Simon 2-descent only returns an upper bound on the 2-Selmer rank

[sagetrac-weigandt]

[See #15608 for a list of open simon_two_descent tickets]

Given an elliptic curve E the method E.simon_two_descent() returns an ordered triple. This consists of a lower bound on the Mordell-Weil rank of E, an integer which is supposed to be the F_2 dimension of the 2-Selmer group of E, and list of points, generating the part of the Mordell-Weil group that has been found.

Sometimes the second entry is larger than the actual 2-Selmer rank as computed by mwrank, and predicted by BSD. The first curve I know of for which this happens is the elliptic curve '438e1' from Cremona's tables.

sage: E=EllipticCurve('438e1')
sage: E.simon_two_descent()
(0, 3, [(13 : -7 : 1)])
sage: E.selmer_rank() #uses mwrank
1
sage: E.sha().an()
1

The explanation for this is that E.simon_two_descent(), unlike Cremona's mwrank, does not do a second descent and therefore only determines an upper bound on the 2-Selmer rank.

Depends on #11005
Depends on #9322

Upstream: Reported upstream. No feedback yet.

CC: @JohnCremona @williamstein @rlmill

Component: elliptic curves

Keywords: simon_two_descent

Author: Peter Bruin

Branch/Commit: 732191d

Reviewer: Chris Wuthrich

Issue created by migration from https://trac.sagemath.org/ticket/10735

[chriswuthrich]

comment:2

This is a bug in Simon's script indeed. I have emailed him about this one, too, as it happens with the later version of his file in gp, too.

[chriswuthrich]

Upstream: Reported upstream. No feedback yet.

[JohnCremona]

Description changed:

--- 
+++ 
@@ -1,3 +1,5 @@
+[See #15608 for a list of open simon_two_descent tickets]
+
 Given an elliptic curve E the method E.simon_two_descent() returns an ordered triple. This consists of a lower bound on the Mordell-Weil rank of E, an integer which is supposed to be the F_2 dimension of the 2-Selmer group of E, and list of points, generating the part of the Mordell-Weil group that has been found.
 
 Sometimes the second entry is larger than the actual 2-Selmer rank as computed by mwrank, and predicted by BSD. The first curve I know of for which this happens is the elliptic curve '438e1' from Cremona's tables.

[pjbruin]

Dependencies: #11005

[pjbruin]

comment:6

The problem still exists after #11005, which upgrades Simon's scripts to the latest version:

sage: E=EllipticCurve('438e1')
sage: E.simon_two_descent()
(0, 3, [])

(Note: the torsion point (13, -7) is no longer returned since #13593.)

[pjbruin]

comment:7

It seems to me that the bug is not caused by failing to detect non-solubility at the real place. In fact, Simon's script computes the same 2-isogeny Selmer ranks as mwrank, but deduces an incorrect 2-Selmer rank from these.

Output of mwrank:

Curve [1,0,1,-130,-556] :	
1 points of order 2:
[13:-7:1]

Using 2-isogenous curve [0,-314,0,73,0] (minimal model [1,0,1,-2050,-35884])
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 2
rk(S^{phi}(E'))=	3
rk(S^{phi'}(E))=	1

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 0
rk(phi'(S^{2}(E)))=	1
rk(phi(S^{2}(E')))=	1
rk(S^{2}(E))=	1
rk(S^{2}(E'))=	3

Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
-------------------------------------------------------
1. E(Q)/phi(E'(Q))
-------------------------------------------------------
(c,d)  =(157,6144)
(c',d')=(-314,73)
This component of the rank is 0
-------------------------------------------------------
2. E'(Q)/phi'(E(Q))
-------------------------------------------------------
This component of the rank is 0

-------------------------------------------------------
Summary of results:
-------------------------------------------------------
	rank(E) = 0
	#E(Q)/2E(Q) = 2

Information on III(E/Q):
	#III(E/Q)[phi']    = 1
	#III(E/Q)[2]       = 1

Information on III(E'/Q):
	#phi'(III(E/Q)[2]) = 1
	#III(E'/Q)[phi]    = 4
	#III(E'/Q)[2]      = 4


Used descent via 2-isogeny with isogenous curve E' = [1,0,1,-2050,-35884]
Rank = 0
Rank of S^2(E)  = 1
Rank of S^2(E') = 3
Rank of S^phi(E') = 3
Rank of S^phi'(E) = 1

Processing points found during 2-descent...done:
  now regulator = 1


Regulator = 1

The rank and full Mordell-Weil basis have been determined unconditionally.
 (0.098 seconds)

Output of simon_two_descent:

ellrank([1,0,1,-130,-556]);
 Elliptic curve: Y^2 = x^3 + x^2 - 2072*x - 35568
 E[2] = [[0], [52, 0]]
 Elliptic curve: Y^2 = x^3 + 157*x^2 + 6144*x
  Algorithm of 2-descent via isogenies
 trivial points on E(Q) = [[0, 0], [1, 1, 0], [0, 0], [0, 0]]

  #K(b,2)gen          = 3
  K(b,2)gen = [-1, 2, 3]~
  quartic ELS: Y^2 = -x^4 + 157*x^2 - 6144
  no point found on the quartic
  quartic ELS: Y^2 = 2*x^4 + 157*x^2 + 3072
  no point found on the quartic
  quartic ELS: Y^2 = -2*x^4 + 157*x^2 - 3072
  no point found on the quartic
  quartic ELS: Y^2 = 3*x^4 + 157*x^2 + 2048
  no point found on the quartic
  quartic ELS: Y^2 = -3*x^4 + 157*x^2 - 2048
  no point found on the quartic
  point on the quartic
 points on E(Q) = [[0, 0]]

[E(Q):phi'(E'(Q))] >= 2
#S^(phi')(E'/Q)     = 8  # agrees with mwrank
#III(E'/Q)[phi']   <= 4

  #K(a^2-4b,2)gen     = 2
  K(a^2-4b,2)gen     = [-1, 73]~
 trivial points on E'(Q) = [[0, 0], [1, 1, 0], [0, 0], [0, 0]]

  point on the quartic
 points on E'(Q) = [[0, 0]]
 points on E(Q) = [[0, 0]]

[E'(Q):phi(E(Q))]   = 2
#S^(phi)(E/Q)       = 2  # agrees with mwrank
#III(E/Q)[phi]      = 1

#III(E/Q)[2]       <= 4
#E(Q)[2]            = 2
#E(Q)/2E(Q)        >= 2

0 <= rank          <= 2

points = [[0, 0]]
v =  [0, 3, [[13, -7]]]

The 3 in the last line, which should be the rank of the 2-Selmer group according to the documentation, is the result of computing (rank of S^(phi)(E/Q)) + (rank of S^(phi')(E'/Q)) + (rank of E(Q)[2]) - 2 = 1 + 3 + 1 - 2 = 3. There must be something wrong with this formula, as it is symmetric in E and *E' * (in this particular case, since E(Q)[2] and E'(Q)[2] both have rank 1) while the 2-Selmer ranks of E and *E' * are in fact different (1 and 3, respectively).