Update the rank function of elliptic curves to use ellrank in pari

src/doc/en/reference/references/index.rst

@@ -3038,6 +3038,9 @@ REFERENCES:
 
 .. [Har1994] Frank Harary. *Graph Theory*. Reading, MA: Addison-Wesley, 1994.
 
+.. [Har2009] Harvey, David. *Efficient computation of p-adic heights*.
+             LMS J. Comput. Math. **11** (2008), 40–59.
+
 .. [Harako2020] Shuichi Harako. *The second homology group of the commutative
                 case of Kontsevich's symplectic derivation Lie algebra*.
                 Preprint, 2020, :arxiv:`2006.06064`.
@@ -4737,17 +4740,21 @@ REFERENCES:
               sharpening of the Parikh mapping*. Theoret. Informatics Appl. 35
               (2001) 551-564.
 
+.. [MST2006] Barry Mazur, William Stein, John Tate.
+             *Computation of p-adic heights and log convergence*.
+             Doc. Math. 2006, Extra Vol., 577-614.
+
 .. [MSZ2013] Michael Maschler, Solan Eilon, and Zamir Shmuel. *Game
              Theory*. Cambridge: Cambridge University Press,
              (2013). ISBN 9781107005488.
 
-.. [MT1991] Mazur, B., & Tate, J. (1991). The `p`-adic sigma
-            function. Duke Mathematical Journal, 62(3), 663-688.
+.. [MT1991] Mazur, B., & Tate, J. (1991). *The `p`-adic sigma
+            function*. Duke Mathematical Journal, **62** (3), 663-688.
 
-.. [MTT1986] \B. Mazur, J. Tate, and J. Teitelbaum, On `p`-adic
-             analogues of the conjectures of Birch and
-             Swinnerton-Dyer, Inventiones Mathematicae 84, (1986),
-             1-48.
+.. [MTT1986] \B. Mazur, J. Tate, and J. Teitelbaum,
+             *On `p`-adic analogues of the conjectures of Birch and
+             Swinnerton-Dyer*,
+             Inventiones Mathematicae **84**, (1986), 1-48.
 
 .. [Mu1997] Murty, M. Ram. *Congruences between modular forms*. In "Analytic
             Number Theory" (ed. Y. Motohashi), London Math. Soc. Lecture Notes

src/sage/schemes/elliptic_curves/BSD.py

@@ -3,6 +3,7 @@
 
 from sage.arith.misc import prime_divisors
 from sage.rings.integer_ring import ZZ
+from sage.rings.rational_field import QQ
 from sage.rings.infinity import Infinity
 from sage.rings.number_field.number_field import QuadraticField
 from sage.functions.other import ceil
@@ -89,11 +90,17 @@ def simon_two_descent_work(E, two_tor_rk):
         sage: from sage.schemes.elliptic_curves.BSD import simon_two_descent_work
         sage: E = EllipticCurve('14a')
         sage: simon_two_descent_work(E, E.two_torsion_rank())
+        doctest:warning
+        ...
+        DeprecationWarning: Use E.rank(algorithm="pari") instead, as this script has been ported over to pari.
+        See https://github.com/sagemath/sage/issues/35621 for details.
         (0, 0, 0, 0, [])
         sage: E = EllipticCurve('37a')
         sage: simon_two_descent_work(E, E.two_torsion_rank())
         (1, 1, 0, 0, [(0 : 0 : 1)])
     """
+    from sage.misc.superseded import deprecation
+    deprecation(35621, 'Use the two-descent in pari instead, as this script has been ported over to pari.')
     rank_lower_bd, two_sel_rk, gens = E.simon_two_descent()
     rank_upper_bd = two_sel_rk - two_tor_rk
     gens = [P for P in gens if P.additive_order() == Infinity]
@@ -140,6 +147,55 @@ def mwrank_two_descent_work(E, two_tor_rk):
     sha2_upper_bd = MWRC.selmer_rank() - two_tor_rk - rank_lower_bd
     return rank_lower_bd, rank_upper_bd, sha2_lower_bd, sha2_upper_bd, gens
 
