurve('53a1').sha().an_padic(5) # rank 1 (long time)
1 + O(5)
Cases that use a twist to a lower conductor
sage: EllipticCurve('99a1').sha().an_padic(5)
1 + O(5)
sage: EllipticCurve('240d3').sha().an_padic(5) # sha has 4 elements here
4 + O(5)
sage: EllipticCurve('448c5').sha().an_padic(7,prec=4, use_twists=False)
2 + 7 + O(7^6)
sage: E = EllipticCurve([-19,34]) # trac 6455
sage: E.sha().an_padic(5) # long time
1 + O(5)
Compute a provably correct bound on the order of the Shafarevich-Tate group of this curve. The bound is a either False (no bound) or a list B of primes such that any divisor of Sha is in this list.
EXAMPLES:
sage: EllipticCurve('37a').sha().bound()
([2], 1)
Returns a list of primes such that the theorems of Kato’s [Ka] and others (e.g., as explained in a paper/thesis of Grigor Grigorov [Gri]) imply that if divides the order of Sha(E) then is in the list.
If , then this function gives no information, so it returns False.
THEOREM (Kato): Suppose and is a prime such that
- does not have additive reduction at ,
- the mod- representation is surjective.
Then divides .
EXAMPLES:
sage: E = EllipticCurve([0, -1, 1, -10, -20]) # 11A = X_0(11)
sage: E.sha().bound_kato()
[2, 3, 5]
sage: E = EllipticCurve([0, -1, 1, 0, 0]) # X_1(11)
sage: E.sha().bound_kato()
[2, 3, 5]
sage: E = EllipticCurve([1,1,1,-352,-2689]) # 66B3
sage: E.sha().bound_kato()
[2, 3]
For the following curve one really has that 25 divides the order of Sha (by Grigorov-Stein paper [GS]):
sage: E = EllipticCurve([1, -1, 0, -332311, -73733731]) # 1058D1
sage: E.sha().bound_kato() # long time (about 1 second)
[2, 3, 5, 23]
sage: E.galois_representation().non_surjective() # long time (about 1 second)
[]
For this one, Sha is divisible by 7:
sage: E = EllipticCurve([0, 0, 0, -4062871, -3152083138]) # 3364C1
sage: E.sha().bound_kato() # long time (< 10 seconds)
[2, 3, 7, 29]
No information about curves of rank > 0:
sage: E = EllipticCurve([0, 0, 1, -1, 0]) # 37A (rank 1)
sage: E.sha().bound_kato()
False
REFERENCES:
Given a fundamental discriminant that satisfies the Heegner hypothesis for , return a list of primes so that Kolyvagin’s theorem (as in Gross’s paper) implies that any prime divisor of Sha is in this list.
INPUT:
OUTPUT:
REMARKS:
EXAMPLES:
sage: E = EllipticCurve('37a')
sage: E.sha().bound_kolyvagin()
([2], 1)
sage: E = EllipticCurve('141a')
sage: E.sha().an()
1
sage: E.sha().bound_kolyvagin()
([2, 7], 49)
We get no information the curve has rank 2.:
sage: E = EllipticCurve('389a')
sage: E.sha().bound_kolyvagin()
(0, 0)
sage: E = EllipticCurve('681b')
sage: E.sha().an()
9
sage: E.sha().bound_kolyvagin()
([2, 3], 9)
Returns a provable upper bound for the order of . In particular, if this algorithm does not fail, then it proves that the -primary part of Sha is finite.
INPUT: p – a prime > 2
OUTPUT: integer – power of that bounds the order of from above
The result is a proven upper bound on the order of . So in particular it proves it finiteness even if the rank of the curve is larger than 1. Note also that this bound is sharp if one assumes the main conjecture of Iwasawa theory of elliptic curves (and this is known in certain cases).
Currently the algorithm is only implemented when certain conditions are verified.
EXAMPLES:
sage: e = EllipticCurve('11a3')
sage: e.sha().p_primary_bound(3)
0
sage: e.sha().p_primary_bound(7)
0
sage: e.sha().p_primary_bound(11)
0
sage: e.sha().p_primary_bound(13)
0
sage: e = EllipticCurve('389a1')
sage: e.sha().p_primary_bound(5)
0
sage: e.sha().p_primary_bound(7)
0
sage: e.sha().p_primary_bound(11)
0
sage: e.sha().p_primary_bound(13)
0
sage: e = EllipticCurve('858k2')
sage: e.sha().p_primary_bound(3) # long time
0
# checks for trac 6406
sage: e.sha().p_primary_bound(7)
...
ValueError: The mod-p Galois representation is not surjective. Current knowledge about Euler systems does not provide an upper bound in this case. Try an_padic for a conjectural bound.
sage: e.sha().an_padic(7)
7^2 + O(7^6)
sage: e = EllipticCurve('11a3')
sage: e.sha().p_primary_bound(5)
...
ValueError: The mod-p Galois representation is not surjective. Current knowledge about Euler systems does not provide an upper bound in this case. Try an_padic for a conjectural bound.
sage: e.sha().an_padic(5)
1 + O(5^2)
This returns the 2-rank, i.e. the -dimension of the 2-torsion part of Sha, provided we can determine the rank of . EXAMPLE:
sage: sh = EllipticCurve('571a1').sha()
sage: sh.two_selmer_bound()
2
sage: sh.an()
4
sage: sh = EllipticCurve('66a1').sha()
sage: sh.two_selmer_bound()
0
sage: sh.an()
1
sage: sh = EllipticCurve('960d1').sha()
sage: sh.two_selmer_bound()
2
sage: sh.an()
4