Tate-Shafarevich group

If \(E\) is an elliptic curve over a global field \(K\), the Tate-Shafarevich group is the subgroup of elements in \(H^1(K,E)\) which map to zero under every global-to-local restriction map \(H^1(K,E) \to H^1(K_v,E)\), one for each place \(v\) of \(K\).

The group is usually denoted by the Russian letter Sha (Ш), in this document it will be denoted by \(Sha\).

\(Sha\) is known to be an abelian torsion group. It is conjectured that the Tate-Shafarevich group is finite for any elliptic curve over a global field. But it is not known in general.

A theorem of Kolyvagin and Gross-Zagier using Heegner points shows that if the L-series of an elliptic curve \(E/\QQ\) does not vanish at 1 or has a simple zero there, then \(Sha\) is finite.

A theorem of Kato, together with theorems from Iwasawa theory, allows for certain primes \(p\) to show that the \(p\)-primary part of \(Sha\) is finite and gives an effective upper bound for it.

The (\(p\)-adic) conjecture of Birch and Swinnerton-Dyer predicts the order of \(Sha\) from the leading term of the (\(p\)-adic) L-series of the elliptic curve.

Sage can compute a few things about \(Sha\). The commands an, an_numerical and an_padic compute the conjectural order of \(Sha\) as a real or \(p\)-adic number. With p_primary_bound one can find an upper bound of the size of the \(p\)-primary part of \(Sha\). Finally, if the analytic rank is at most 1, then bound_kato and bound_kolyvagin find all primes for which the theorems of Kato and Kolyvagin respectively do not prove the triviality the \(p\)-primary part of \(Sha\).

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: S = E.sha()
sage: S.bound_kato()
[2]
sage: S.bound_kolyvagin()
([2, 5], 1)
sage: S.an_padic(7,3)
1 + O(7^5)
sage: S.an()
1
sage: S.an_numerical()
1.00000000000000

sage: E = EllipticCurve('389a')
sage: S = E.sha(); S
Tate-Shafarevich group for the Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
sage: S.an_numerical()
1.00000000000000
sage: S.p_primary_bound(5)
0
sage: S.an_padic(5)
1 + O(5)
sage: S.an_padic(5,prec=4)  # long time (2s on sage.math, 2011)
1 + O(5^3)

AUTHORS:

• William Stein (2007) – initial version
• Chris Wuthrich (April 2009) – reformat docstrings
• Aly Deines, Chris Wuthrich, Jeaninne Van Order (2016-03): Added functionality that tests the Skinner-Urban condition.

class sage.schemes.elliptic_curves.sha_tate.Sha(E)

Bases: sage.structure.sage_object.SageObject

The Tate-Shafarevich group associated to an elliptic curve.

If \(E\) is an elliptic curve over a global field \(K\), the Tate-Shafarevich group is the subgroup of elements in \(H^1(K,E)\) which map to zero under every global-to-local restriction map \(H^1(K,E) \to H^1(K_v,E)\), one for each place \(v\) of \(K\).

EXAMPLES:

sage: E = EllipticCurve('571a1')
sage: E._set_gens([])   # curve has rank 0, but non-trivial Sha[2]
sage: S = E.sha()
sage: S.bound_kato()
[2]
sage: S.bound_kolyvagin()
([2], 1)
sage: S.an_padic(7,3)
4 + O(7^5)
sage: S.an()
4
sage: S.an_numerical()
4.00000000000000

sage: E = EllipticCurve('389a')
sage: S = E.sha(); S
Tate-Shafarevich group for the Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
sage: S.an_numerical()
1.00000000000000
sage: S.p_primary_bound(5)  # long time
0
sage: S.an_padic(5)         # long time
1 + O(5)
sage: S.an_padic(5,prec=4)  # very long time
1 + O(5^3)

an(use_database=False, descent_second_limit=12)

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 cannot 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()
Traceback (most recent call last):
...
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 descent_second_limit (in the above example, set to the default, 12) option to try again:

sage: E.sha().an(descent_second_limit=16)  # long time (2s on sage.math, 2011)
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 did not transform it to be minimal, it would give nonsense:

sage: E = EllipticCurve([0,-432*6^2])
sage: E.sha().an()
1

See trac ticket #10096: this used to give the wrong result 6.0000 before since the minimal model was not used:

sage: E = EllipticCurve([1215*1216,0]) # non-minimal model
sage: E.sha().an()  # long time (2s on sage.math, 2011)
1.00000000000000
sage: E.minimal_model().sha().an()  # long time (1s on sage.math, 2011)
1.00000000000000

an_numerical(prec=None, use_database=True, proof=None)

Return the numerical analytic order of \(Sha\), which is a floating point number in all cases.

INPUT:

• prec - integer (default: 53) bits precision – used for the L-series computation, period, regulator, etc.
• use_database - whether the rank and generators should be looked up in the database if possible. Default is True
• proof - bool or None (default: None, see proof.[tab] or sage.structure.proof) proof option passed onto regulator and rank computation.

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()
1.00000000000000
sage: EllipticCurve('389a').sha().an_numerical()
1.00000000000000
sage: EllipticCurve('66b3').sha().an_numerical()
4.00000000000000
sage: EllipticCurve('5077a').sha().an_numerical()
1.00000000000000

A rank 4 curve:

sage: EllipticCurve([1, -1, 0, -79, 289]).sha().an_numerical()  # long time (3s on sage.math, 2011)
1.00000000000000

A rank 5 curve:

sage: EllipticCurve([0, 0, 1, -79, 342]).sha().an_numerical(prec=10, proof=False)  # long time (22s on sage.math, 2011)
1.0

See trac ticket #1115:

sage: sha = EllipticCurve('37a1').sha()
sage: [sha.an_numerical(prec) for prec in range(40,100,10)]  # long time (3s on sage.math, 2013)
[1.0000000000,
1.0000000000000,
1.0000000000000000,
1.0000000000000000000,
1.0000000000000000000000,
1.0000000000000000000000000]

an_padic(p, prec=0, use_twists=True)

Returns the conjectural order of \(Sha(E/\QQ)\), according to the \(p\)-adic analogue of the Birch and Swinnerton-Dyer conjecture as formulated in [MTT1986] and [BP1993].

INPUT:

• p - a prime > 3
• prec (optional) - the precision used in the computation of the \(p\)-adic L-Series
• use_twists (default = True) - If True the algorithm may change to a quadratic twist with minimal conductor to do the modular symbol computations rather than using the modular symbols of the curve itself. If False it forces the computation using the modular symbols of the curve itself.

OUTPUT: \(p\)-adic number - that conjecturally equals \(\# Sha(E/\QQ)\).

If prec is set to zero (default) then the precision is set so that at least the first \(p\)-adic digit of conjectural \(\# Sha(E/\QQ)\) is determined.

EXAMPLES:

Good ordinary examples:

sage: EllipticCurve('11a1').sha().an_padic(5)    # rank 0
1 + O(5^22)
sage: EllipticCurve('43a1').sha().an_padic(5)    # rank 1
1 + O(5)
sage: EllipticCurve('389a1').sha().an_padic(5,4) # rank 2, long time (2s on sage.math, 2011)
1 + O(5^3)
sage: EllipticCurve('858k2').sha().an_padic(7)   # rank 0, non trivial sha, long time (10s on sage.math, 2011)
7^2 + O(7^24)
sage: EllipticCurve('300b2').sha().an_padic(3)   # 9 elements in sha, long time (2s on sage.math, 2011)
3^2 + O(3^24)
sage: EllipticCurve('300b2').sha().an_padic(7, prec=6)  # long time
2 + 7 + O(7^8)

Exceptional cases:

sage: EllipticCurve('11a1').sha().an_padic(11) # rank 0
1 + O(11^22)
sage: EllipticCurve('130a1').sha().an_padic(5) # rank 1
1 + O(5)

Non-split, but rank 0 case (trac ticket #7331):

sage: EllipticCurve('270b1').sha().an_padic(5) # rank 0, long time (2s on sage.math, 2011)
1 + O(5^22)

The output has the correct sign:

sage: EllipticCurve('123a1').sha().an_padic(41) # rank 1, long time (3s on sage.math, 2011)
1 + O(41)

Supersingular cases:

sage: EllipticCurve('34a1').sha().an_padic(5) # rank 0
1 + O(5^22)
sage: EllipticCurve('53a1').sha().an_padic(5) # rank 1, long time (11s on sage.math, 2011)
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)  # long time (2s on sage.math, 2011)
2 + 7 + O(7^6)
sage: EllipticCurve([-19,34]).sha().an_padic(5)  # see trac #6455, long time (4s on sage.math, 2011)
1 + O(5)

Test for trac ticket #15737:

sage: E = EllipticCurve([-100,0])
sage: s = E.sha()
sage: s.an_padic(13)
1 + O(13^20)

bound()

Compute a provably correct bound on the order of the Tate-Shafarevich group of this curve. The bound is either False (no bound) or a list B of primes such that any prime divisor of the order of \(Sha\) is in this list.

EXAMPLES:

sage: EllipticCurve('37a').sha().bound()
([2], 1)

bound_kato()

Returns a list of primes \(p\) such that the theorems of Kato’s [Kat2004] and others (e.g., as explained in a thesis of Grigor Grigorov [Gri2005]) imply that if \(p\) divides the order of \(Sha(E/\QQ)\) then \(p\) is in the list.

If \(L(E,1) = 0\), then this function gives no information, so it returns False.

THEOREM: Suppose \(L(E,1) \neq 0\) and \(p \neq 2\) is a prime such that

• \(E\) does not have additive reduction at \(p\),
• either the \(p\)-adic representation is surjective or has its image contained in a Borel subgroup.

Then \({ord}_p(\#Sha(E))\) is bounded from above by the \(p\)-adic valuation of \(L(E,1)\cdot\#E(\QQ)_{tor}^2 / (\Omega_E \cdot \prod c_v)\).

If the L-series vanishes, the method p_primary_bound can be used instead.

EXAMPLES:

sage: E = EllipticCurve([0, -1, 1, -10, -20])   # 11A  = X_0(11)
sage: E.sha().bound_kato()
[2]
sage: E = EllipticCurve([0, -1, 1, 0, 0])       # X_1(11)
sage: E.sha().bound_kato()
[2]
sage: E = EllipticCurve([1,1,1,-352,-2689])     # 66B3
sage: E.sha().bound_kato()
[2]

For the following curve one really has that 25 divides the order of \(Sha\) (by [GJPST2009]):

sage: E = EllipticCurve([1, -1, 0, -332311, -73733731])   # 1058D1
sage: E.sha().bound_kato()                 # long time (about 1 second)
[2, 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, 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

bound_kolyvagin(D=0, regulator=None, ignore_nonsurj_hypothesis=False)

Given a fundamental discriminant \(D \neq -3,-4\) that satisfies the Heegner hypothesis for \(E\), 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:

• D - (optional) a fundamental discriminant < -4 that satisfies the Heegner hypothesis for \(E\); if not given, use the first such \(D\)
• regulator – (optional) regulator of \(E(K)\); if not given, will be computed (which could take a long time)
• ignore_nonsurj_hypothesis (optional: default False) – If True, then gives the bound coming from Heegner point index, but without any hypothesis on surjectivity of the mod-\(p\) representation.

OUTPUT:

• list – a list of primes such that if \(p\) divides \(Sha(E/K)\), then \(p\) is in this list, unless \(E/K\) has complex multiplication or analytic rank greater than 2 (in which case we return 0).
• index – the odd part of the index of the Heegner point in the full group of \(K\)-rational points on E. (If \(E\) has CM, returns 0.)

REMARKS:

1. We do not have to assume that the Manin constant is 1 (or a power of 2). If the Manin constant were divisible by a prime, that prime would get included in the list of bad primes.
2. We assume the Gross-Zagier theorem is true under the hypothesis that \(gcd(N,D) = 1\), instead of the stronger hypothesis \(gcd(2\cdot N,D)=1\) that is in the original Gross-Zagier paper. That Gross-Zagier is true when \(gcd(N,D)=1\) is “well-known” to the experts, but does not seem to written up well in the literature.
3. Correctness of the computation is guaranteed using interval arithmetic, under the assumption that the regulator, square root, and period lattice are computed to precision at least \(10^{-10}\), i.e., they are correct up to addition or a real number with absolute value less than \(10^{-10}\).

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 when 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)

p_primary_bound(p)

Return a provable upper bound for the order of the \(p\)-primary part \(Sha(E)(p)\) of the Tate-Shafarevich group.

INPUT:

• p – a prime > 2

OUTPUT:

• e – a non-negative integer such that \(p^e\) is an upper bound for the order of \(Sha(E)(p)\)

In particular, if this algorithm does not fail, then it proves that the \(p\)-primary part of \(Sha\) is finite. This works also for curves of rank > 1.

Note also that this bound is sharp if one assumes the main conjecture of Iwasawa theory of elliptic curves. One may use the method p_primary_order for checking if the extra conditions hold under which the main conjecture is known by the work of Skinner and Urban. This then returns the provable \(p\)-primary part of the Tate-Shafarevich group,

Currently the algorithm is only implemented when the following conditions are verified:

• The \(p\)-adic Galois representation must be surjective or must have its image contained in a Borel subgroup.
• The reduction at \(p\) is not allowed to be additive.
• If the reduction at \(p\) is non-split multiplicative, then the rank must be 0.
• If \(p = 3\), then the reduction at 3 must be good ordinary or split multiplicative, and the rank must be 0.

ALGORITHM:

The algorithm is described in [SW2013]. The results for the reducible case can be found in [Wu2004]. The main ingredient is Kato’s result on the main conjecture in Iwasawa theory.

EXAMPLES:

sage: e = EllipticCurve('11a3')
sage: e.sha().p_primary_bound(3)
0
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('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 (10s on sage.math, 2011)
0

Some checks for trac ticket #6406 and trac ticket #16959:

sage: e.sha().p_primary_bound(7)  # long time
2

sage: E = EllipticCurve('608b1')
sage: E.sha().p_primary_bound(5)
Traceback (most recent call last):
...
ValueError: The p-adic Galois representation is not surjective or reducible. 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)           # long time
1 + O(5^22)

sage: E = EllipticCurve("5040bi1")
sage: E.sha().p_primary_bound(5)    # long time
0

p_primary_order(p)

Return the order of the \(p\)-primary part of the Tate-Shafarevich group.

This uses the result of Skinner and Urban [SU2014] on the main conjecture in Iwasawa theory. In particular the elliptic curve must have good ordinary reduction at \(p\), the residual Galois representation must be surjective. Furthermore there must be an auxiliary prime \(\ell\) dividing the conductor of the curve exactly once such that the residual representation is ramified at \(p\).

INPUT:

• \(p\) – an odd prime

OUTPUT:

• \(e\) – a non-negative integer such that \(p^e\) is the order of the \(p\)-primary order if the conditions are satisfied and raises a ValueError otherwise.

EXAMPLES:

sage: E = EllipticCurve("389a1")  # rank 2
sage: E.sha().p_primary_order(5)
0
sage: E = EllipticCurve("11a1")
sage: E.sha().p_primary_order(7)
0
sage: E.sha().p_primary_order(5)
Traceback (most recent call last):
...
ValueError: The order is not provably known using Skinner-Urban.
Try running p_primary_bound to get a bound.

two_selmer_bound()

This returns the 2-rank, i.e. the \(\GF{2}\)-dimension of the 2-torsion part of \(Sha\), provided we can determine the rank of \(E\).

EXAMPLES:

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