+def pari_two_descent_work(E):
+    r"""
+    Prepare the output from pari by two-isogeny.
+
+    INPUT:
+
+    - ``E`` -- an elliptic curve
+
+    OUTPUT: A tuple of 5 elements with the first 4 being integers.
+
+    - a lower bound on the rank
+
+    - an upper bound on the rank
+
+    - a lower bound on the rank of Sha[2]
+
+    - an upper bound on the rank of Sha[2]
+
+    - a list of the generators found
+
+    EXAMPLES::
+
+        sage: from sage.schemes.elliptic_curves.BSD import pari_two_descent_work
+        sage: E = EllipticCurve('14a')
+        sage: pari_two_descent_work(E)
+        (0, 0, 0, 0, [])
+        sage: E = EllipticCurve('37a')
+        sage: pari_two_descent_work(E) # random, up to sign
+        (1, 1, 0, 0, [(0 : -1 : 1)])
+        sage: E = EllipticCurve('210e7')
+        sage: pari_two_descent_work(E)
+        (0, 2, 0, 2, [])
+        sage: E = EllipticCurve('66b3')
+        sage: pari_two_descent_work(E)
+        (0, 0, 2, 2, [])
+
+    """
+    ep = E.pari_curve()
+    lower, rank_upper_bd, s, pts = ep.ellrank()
+    gens = sorted([E.point([QQ(x[0]),QQ(x[1])], check=True) for x in pts])
+    gens = E.saturation(gens)[0]
+    # this is explained in the pari-gp documentation:
+    # s is the dimension of Sha[2]/2Sha[4],
+    # which is a lower bound for dim Sha[2]
+    # dim Sha[2] = dim Sel2 - rank E(Q) - dim tors
+    # rank_upper_bd = dim Sel_2 - dim tors - s
+    sha_upper_bd = rank_upper_bd - len(gens) + s
+    return len(gens), rank_upper_bd, s, sha_upper_bd, gens
+
 
 def native_two_isogeny_descent_work(E, two_tor_rk):
     """
@@ -254,7 +310,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
       - 2: print information about remaining primes
 
     - ``two_desc`` -- string (default ``'mwrank'``), what to use for the
-      two-descent. Options are ``'mwrank', 'simon', 'sage'``
+      two-descent. Options are ``'mwrank', 'pari', 'sage'``
 
     - ``proof`` -- bool or None (default: None, see
       proof.elliptic_curve or sage.structure.proof). If False, this
@@ -317,7 +373,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
         True for p = 3 by Kolyvagin bound
         True for p = 5 by Kolyvagin bound
         []
-        sage: E.prove_BSD(two_desc='simon')
+        sage: E.prove_BSD(two_desc='pari')
         []
 
     A rank two curve::
@@ -433,6 +489,15 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
         p = 2: True by 2-descent
         True for p not in {2} by Kolyvagin.
         []
+
+    ::
+
+        sage: E = EllipticCurve('66b3')
+        sage: E.prove_BSD(two_desc="pari",verbosity=1)
+        p = 2: True by 2-descent
+        True for p not in {2} by Kolyvagin.
+        []
+
     """
     if proof is None:
         from sage.structure.proof.proof import get_flag
@@ -461,8 +526,8 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
 
     if two_desc == 'mwrank':
         M = mwrank_two_descent_work(BSD.curve, BSD.two_tor_rk)
-    elif two_desc == 'simon':
-        M = simon_two_descent_work(BSD.curve, BSD.two_tor_rk)
+    elif two_desc == 'pari':
+        M = pari_two_descent_work(BSD.curve)
     elif two_desc == 'sage':
         M = native_two_isogeny_descent_work(BSD.curve, BSD.two_tor_rk)
     else:

src/sage/schemes/elliptic_curves/constructor.py

@@ -308,7 +308,7 @@ class EllipticCurveFactory(UniqueFactory):
         TypeError: invalid input to EllipticCurve constructor
     """
     def create_key_and_extra_args(self, x=None, y=None, j=None, minimal_twist=True, **kwds):
-        """
+        r"""
         Return a ``UniqueFactory`` key and possibly extra parameters.
 
         INPUT: See the documentation for :class:`EllipticCurveFactory`.
@@ -329,7 +329,7 @@ def create_key_and_extra_args(self, x=None, y=None, j=None, minimal_twist=True,
             sage: EllipticCurve.create_key_and_extra_args(j=8000)
             ((Rational Field, (0, 1, 0, -3, 1)), {})
 
-        When constructing a curve over `\\QQ` from a Cremona or LMFDB
+        When constructing a curve over `\QQ` from a Cremona or LMFDB
         label, the invariants from the database are returned as
         ``extra_args``::
 
@@ -454,7 +454,7 @@ def create_key_and_extra_args(self, x=None, y=None, j=None, minimal_twist=True,
         return (R, tuple(R(a) for a in x)), kwds
 
     def create_object(self, version, key, **kwds):
-        """
+        r"""
         Create an object from a ``UniqueFactory`` key.
 
         EXAMPLES::
@@ -466,7 +466,7 @@ def create_object(self, version, key, **kwds):
         .. NOTE::
 
             Keyword arguments are currently only passed to the
-            constructor for elliptic curves over `\\QQ`; elliptic
+            constructor for elliptic curves over `\QQ`; elliptic
             curves over other fields do not support them.
         """
         R, x = key
@@ -494,7 +494,7 @@ def create_object(self, version, key, **kwds):
 
 
 def EllipticCurve_from_Weierstrass_polynomial(f):
-    """
+    r"""
     Return the elliptic curve defined by a cubic in (long) Weierstrass
     form.
 
@@ -534,7 +534,7 @@ def EllipticCurve_from_Weierstrass_polynomial(f):
     return EllipticCurve(coefficients_from_Weierstrass_polynomial(f))
 
 def coefficients_from_Weierstrass_polynomial(f):
-    """
+    r"""
     Return the coefficients `[a_1, a_2, a_3, a_4, a_6]` of a cubic in
     Weierstrass form.
 
@@ -582,7 +582,7 @@ def coefficients_from_Weierstrass_polynomial(f):
 
 
 def EllipticCurve_from_c4c6(c4, c6):
-    """
+    r"""
     Return an elliptic curve with given `c_4` and
     `c_6` invariants.
 
@@ -664,7 +664,7 @@ def EllipticCurve_from_j(j, minimal_twist=True):
 
 
 def coefficients_from_j(j, minimal_twist=True):
-    """
+    r"""
     Return Weierstrass coefficients `(a_1, a_2, a_3, a_4, a_6)` for an
     elliptic curve with given `j`-invariant.
 
@@ -680,7 +680,7 @@ def coefficients_from_j(j, minimal_twist=True):
         sage: coefficients_from_j(1)
         [1, 0, 0, 36, 3455]
 
-    The ``minimal_twist`` parameter (ignored except over `\\QQ` and
+    The ``minimal_twist`` parameter (ignored except over `\QQ` and
     True by default) controls whether or not a minimal twist is
     computed::
 
@@ -1238,7 +1238,7 @@ def EllipticCurve_from_cubic(F, P=None, morphism=True):
 
 
 def tangent_at_smooth_point(C,P):
-    """Return the tangent at the smooth point `P` of projective curve `C`.
+    r"""Return the tangent at the smooth point `P` of projective curve `C`.
 
     INPUT:
 
@@ -1278,7 +1278,7 @@ def tangent_at_smooth_point(C,P):
         return C.tangents(P,factor=False)[0]
 
 def chord_and_tangent(F, P):
-    """Return the third point of intersection of a cubic with the tangent at one point.
+    r"""Return the third point of intersection of a cubic with the tangent at one point.
 
     INPUT:
 
@@ -1352,7 +1352,7 @@ def chord_and_tangent(F, P):
 
 
 def projective_point(p):
-    """
+    r"""
     Return equivalent point with denominators removed
 
     INPUT:
@@ -1384,7 +1384,7 @@ def projective_point(p):
 
 
 def are_projectively_equivalent(P, Q, base_ring):
-    """
+    r"""
     Test whether ``P`` and ``Q`` are projectively equivalent.
 
     INPUT:
@@ -1409,7 +1409,7 @@ def are_projectively_equivalent(P, Q, base_ring):
 
 def EllipticCurves_with_good_reduction_outside_S(S=[], proof=None, verbose=False):
     r"""
-    Return a sorted list of all elliptic curves defined over `Q`
+    Return a sorted list of all elliptic curves defined over `\QQ`
     with good reduction outside the set `S` of primes.
 
     INPUT: