commit 5ea79c7ae0d44686f1df05c4a016652afbe58968
Author: Sebastiano Tronto <sebastiano@tronto.net>
Date: Sun, 14 Jun 2026 09:58:21 +0200
Initial commit
Diffstat:
33 files changed, 47643 insertions(+), 0 deletions(-)
diff --git a/README.md b/README.md
@@ -0,0 +1,9 @@
+# Notes and code from my early research in Kummer Theory for Elliptic curves
+
+This repository contains various bits and pieces and code and research notes
+that I have written at the beginning of my PhD.
+
+I have collected this documentation years after finishing my PhD, so I don't
+remember how all these scripts work, and there may be plenty of mistakes.
+Feel free to send me an email if you have any question, but keep in mind I may
+not be able to help you.
diff --git a/divisibility_reductions/1k-max3-posrank.sage b/divisibility_reductions/1k-max3-posrank.sage
@@ -0,0 +1,1671 @@
+
+# Elliptic curves downloaded from the LMFDB downloaded on 30 January 2019.
+# Below is a list called data. Each entry has the form:
+# [a1,a2,a3,a4,a6] (Weierstrass Coefficients)
+
+
+data = [\
+[1,1,0,-82,-305],\
+[0,0,1,-1,0],\
+[0,0,0,1,6],\
+[0,1,0,0,4],\
+[1,1,1,-55,134],\
+[1,1,0,-6,4],\
+[1,0,1,1,0],\
+[1,1,0,1,1],\
+[1,1,0,-7,5],\
+[1,0,0,-1,2],\
+[1,-1,1,-2,0],\
+[0,1,1,-1,-1],\
+[0,-1,1,-1,-2],\
+[0,0,1,-3,0],\
+[0,0,1,-3,2],\
+[0,-1,1,2,-2],\
+[0,1,1,-12,2],\
+[0,1,0,0,1],\
+[0,1,0,2,0],\
+[1,1,0,3,-3],\
+[0,0,0,-1,1],\
+[0,-1,0,16,0],\
+[0,0,0,1,-1],\
+[0,-1,1,-19,39],\
+[0,-1,0,4,-8],\
+[1,1,1,-102,355],\
+[1,1,0,7,-9],\
+[0,1,1,-441,3422],\
+[1,0,0,-7,9],\
+[0,0,1,-21,40],\
+[0,0,0,-412,3316],\
+[0,-1,0,-9,13],\
+[0,1,0,-57,171],\
+[1,-1,1,25,-26],\
+[0,-1,0,3,-11],\
+[0,-1,0,3,-2],\
+[1,-1,0,13,-11],\
+[0,0,0,-7,-2],\
+[1,1,1,-39,-35],\
+[0,-1,1,19,100],\
+[1,-1,1,-7,8],\
+[1,0,1,-15,22],\
+[1,1,0,-33,61],\
+[1,-1,0,-15,-46],\
+[1,-1,0,-19,37],\
+[0,1,0,11,695],\
+[1,0,1,-7,14],\
+[0,1,0,3,11],\
+[1,0,1,-2,1],\
+[1,-1,1,-2,28],\
+[0,1,0,0,-1],\
+[0,0,1,-12,4],\
+[1,-1,0,21,53],\
+[1,-1,0,-935,11229],\
+[1,1,0,-2819,-58803],\
+[1,1,1,-784,8720],\
+[0,0,0,1,1],\
+[0,0,0,-8,16],\
+[1,1,1,-8,9],\
+[1,-1,0,12,-208],\
+[0,-1,0,-383,3012],\
+[0,1,0,6,-43],\
+[1,1,0,81,-27],\
+[0,-1,0,-10,17],\
+[0,1,1,1815,141239],\
+[1,-1,0,-2,1],\
+[1,-1,0,-4,-2],\
+[0,0,0,-6,5],\
+[1,-1,0,-18,36],\
+[0,-1,0,-13,25],\
+[0,-1,0,-12,24],\
+[0,-1,0,-80,304],\
+[0,0,0,-2,4],\
+[1,-1,1,8,-5],\
+[1,1,1,-9,7],\
+[0,-1,0,-544,-4352],\
+[1,-1,0,90,436],\
+[0,0,1,1,-8],\
+[1,1,1,-120,42282],\
+[1,1,1,9,13],\
+[1,0,0,-185,1401],\
+[0,1,1,-4,-2],\
+[1,1,1,-17,-70],\
+[0,0,0,2,1],\
+[1,1,0,-12,12],\
+[1,1,1,-18,415],\
+[0,-1,0,4,-3],\
+[1,-1,1,-374,2949],\
+[1,-1,1,-171,1904],\
+[1,-1,0,-1414,-44027],\
+[1,-1,1,-12,18],\
+[0,1,1,-1006,11952],\
+[0,0,1,-111,450],\
+[1,-1,0,0,2],\
+[0,1,0,-233,1563],\
+[0,0,0,-8,9],\
+[0,0,1,43,-2088],\
+[1,1,0,-1,-2],\
+[0,1,1,9,344],\
+[1,0,1,-1,1],\
+[0,0,0,-27,27],\
+[1,-1,1,2064,18771],\
+[0,1,1,-16649,821406],\
+[0,1,0,-11,11],\
+[0,0,1,-4,-3],\
+[1,0,1,-10758,428760],\
+[1,0,1,1,1],\
+[0,0,1,-12,9],\
+[0,0,1,242,-333],\
+[1,0,1,-722,7396],\
+[1,0,1,706,-64375],\
+[1,0,1,-17,56],\
+[1,0,1,31,20],\
+[1,-1,0,-3,-1],\
+[1,-1,0,-2430,46732],\
+[0,0,0,1468,-2844],\
+[1,-1,1,-11,27],\
+[1,0,1,6,1],\
+[0,1,0,1,2],\
+[0,-1,1,-2,2],\
+[0,-1,0,-4,5],\
+[0,0,0,-67,226],\
+[1,1,0,-59,-201],\
+[1,-1,0,-70,244],\
+[0,1,0,-1,1],\
+[0,0,1,-5,4],\
+[0,1,1,0,2],\
+[1,1,0,-512,4237],\
+[0,-1,1,1,-1],\
+[1,0,0,31,-192],\
+[0,0,1,-75,256],\
+[0,1,0,-1,31],\
+[0,-1,1,-4,-2],\
+[0,1,1,-10,10],\
+[1,0,1,2,0],\
+[0,-1,0,-432,-3316],\
+[1,1,1,2,-1],\
+[0,-1,1,-83,3818],\
+[1,0,1,-3,0],\
+[0,1,0,84,36],\
+[0,1,1,-156,700],\
+[1,1,1,-56,-135],\
+[0,-1,0,-1,2],\
+[0,-1,1,2,0],\
+[0,-1,1,1,0],\
+[1,1,1,13,177],\
+[1,0,0,-19,33],\
+[1,-1,0,-1,1],\
+[1,0,0,-5,4],\
+[0,0,1,-3,4],\
+[0,-1,0,8,-4],\
+[0,1,0,-3,-2],\
+[0,1,0,1,1],\
+[1,0,0,-1,0],\
+[0,0,0,-12,20],\
+[0,0,0,-2,0],\
+[1,0,0,-12,16],\
+[1,1,0,2,1],\
+[1,-1,1,0,0],\
+[0,-1,1,10,6],\
+[1,-1,0,-80,-256],\
+[0,-1,0,-16,29],\
+[1,0,0,-6,9],\
+[1,0,0,1,25],\
+[0,0,0,-584,5444],\
+[1,-1,0,1,-1],\
+[0,-1,1,-5,6],\
+[1,1,0,-794,8289],\
+[1,-1,0,1,1],\
+[1,0,0,-2,1],\
+[1,-1,0,12,35],\
+[1,0,0,-6,4],\
+[0,0,0,-55,157],\
+[1,0,0,-3,2],\
+[0,0,1,-81,290],\
+[1,1,1,-2,0],\
+[0,1,1,-100,406],\
+[1,1,0,-30,52],\
+[0,-1,0,0,1],\
+[1,-1,0,-20,40],\
+[0,1,0,-21,31],\
+[1,-1,0,0,4],\
+[0,0,1,-3,18],\
+[0,-1,0,55,93],\
+[1,1,1,6,7],\
+[0,0,1,-237,1404],\
+[1,0,0,-2,-1],\
+[1,1,0,2,2],\
+[1,1,0,-3,1],\
+[0,-1,0,-45,133],\
+[1,-1,1,-5,20],\
+[1,0,0,-3,-2],\
+[0,0,1,-3,-2],\
+[0,-1,0,4,4],\
+[1,0,1,1,-5],\
+[0,1,1,1,1],\
+[1,0,0,-213,-1208],\
+[1,1,1,-1,0],\
+[1,0,1,-23,39],\
+[0,1,1,-2376,-61851],\
+[0,-1,0,-21,49],\
+[1,-1,0,0,1],\
+[1,-1,0,-454,5812],\
+[1,-1,1,-12,15],\
+[0,-1,0,-8,16],\
+[0,-1,0,0,4],\
+[0,0,1,-9,10],\
+[0,1,0,-50,129],\
+[1,-1,1,-4,-1],\
+[1,0,0,-24,63],\
+[1,0,0,0,-1],\
+[1,-1,0,-5252,-145223],\
+[0,-1,1,-26,68],\
+[1,-1,1,-1487,-12905],\
+[0,0,0,5,10],\
+[1,-1,1,-2,82],\
+[0,0,1,-597,8820],\
+[0,-1,1,-52,-3863],\
+[0,-1,1,-3,2],\
+[1,0,0,-3,0],\
+[1,1,0,-35,-98],\
+[1,-1,0,-5,7],\
+[1,-1,1,-149,749],\
+[1,0,0,-28,-59],\
+[1,-1,1,-117,141],\
+[0,1,1,-8,8],\
+[0,0,0,-48,196],\
+[0,-1,0,2,1],\
+[1,1,1,-16,-15],\
+[1,1,0,-715,7069],\
+[0,-1,0,-1,197],\
+[0,1,0,-16,16],\
+[1,-1,1,-83,595],\
+[0,1,1,-2,-2],\
+[1,0,0,43,-31],\
+[1,1,1,79,335],\
+[1,-1,1,13,-12],\
+[1,-1,0,-29,-635],\
+[1,-1,0,-2846,59156],\
+[1,0,0,-350,2500],\
+[1,-1,0,1,-3],\
+[1,0,1,-80,-275],\
+[1,1,1,4,-1443],\
+[0,0,1,3,-4],\
+[1,0,1,-8,7],\
+[1,1,1,-5,11],\
+[1,0,1,-174,880],\
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+[1,-1,1,-61,197],\
+[1,-1,0,16,-10],\
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+[1,0,1,-604,-5734],\
+[0,1,1,-11,-16],\
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+[0,0,1,175,-1344],\
+[1,-1,0,-107,454],\
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+[1,1,1,64,258],\
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+[0,-1,0,-1,-1],\
+[1,1,0,-97,281],\
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+[0,0,0,-8,4],\
+[0,1,0,8,89],\
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+[1,1,1,-70,195],\
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+[1,1,1,1,0],\
+[0,0,0,5,42],\
+[0,-1,0,-4,8],\
+[1,1,0,25,-14],\
+[1,0,1,33924,-387702],\
+[1,0,1,-1,148],\
+[1,0,1,-4,-2],\
+[0,1,1,-269,1628],\
+[1,-1,0,1,5],\
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+[1,1,0,-1693,26434],\
+[0,0,0,4,-4],\
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+[1,-1,0,-42,-127],\
+[1,1,0,-108,-432],\
+[1,-1,1,7,-7],\
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+[0,0,0,-192,1028],\
+[1,-1,0,-45,139],\
+[0,1,1,-179,881],\
+[0,1,1,0,0],\
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+[0,-1,0,3,9],\
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+[1,0,1,4,2],\
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+[0,0,0,-56,-4848],\
+[0,0,0,-10,12],\
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+[0,-1,0,2,-7],\
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+[0,0,0,-484,-5324],\
+[0,0,1,1,0],\
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+[0,1,0,-6,4],\
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+[0,-1,1,-6,8],\
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+[1,1,1,-5,0],\
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+[1,0,0,-5391,285606],\
+[1,-1,0,-71370,8011575],\
+[0,0,0,-15843,-1441118],\
+[0,0,1,-13,18],\
+[1,0,1,-3,2],\
+[0,-1,1,-2,0],\
+[0,0,0,-4,1],\
+[1,0,0,-4,3],\
+[1,0,0,0,1],\
+[1,1,1,-15,16],\
+[1,0,1,-5,0],\
+[0,-1,1,0,2],\
+[0,0,0,-7,10],\
+[0,1,1,-4,2],\
+[0,1,1,-2,0],\
+[1,-1,0,-4,4],\
+[0,1,1,-12,12],\
+[0,1,1,1,6],\
+[0,0,0,-19,34],\
+[0,-1,1,-24,54],\
+[0,-1,1,-5,-3]]
diff --git a/divisibility_reductions/jumps.sage b/divisibility_reductions/jumps.sage
@@ -0,0 +1,145 @@
+from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
+
+# Pre-tests on E and P
+def suitable( E, P, ell ):
+ if not is_EllipticCurve(E):
+ print "E is not an elliptic curve"
+ return False
+ if E.base_field() != QQ:
+ print "E is not defined over Q"
+ return False
+ if not P in E:
+ print "P is not in E"
+ return False
+ if P.has_finite_order():
+ print "P has finite order"
+ return False
+ if not is_prime(ell):
+ print "ell is not prime"
+ return False
+ return True
+
+
+# Stupid auxiliary function. Returns higest power of n dividing m.
+def val( n, m ):
+ ret = 0
+ while m % (n^(ret+1)) == 0:
+ ret += 1
+ return ret
+
+def test_jump_kl2_label( label, ell ):
+ E = EllipticCurve( label )
+ if E.rank() < 1:
+ print "E has rank 0"
+ return
+ P = E.gens()[0]
+ test_jump_kl2( E, P, ell )
+
+def test_jump_kl2( E, P, ell ):
+ if not suitable( E, P, ell ):
+ return
+
+ flag = True
+ for p in Primes():
+ if p > 10^3:
+ break
+ if p == ell:
+ # print "Skipping", p, "because = ell"
+ continue
+ if E.discriminant() % p == 0:
+ # print "Skipping", p, "because E has bad reduction"
+ continue
+ if P.reduction(p).order() % ell != 0:
+ # print "Skipping", p, "because red of P is infinitely ell-divisible"
+ continue
+
+ # print "Working with prime p =", p
+
+ F = FiniteField(p)
+
+ for i in range(3):
+ # print "Torsion level", i, "..."
+
+ # We can build the division fields for increasing powers of l
+ # incrementally. To get a division field, we first compute the
+ # splitting field of the division polynomial. We may be off by
+ # a degree 2 extension. If so, by finite field magic we know
+ # exactly which degree 2 extension we need: the unique one!
+ R.<x> = PolynomialRing(F)
+ F.<a> = E.reduction(p).division_polynomial(ell^i).splitting_field()
+ E_red = E.reduction(p).base_extend(F)
+ # extend if necessary
+ k = val(ell,E_red.gens()[0].order())
+ h = val(ell,E_red.order()) - k
+ if k < i or h < i:
+ F.<a> = F.extension(2)
+ E_red = E_red.base_extend(F)
+
+ P_red = E_red.point(P.reduction(p))
+
+ flag = False
+ if P_red.is_divisible_by(ell):
+ # print "P ell-divisible in this torsion level"
+ flag = True
+ break
+
+ if not flag:
+ print "Point not divisible mod", p
+ print "Stopping here"
+ break
+ if flag:
+ print "*********************************"
+ print "*** Candidate counterexample! ***"
+ print "*********************************"
+
+# Wrapper
+def test_jump_den_label( label, ell ):
+ E = EllipticCurve(label)
+ if E.rank() < 1:
+ print "E has rank 0"
+ return
+ P = E.gens()[0]
+ test_jump_den( E, P, ell )
+
+def test_jump_den( E, P, ell ):
+ if not suitable( E, P, ell ):
+ return
+
+ n_primes = 0
+ inf_divisible = 0
+ tot_l_part = 0
+ tot_non_l_part = 0
+
+ for p in Primes():
+ if p > 5*10^3:
+ break
+ if p == ell or E.discriminant() % p == 0:
+ continue
+
+ n_primes += 1
+
+ E_red = E.reduction(p)
+ N = E_red.order()
+ k = val(ell,E_red.gens()[0].order())
+ h = val(ell,N) - k
+ temp_non_l_part = (N / (ell^(h+k)))-1
+ tot_non_l_part += temp_non_l_part
+ temp_l_part = N - temp_non_l_part
+ tot_l_part += temp_l_part
+
+ if P.reduction(p).order() % ell != 0:
+ inf_divisible += 1
+
+ found = RDF( inf_divisible / n_primes )
+ expected = RDF( tot_non_l_part / (tot_l_part+tot_non_l_part) )
+ print "Found: %0.3f, expected: %0.3f"%(found, expected)
+ if abs(found-expected) > 0.05:
+ print "Unexpected density! Checking divisibility in reductions..."
+ test_jump_kl2( E, P, ell )
+
+def test_from_file( filename ):
+ attach(filename)
+ for coord in data:
+ label = EllipticCurve(coord).label()
+ print "Trying curve", label, "with ell =", 3
+ test_jump_den_label(label,3)
diff --git a/divisibility_reductions/mod3mod9.sage b/divisibility_reductions/mod3mod9.sage
@@ -0,0 +1,10 @@
+# Curves with surjective mod3 represetation but not surjective mod 9
+
+data = [\
+ [ 0, 0, 0, -27, -42 ],
+ [ 0, 0, 0, -162, 792 ],
+ [ 0, 0, 1, -135, -604 ],
+ [ 0, 0, 0, -5427, 153882 ],
+ [ 0, 0, 0, -201042, 34695912 ],
+ [ 0, 0, 0, -1126035, 459913278 ],
+ [ 0, 0, 1, -1127379978, -14569799990728 ] ]
diff --git a/divisibility_reductions/test_div_1.sage b/divisibility_reductions/test_div_1.sage
@@ -0,0 +1,223 @@
+from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
+
+# Pre-tests on E and P
+def suitable( E, P, ell ):
+ if not is_EllipticCurve(E):
+ print "E is not an elliptic curve"
+ return False
+ if E.base_field() != QQ:
+ print "E is not defined over Q"
+ return False
+ if not P in E:
+ print "P is not in E"
+ return False
+ if P.has_finite_order():
+ print "P has finite order"
+ return False
+ if not is_prime(ell):
+ print "ell is not prime"
+ return False
+ return True
+
+
+# Stupid auxiliary function. Returns higest power of n dividing m.
+def val( n, m ):
+ ret = 0
+ while m % (n^(ret+1)) == 0:
+ ret += 1
+ return ret
+
+# Valuation of l-divisibility of the point P
+# The parameters h and k describe the ell-part of E(F) (k>=h)
+def divisibility( P, ell, k, h ):
+ v = 0
+ while v != k-1 and P.is_divisible_by(ell^(v+1)):
+ v += 1
+ return v
+
+# label is the Cremona label of an elliptic curve over Q of rank >= 1.
+# ell is a rational prime.
+def test_label( label, ell ):
+ E = EllipticCurve(label)
+ if E.rank() < 1:
+ print "E has rank 0"
+ return
+ P = E.gens()[0]
+ test( E, P, ell )
+
+# E is an elliptic curve over Q and P a point of infinite order on E.
+# ell is a rational prime.
+def test( E, P, ell ):
+ if not suitable( E, P, ell ):
+ return
+
+ # Setting up for small primes cases
+ print "Computations for small primes starting..."
+ small_primes = []
+ vl = []
+ lpart = []
+ for p in Primes():
+ if p > 100:
+ break
+ if p == ell:
+ print "Skipping", p, "because = ell"
+ continue
+ if E.discriminant() % p == 0:
+ print "Skipping", p, "because E has bad reduction"
+ continue
+ if P.reduction(p).order() % ell != 0:
+ print "Skipping", p, "because red of P is infinitely ell-divisible"
+ continue
+ small_primes.append(p)
+ print "Working with prime p =", p
+
+ # "torsion level"
+ F = FiniteField(p)
+ vlj = []
+ lpj = []
+ for i in range(3): ### I WOULD LIKE TO CHANGE THIS TO SOMETHING BIGGER
+ print "Torsion level", i, "..."
+
+ # We can build the division fields for increasing powers of l
+ # incrementally. To get a division field, we first compute the
+ # splitting field of the division polynomial. We may be off by
+ # a degree 2 extension. If so, by finite field magic we know
+ # exactly which degree 2 extension we need: the unique one!
+ R.<x> = PolynomialRing(F)
+ F.<a> = E.reduction(p).division_polynomial(ell^i).splitting_field()
+ E_red = E.reduction(p).base_extend(F)
+ # extend if necessary
+ k = val(ell,E_red.gens()[0].order())
+ h = val(ell,E_red.order()) - k
+ if k < i or h < i:
+ F.<a> = F.extension(2)
+ E_red = E_red.base_extend(F)
+
+ P_red = E_red.point(P.reduction(p))
+
+ print "[Torsion field computed]"
+
+ # l-part of the abelian group E(F_i)
+ #gg = E_red.abelian_group().gens()
+ #if len( gg ) == 1:
+ # lpj.append( ( val(ell,gg[0].order()), 0 ) )
+ #else:
+ # lpj.append( (val(ell,gg[0].order()), val(ell,gg[1].order())) )
+ k = val(ell,E_red.gens()[0].order())
+ h = val(ell,E_red.order()) - k
+
+ lpj.append( ( k, h ) )
+ vlj.append(divisibility(P_red,ell,k,h))
+
+ vl.append(vlj)
+ lpart.append(lpj)
+
+ # Output
+ print "Done!"
+ for i in range(3):
+ print ""
+ print "******************************************************"
+ print "Torsion Level:", i
+ print ""
+ rows = [small_primes, [vl[j][i] for j in range(len(small_primes))],
+ [lpart[j][i] for j in range(len(small_primes))] ]
+ print table(rows)
+ print ""
+
+# Wrapper for densities(E,P,ell)
+def densities_label( label, ell ):
+ E = EllipticCurve(label)
+ if E.rank() < 1:
+ print "E has rank 0"
+ return
+ P = E.gens()[0]
+ densities( E, P, ell )
+
+def ratio( h, k, d, ell ):
+ x1 = max(k-d,0)
+ y1 = max(h-d,0)
+ x2 = max(k-d-1,0)
+ y2 = max(h-d-1,0)
+ up = ell^(x1+y1)-ell^(x2+y2)
+ down = ell^(h+k)
+ if d == 0:
+ return RDF((up+1)/down)
+ return RDF(up/down)
+
+
+# Computes the densities dens[n] of primes such that P is ell^n-divisible mod p
+def densities( E, P, ell ):
+ if not suitable( E, P, ell ):
+ return
+
+ n_primes = 0
+ divisible = []
+ inf_divisible = 0
+
+ # Array for counting divisibility with specified l-part
+ div_part = [[[0 for i in range(10)] for j in range(10)] for k in range(10)]
+ # total number of elements in the reductions that (do not) form the l-parts
+ tot_l_part = 0
+ tot_non_l_part = 0
+
+ for p in Primes():
+ if p > 10^2:
+ break
+ if p == ell or E.discriminant() % p == 0:
+ continue
+
+ n_primes +=1
+ #print "Working with prime", p
+
+ E_red = E.reduction(p)
+ N = E_red.order()
+ k = val(ell,E_red.gens()[0].order())
+ h = val(ell,N) - k
+ temp_non_l_part = (N / (ell^(h+k)))-1
+ tot_non_l_part += temp_non_l_part
+ temp_l_part = N - temp_non_l_part
+ tot_l_part += temp_l_part
+ # Debug
+ # print "Group structure at", p, ":"
+ # print E_red.abelian_group()
+ # print "Our result:", N, (k,h), temp_l_part, temp_non_l_part
+
+ if P.reduction(p).order() % ell != 0:
+ inf_divisible += 1
+ continue
+
+ n = divisibility( P.reduction(p), ell, \
+ val(ell,E.reduction(p).gens()[0].order()), 0 ) # Wrong parameters but ok
+ while len(divisible) < n+1:
+ divisible.append(0)
+ divisible[n] += 1
+ div_part[k][h][n] += 1
+
+ N = len(divisible)
+ divtotal = [0]*N
+ divtotal[N-1] = divisible[N-1] + inf_divisible
+ for i in range(2,N):
+ divtotal[N-i] = divisible[N-i] + divtotal[N-i+1]
+
+ print "Tested primes:", n_primes
+ for i in range(10):
+ for j in range(10):
+ s = 0
+ for l in range(10):
+ s += div_part[i][j][l]
+ if s != 0:
+ print "l-part (%d,%d):"%(i,j)
+ for l in range(10):
+ if div_part[i][j][l] != 0:
+ print "Exactly %d^%d-divisible: %d, expected %0.3f"%\
+ (ell,l,div_part[i][j][l],ratio(i,j,l,ell))
+
+ for n in range(1,N):
+ print "At least %d^%d-divisble: %0.3f density (%d times)"%(ell,n,\
+ RDF(divtotal[n]/n_primes),divtotal[n])
+ print "Infinitely-divisble: %0.3f density (%d times), expected %0.3f"%\
+ (RDF(inf_divisible/n_primes),inf_divisible,\
+ RDF(tot_non_l_part/(tot_l_part+tot_non_l_part)))
+
+
+
diff --git a/l-multiples_l-torsion/.list-3-100k.swp b/l-multiples_l-torsion/.list-3-100k.swp
Binary files differ.
diff --git a/l-multiples_l-torsion/.test4.gp.swp b/l-multiples_l-torsion/.test4.gp.swp
Binary files differ.
diff --git a/l-multiples_l-torsion/.test9.gp.swp b/l-multiples_l-torsion/.test9.gp.swp
Binary files differ.
diff --git a/l-multiples_l-torsion/.test_l-multiple.sage.swp b/l-multiples_l-torsion/.test_l-multiple.sage.swp
Binary files differ.
diff --git a/l-multiples_l-torsion/100k-3.gp b/l-multiples_l-torsion/100k-3.gp
@@ -0,0 +1,9389 @@
+
+\\ Elliptic curves downloaded from the LMFDB downloaded on 10 January 2019.
+\\ Below is a list called data. Each entry has the form:
+\\ [a1,a2,a3,a4,a6] (Weierstrass Coefficients)
+
+
+data = [\
+[0,0,1,-54,-88],\
+[0,1,1,13,42],\
+[0,1,0,-2,1],\
+[0,1,1,-83,244],\
+[1,-1,0,-96,1088],\
+[0,0,0,27,-27],\
+[0,1,1,-1423,-21113],\
+[1,-1,1,-6404,199847],\
+[0,0,1,-6924,221760],\
+[0,1,1,-27,55],\
+[0,0,1,-534,4752],\
+[1,-1,0,-6,8],\
+[1,0,0,-2,4],\
+[1,0,1,-621,-7064],\
+[0,1,1,-7,5],\
+[0,1,1,3,2],\
+[0,0,1,-210,1134],\
+[0,0,0,-9,9],\
+[1,-1,0,-96,386],\
+[1,-1,1,-29,37],\
+[1,0,1,800,1070],\
+[0,0,1,-18,29],\
+[1,0,0,-3473,-79127],\
+[1,-1,0,-177,197],\
+[0,0,1,-24,45],\
+[1,0,0,-302,-2036],\
+[0,1,1,5,6],\
+[0,1,0,-13,15],\
+[1,0,0,-57,161],\
+[0,0,1,240,1233],\
+[0,0,0,-24,-284],\
+[0,0,1,-84,315],\
+[0,0,1,-246,2043],\
+[0,0,1,0,20],\
+[1,-1,0,-177,953],\
+[1,0,0,-22,36],\
+[1,0,0,27,1],\
+[1,0,0,-36,-440],\
+[0,0,1,-16860,842625],\
+[0,0,0,-264,1636],\
+[1,-1,0,-4041,99891],\
+[0,1,1,-23,41],\
+[0,1,1,15,-69],\
+[1,0,1,714,-16080],\
+[0,0,0,-57,169],\
+[0,1,0,-93,316],\
+[1,-1,0,-81,783],\
+[1,0,0,167,225],\
+[1,0,0,-8,64],\
+[0,1,1,-163,1721],\
+[0,1,0,-77,236],\
+[0,1,1,-253,441],\
+[1,0,1,-2801,49924],\
+[1,0,1,-26,46],\
+[1,-1,0,-1332,19062],\
+[1,-1,0,-221427,40159989],\
+[0,1,0,-188,932],\
+[1,-1,0,-27,99],\
+[1,-1,0,-1197,16245],\
+[0,1,1,-7112371,-7584550689],\
+[1,-1,0,-522,4590],\
+[0,0,0,105,142],\
+[0,1,1,7,12],\
+[1,0,1,-901,-10648],\
+[0,0,1,-2268,41573],\
+[0,1,0,-228,1252],\
+[0,0,0,-135,702],\
+[0,1,1,-457,3560],\
+[0,1,1,-3275,-57546],\
+[1,-1,1,-455,3647],\
+[1,0,1,-14605,702472],\
+[0,0,0,-72,36],\
+[1,0,0,15753,666801],\
+[0,0,1,-1506,24354],\
+[1,0,0,-133932,-25444656],\
+[1,-1,1,-234509,43769909],\
+[1,0,0,3149,66081],\
+[1,-1,0,-25020,1529550],\
+[0,1,1,4415,-63694],\
+[1,-1,1,-1094,133589],\
+[0,0,1,-102,252],\
+[1,-1,0,-1122,11348],\
+[1,-1,0,-4986,136836],\
+[0,0,0,-63,198],\
+[1,0,0,12,-8],\
+[0,0,1,-270570,54171090],\
+[1,-1,1,-3380,76439],\
+[0,0,1,1734,10044],\
+[0,1,1,-333,2244],\
+[0,1,1,-36581,2679900],\
+[1,-1,1,-338,12881],\
+[1,0,1,3887,-78212],\
+[1,0,1,-256,2854],\
+[0,0,0,33,-26],\
+[1,0,0,67,113],\
+[0,0,1,348,2835],\
+[0,0,1,-894,23535],\
+[0,0,1,-378,4529],\
+[0,0,0,-62769,6052921],\
+[1,-1,0,-219660,39865616],\
+[1,-1,0,-162,3024],\
+[0,0,1,-1771248,907333974],\
+[0,1,1,-159549,-38239046],\
+[1,-1,0,-54,360],\
+[0,0,1,-486,4124],\
+[0,0,1,6,-468],\
+[1,-1,0,162,2808],\
+[1,-1,0,-24723,1502433],\
+[1,-1,0,54,288],\
+[1,0,1,-2175,-36590],\
+[1,0,1,-654,4152],\
+[0,1,1,-163,748],\
+[1,0,1,562,-105344],\
+[0,1,1,243,-4705],\
+[0,1,1,-28027,1796670],\
+[0,0,1,726,13545],\
+[1,0,0,-701,-5095],\
+[0,0,1,-90552,10509777],\
+[0,0,0,-45,117],\
+[1,0,1,121,-4894],\
+[0,1,0,-108,388],\
+[0,1,1,-4781,136281],\
+[0,0,0,15,-23],\
+[0,0,1,1302,3528],\
+[1,-1,0,-2718,54972],\
+[1,-1,0,-10950,443780],\
+[1,-1,0,-26802,1695654],\
+[1,-1,0,-3957,106037],\
+[0,1,1,7,-53],\
+[0,0,1,-12810,557235],\
+[0,0,1,-1037190,406570248],\
+[1,-1,0,-219888,39742218],\
+[0,0,0,-324,2241],\
+[0,0,0,-1992,39076],\
+[1,-1,1,2785,6023],\
+[1,0,1,-1664,30862],\
+[0,0,1,-395274,95652315],\
+[0,0,1,-3684,84330],\
+[1,-1,1,526,30449],\
+[1,-1,1,-1352,-12149],\
+[1,0,1,-31421,2098200],\
+[1,-1,1,4180,-50641],\
+[0,1,1,-863,6705],\
+[0,0,0,-489,7369],\
+[1,0,0,-2031,88145],\
+[0,0,1,-318,2178],\
+[0,1,1,579,83810],\
+[0,0,0,-15375,733750],\
+[0,1,1,-193,970],\
+[0,1,1,-157,-705],\
+[1,0,0,-2066,-41404],\
+[0,0,1,1410,4914],\
+[1,-1,0,243,6399],\
+[1,-1,1,148,2351],\
+[1,-1,1,795451,364884797],\
+[1,-1,1,-3155,72947],\
+[0,0,0,483,5749],\
+[1,0,1,-3806,-60208],\
+[1,-1,0,-2151,38151],\
+[0,1,1,-8675,-314021],\
+[0,0,1,-858,9828],\
+[1,-1,0,288,-4050],\
+[1,0,1,1031,19676],\
+[1,0,0,58,-22684],\
+[1,-1,1,-73580,-7523089],\
+[1,0,0,288,-27584],\
+[1,-1,1,-299,2107],\
+[0,0,1,-7572,253620],\
+[0,0,1,-177390,28599392],\
+[1,-1,1,-7184,234739],\
+[0,0,0,-420,3313],\
+[1,0,0,-58822,-5496476],\
+[1,0,0,-288,-27712],\
+[1,-1,1,1498,248501],\
+[1,-1,1,-167,-9241],\
+[1,0,0,-65738,-6492380],\
+[1,-1,1,-63797,6225869],\
+[1,0,0,1427,-22511],\
+[1,-1,1,-78950,8557813],\
+[1,-1,0,-5364,152748],\
+[1,-1,1,-50,-1159],\
+[1,0,0,49377,14082265],\
+[1,-1,0,-2547,51705],\
+[0,1,1,-557,4545],\
+[0,1,1,-55,209],\
+[1,-1,1,-9842,10768241],\
+[0,1,1,-79,-436],\
+[1,-1,0,-108,1368],\
+[0,0,0,267,1573],\
+[1,0,1,-1888241,-998908100],\
+[0,1,1,887,4318],\
+[1,0,1,16389,-3528890],\
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diff --git a/l-multiples_l-torsion/400k-3.gp b/l-multiples_l-torsion/400k-3.gp
@@ -0,0 +1,27648 @@
+
+\\ Elliptic curves downloaded from the LMFDB downloaded on 11 January 2019.
+\\ Below is a list called data. Each entry has the form:
+\\ [a1,a2,a3,a4,a6] (Weierstrass Coefficients)
+
+
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diff --git a/l-multiples_l-torsion/bunch_of_curves.gp b/l-multiples_l-torsion/bunch_of_curves.gp
@@ -0,0 +1,2022 @@
+
+\\ Elliptic curves downloaded from the LMFDB downloaded on 31 January 2019.
+\\ Below is a list called data. Each entry has the form:
+\\ [a1,a2,a3,a4,a6] (Weierstrass Coefficients)
+
+\\~2000 curves with rank 1 and conductor <1000
+
+data = [\
+[0,0,1,-1,0],\
+[0,1,1,0,0],\
+[1,-1,1,0,0],\
+[0,-1,1,-2,2],\
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+[0,0,0,8,0],\
+[1,1,0,3,-3],\
+[1,0,1,-15,22],\
+[1,-1,0,-2,2],\
+[1,0,0,1,25],\
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+[1,-1,1,-133,702],\
+[0,0,1,-2,-1],\
+[0,1,0,-48,-140],\
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+[0,0,0,-1451,21274],\
+[0,0,0,-91,330],\
+[0,0,0,-11,-6],\
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diff --git a/l-multiples_l-torsion/l-multiples_l-torsion.pdf b/l-multiples_l-torsion/l-multiples_l-torsion.pdf
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diff --git a/l-multiples_l-torsion/l-multiples_l-torsion.tex b/l-multiples_l-torsion/l-multiples_l-torsion.tex
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+\newcommand{\qbar}{\overline{\mathbb{Q}}}
+\newcommand{\zbar}{\overline{\mathbb{Z}}}
+
+\newcommand{\pro}{\mathbb{P}}
+\newcommand{\aff}{\mathbb{A}}
+\newcommand{\quat}{\mathbb{H}}
+\newcommand{\rea}{\mathbb{R}}
+\newcommand{\kiu}{\mathbb{Q}}
+\newcommand{\F}{\mathbb{F}}
+\newcommand{\zee}{\mathbb{Z}}
+\newcommand{\ow}{\mathcal{O}}
+\newcommand{\mcx}{\mathcal{X}}
+\newcommand{\mcy}{\mathcal{Y}}
+\newcommand{\mcs}{\mathcal{S}}
+\newcommand{\mca}{\mathcal{A}}
+\newcommand{\mcb}{\mathcal{B}}
+\newcommand{\mcf}{\mathcal{F}}
+\newcommand{\mcg}{\mathcal{G}}
+\newcommand{\mct}{\mathcal{T}}
+\newcommand{\mcq}{\mathcal{Q}}
+\newcommand{\mcr}{\mathcal{R}}
+\newcommand{\adl}{\mathbf{A}}
+\newcommand{\mbk}{\mathbf{k}}
+\newcommand{\m}{\mathfrak{m}}
+\newcommand{\p}{\mathfrak{p}}
+
+\newcommand{\kbar}{\overline{K}}
+
+\newtheorem{lemma}{Lemma}
+\newtheorem{proposition}[lemma]{Proposition}
+\newtheorem{corollary}[lemma]{Corollary}
+\theoremstyle{definition}
+\newtheorem{remark}[lemma]{Remark}
+
+\title{Rational $\ell$-multiples of points over the $\ell$-torsion field}
+\author{Sebastiano Tronto}
+
+\begin{document}
+
+\maketitle
+
+Let $\ell$ be a rational prime and let $A$ be an abelian variety of dimension $d$ over a number field $K$. Let $K_\ell=K(A[\ell])$ and let $\mathcal{T}_1=\gal(K_\ell\,|\,K)$.
+
+\begin{remark}
+For $m\geq 1$ we have $\#\gl_m(\F_\ell)=\prod_{i=0}^{\ell-1}(\ell^m-\ell^i)$. In fact, elements of $\gl_m(\F_\ell)$ are in bijection with bases of $\F^m_\ell$, and counting basis of a vector space over a finite field is a simple combinatorics exercise: first we pick any non-zero vector ($\ell^n-1$ possibilities), then we pick any vector that is not in the $\F_\ell$-span on the first one ($\ell^n-\ell$ possibilities), then a third one that is not in the span of the first two...
+
+In particular, $v_\ell\left(\#\gl_m(\F_\ell)\right)=\frac{1}{2}m(m-1)$.
+\end{remark}
+
+Compare the next lemma with \cite{jr}, Lemma 3.7.
+
+\begin{lemma}
+There is an exact sequence
+\begin{align*}
+0\to \ell A(K) \to A(K)\cap \ell A(K_{\ell})\to H^1(\mathcal{T}_1,A[\ell]).
+\end{align*}
+In particular, if $H^1(\mathcal{T}_1,A[\ell])=0$ we have $A(K)\cap \ell A(K_\ell)=\ell A(K)$.
+\begin{proof}
+Consider the short exact sequence of $\mathcal{T}_1$-modules
+\begin{align*}
+0\to A[\ell](K_{\ell})\to A(K_{\ell})\to \ell A(K_{\ell})\to 0
+\end{align*}
+and the induced long exact sequence in cohomology (i.e. take $H^*(\mathcal{T}_1,-)$)
+\begin{align*}
+0\to A[\ell](K)\to A(K)\to A(K)\cap\ell A(K_{\ell})\to H^1(\mathcal{T}_1,A[\ell])\to\cdots
+\end{align*}
+and the thesis follows by noticing that $A(K)/A[\ell](K)\cong \ell A(K)$.
+\end{proof}
+\end{lemma}
+
+This leads us to study the group $H^1(\mathcal{T}_1,A[\ell])$. In particular, we would like to know in which cases it is trivial (and so we are happy). In particular, the case of elliptic curves has been completely solved if $K=\kiu$, and there are rather complete results if $K\cap \kiu(\zeta_\ell)=\kiu$ (see \cite{lawson}).
+
+It is in fact possible that some point $\alpha\in A(K)$ is not $\ell$-divisible in $A(K)$, but becomes $\ell$-divisible in $A(K_\ell)$. The smallest example I have found is the point
+$(23769/400, 3529853/8000)$ on the elliptic curve over $\kiu$ with Cremona Label $17739g1$. A gp script that finds all the 12 examples of elliptic curves with conductor $<10^5$ with a generator of the free part of the group of rational points that becomes $3$-divisible over $K_3$ can be found in \texttt{test3.gp}.
+\\
+
+The question now becomes: ``How much'' can a point $\alpha\in A(K)$ become $\ell$-divisible in $A(K_{\ell^\infty})$? That is, can we find an (explicit) $N$ such that there is no $\beta\in A(K_{\ell^\infty})$ with $\alpha=\ell^n\beta$ for $n\geq N$?
+
+%\textbf{We restrict to the case where $A=E$ is an elliptic curve without complex multiplication}. Following the proof of Theorem 5.2 of \cite{jr} (using Lemma 3.6 of the same article), we see that $A(K)\cap \ell A(K_{\ell^n})=A(K)\cap \ell A(K_{\ell^{n-1}})$ for any $n\geq n_0$, where $n_0$ is such that $A$ satisfies maximal growth of the torsion part starting from $n_0$. So we can replace $K_{\ell^{\infty}}$ with the finite extension $K_{\ell^{n_0}}$.
+
+%Second question: work in progress! (Latest idea: use heights and the fact that that $P$ does not become ``more divisible'' by $\ell$ after a certain point; see descent part in the proof of Mordell-Weil).
+
+(If we could say $A(K)\cap\ell A(K_{\ell^\infty})=A(K)\cap\ell A(K_{\ell^{n_0}})$ for some $n_0$, then we could use Petsche's results to explicitly bound $N$ in terms of the height of $P$)
+
+\begin{thebibliography}{[99]}
+\bibitem{jr} R. Jones, J. Rouse, \emph{Galois Theory of Iterated Endomorphisms}, preprint(?).
+%\bibitem{lom} D. Lombardo, A. Perucca, \emph{Reductions of Points on Algebraic Groups}, preprint.
+%\bibitem{milne-ft} J. S. Milne, \emph{Fields and Galois Theory}, Online notes.
+\bibitem{lawson} T. Lawson, C. Wuthrich, \emph{Vanishing of some Galois cohomology groups for elliptic curves}, preprint(?).
+\end{thebibliography}
+
+\end{document}
+\ No newline at end of file
diff --git a/l-multiples_l-torsion/test3-extended.gp b/l-multiples_l-torsion/test3-extended.gp
@@ -0,0 +1,40 @@
+read("400k-3.gp");
+/* Testing all ECs with a 3-torsion point, conductor < 4*10^5 and rank > 0 */
+/* Taken from LMFBD */
+
+S = List();
+
+{
+for( i = 1, length( data ),
+ if( i % 100 == 1, print( "Tested ", i-1, " curves." ); );
+ coeffs = data[i];
+ E = ellinit( coeffs );
+ f3 = elldivpol( E, 3 ); /* 3-division polynomial */
+ eqn = y^2 + E[1]*x*y + E[3]*y - x^3 - E[2]*x^2 - E[4]*x - E[5]; /* eqn of E */
+ res = substpol( polresultant( f3, eqn ), y, x ); /* Keep x for consistency */
+ K3 = nfinit( polredbest( nfsplitting( res, 6 ) ) );
+ E_ext = ellinit( coeffs, K3 );
+ Egens = ellgenerators(E);
+ for( j = 1, length( Egens ),
+ P = Egens[j];
+ if( ellisdivisible( substpol(E_ext,x,y), P, 3, &Q ),
+ cremona_label = ellidentify(E)[1][1];
+ listput( S, [cremona_label, P, Q] );
+ write( "pari-output-ext", cremona_label );
+ write( "pari-output-ext", P );
+ write( "pari-output-ext", Q );
+ print( "Found EC ", cremona_label, " with point ", P );
+ /* Check for 3-divisibility up to torsion */
+ T = elltors(E)[3][1];
+ if( ellisdivisible( E, elladd(E,P,T), 3 ),
+ print( "P+T_1 is 3-divisible!" );
+ );
+ T = ellmul(E,T,2);
+ if( ellisdivisible( E, elladd(E,P,T), 3 ),
+ print( "P+T_2 is 3-divisible!" );
+ );
+ );
+ );
+);
+}
+
diff --git a/l-multiples_l-torsion/test4.gp b/l-multiples_l-torsion/test4.gp
@@ -0,0 +1,36 @@
+read("bunch_of_curves.gp");
+
+{
+for( i = 1, length( data ),
+ print( "Testing curve with coefficients ", data[i] );
+ E = ellinit( data[i] );
+ f4 = elldivpol( E, 4 ); /* 4-division polynomial */
+ eqn = y^2 + E[1]*x*y + E[3]*y - x^3 - E[2]*x^2 - E[4]*x - E[5]; /* eqn of E */
+ res = substpol( polresultant( f4, eqn ), y, x ); /* Keep x for consistency */
+ print( "Computing K4..." );
+ K4 = nfinit( nfsplitting( res, 96 ) );
+ print( "Computed!" );
+ E_ext = ellinit( E[1..5], K4 );
+ Egens = ellgenerators(E);
+ for( j = 1, length( Egens ),
+ P = Egens[j];
+ if( ellisdivisible( substpol(E_ext,x,y), P, 2, &Q ),
+ print( "Found EC ", cremona_label, " with point ", P,
+ " and 2-division point ", Q, " over K4" );
+ if( ellisdivisible( substpol(E_ext,x,y), P, 4, &Q ),
+ print( "Even 4-divisible! 4-division point: ", Q );
+ );
+ /* Check for 3-divisibility up to torsion */
+ /* T = elltors(E)[3][1];
+ if( ellisdivisible( E, elladd(E,P,T), 3 ),
+ print( "P+T_1 is 3-divisible!" );
+ );
+ T = ellmul(E,T,2);
+ if( ellisdivisible( E, elladd(E,P,T), 3 ),
+ print( "P+T_2 is 3-divisible!" );
+ ); */
+ );
+ );
+);
+}
+
diff --git a/l-multiples_l-torsion/test9.gp b/l-multiples_l-torsion/test9.gp
@@ -0,0 +1,34 @@
+/* Candidate counterex mod3 not surj, unexpected density, rank >=1 */
+label = [ "220a3", "220a4", "660c3", "660c4", "770f3", "770f4" ];
+
+{
+for( i = 1, length( data ),
+ print( "Testing curve ", i, ", label ", label[i] );
+ E = ellinit( label[i] );
+ f9 = elldivpol( E, 9 ); /* 3-division polynomial */
+ eqn = y^2 + E[1]*x*y + E[3]*y - x^3 - E[2]*x^2 - E[4]*x - E[5]; /* eqn of E */
+ res = substpol( polresultant( f9, eqn ), y, x ); /* Keep x for consistency */
+ print( "Computing K9..." );
+ K9 = nfinit( nfsplitting( res, 3^5*2 ) ); /* mo3 is at most 6 */
+ print( "Computed!" );
+ E_ext = ellinit( E[1..5], K9 );
+ Egens = ellgenerators(E);
+ for( j = 1, length( Egens ),
+ P = Egens[j];
+ if( ellisdivisible( substpol(E_ext,x,y), P, 3, &Q ),
+ print( "Found EC ", cremona_label, " with point ", P,
+ " and 3-division point ", Q, " over K9" );
+ /* Check for 3-divisibility up to torsion */
+ /* T = elltors(E)[3][1];
+ if( ellisdivisible( E, elladd(E,P,T), 3 ),
+ print( "P+T_1 is 3-divisible!" );
+ );
+ T = ellmul(E,T,2);
+ if( ellisdivisible( E, elladd(E,P,T), 3 ),
+ print( "P+T_2 is 3-divisible!" );
+ ); */
+ );
+ );
+);
+}
+
diff --git a/misc_code/2_division.sage b/misc_code/2_division.sage
@@ -0,0 +1,126 @@
+
+R1.<x> = PolynomialRing(QQ)
+R2.<x,y> = PolynomialRing(QQ)
+
+def extended_field( f, A, B, deg_mult ):
+ # f: a polynomial whose roots are the x-coordinates of some points of an
+ # elliptic curve E: y^2 = x^3 + Ax + B.
+ # return value: a field containing the x and y coordinates of those points
+ # deg_mult: a positive integer known to be a multiple of the degree of the
+ # extended field.
+ #
+ # This function uses the properties of resultants (I can provide a pdf
+ # explaining how it works).
+ #
+ # It is useful to compute, e.g., the fields obtained by adjoining the
+ # coordinates of the n-division points of a point (using the n-uplication
+ # formulas to get the required polynomials).
+ #
+ # When used to compute the 2-division fields, it gives the same output as
+ # E.division_field(2).
+
+
+ g = y^2 - x^3 - A*x - B
+ res = f.resultant(g,x)
+ res = res.subs(y=x)
+
+ #K.<b> = f.splitting_field()
+ #print aux
+ print "+++ Computing splitting field of the following: +++"
+
+ pol = R1(res*f)
+ print pol
+
+ K.<b> = NumberField( R( pari(pol).nfsplitting(deg_mult) ) )
+
+ return K
+
+L = []
+
+for A in range(1,9):
+ for B in range(1,9):
+
+ print "Current list of examples:", len(L), "elements. List:"
+ print L
+
+ E = EllipticCurve([0,0,0,A,B])
+ print "*************************"
+ print E
+ print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens()
+ print "CM:", E.has_cm()
+
+ rep = E.galois_representation()
+ print "mod 2 rep is surjective:", rep.is_surjective(2)
+
+ if E.rank() == 0:
+ print "Stopping because rank 0"
+ print ""
+ continue
+ if len(E.gens()) == 0:
+ print "Stopping because no points of infinite order found"
+ print ""
+ continue
+ if E.has_cm():
+ print "Stopping because CM"
+ print ""
+ print ""
+ continue
+ if not rep.is_surjective(2):
+ print "Stopping because mod 2 rep is not surjective"
+ print ""
+ continue
+
+ K_2.<a> = E.division_field(2)
+ print "2-division field:", K_2
+ P = E(0)
+ flag = False
+ for P in E.gens():
+ if len(P.division_points(2)) == 0:
+ flag = True
+ break
+ if not flag:
+ print "Stopping because the points found are 2-divisible"
+ print ""
+ continue
+ print "Taking the 2-division of P =", P
+
+ # The following polynomial is derived from the duplication formula
+ # (Silverman, p.54) using the x-coordinate of the 2-division point as
+ # an indeterminate.x
+ f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0])
+ #print f_P.roots()
+
+ M = extended_field( f_P, A, B, 24 ) # The 2-division field of P
+ print "2-division field of P:", M
+
+ if M.degree() != 24:
+ print "Stopping because 2-division of P is too small"
+ print ""
+ continue
+
+ div_pol_4 = E.division_polynomial(4)
+
+ #if div_pol_4.splitting_field('zz').degree() < 48:
+ # print "Stopping because splitting field of div_pol_4 < 48"
+ # exit()
+
+ K_4 = extended_field( div_pol_4, A, B, 96 )
+ print "4-division field:", K_4
+
+ if K_4.degree() != 96:
+ print "Stopping because mod 4 representation not surjective"
+ print ""
+ continue
+
+ if len(f_P.roots(ring=K_4)) == 0:
+ print "Stopping because M is not contained in K_4"
+ print ""
+ continue
+
+ print "----------------"
+ print "|Example Found!|"
+ print "----------------"
+ L.append((A,B))
+
+ print "*************************"
+ print ""
diff --git a/misc_code/2_division_counterex.sage b/misc_code/2_division_counterex.sage
@@ -0,0 +1,59 @@
+
+R1.<x> = PolynomialRing(QQ)
+R2.<x,y> = PolynomialRing(QQ)
+
+def extended_field( f, A, B, deg_mult ):
+ # f: a polynomial whose roots are the x-coordinates of some points of an
+ # elliptic curve E: y^2 = x^3 + Ax + B.
+ # return value: a field containing the x and y coordinates of those points
+ # deg_mult: a positive integer known to be a multiple of the degree of the
+ # extended field.
+ #
+ # This function uses the properties of resultants (I can provide a pdf
+ # explaining how it works).
+ #
+ # It is useful to compute, e.g., the fields obtained by adjoining the
+ # coordinates of the n-division points of a point (using the n-uplication
+ # formulas to get the required polynomials).
+ #
+ # When used to compute the 2-division fields, it gives the same output as
+ # E.division_field(2).
+
+
+ g = y^2 - x^3 - A*x - B
+ res = f.resultant(g,x)
+ res = res.subs(y=x)
+
+ #print "Splitting field of division pol:"
+ #print R1(f).splitting_field('r')
+ K.<b> = (R1(res*f)).splitting_field(degree_multiple=deg_mult)
+
+ return K
+
+
+A = -3
+B = 17/4
+
+E = EllipticCurve([0,0,0,A,B])
+print E
+print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens()
+print "CM:", E.has_cm()
+
+rep = E.galois_representation()
+print "mod 2 rep is surjective:", rep.is_surjective(2)
+
+K_2.<a> = E.division_field(2)
+print "2-division field:", K_2
+P = [4,15/2]
+
+# The following polynomial is derived from the duplication formula
+# (Silverman, p.54) using the x-coordinate of the 2-division point as
+# an indeterminate.x
+f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0])
+
+M = extended_field( f_P, A, B, 24 ) # The 2-division field of P
+print "2-division field of P:", M
+
+if M.degree() != 24:
+ print "Stopping because 2-division of P is too small"
+ exit()
diff --git a/misc_code/2_division_onecurve.sage b/misc_code/2_division_onecurve.sage
@@ -0,0 +1,101 @@
+
+R1.<x> = PolynomialRing(QQ)
+R2.<x,y> = PolynomialRing(QQ)
+
+def extended_field( f, A, B, deg_mult ):
+ # f: a polynomial whose roots are the x-coordinates of some points of an
+ # elliptic curve E: y^2 = x^3 + Ax + B.
+ # return value: a field containing the x and y coordinates of those points
+ # deg_mult: a positive integer known to be a multiple of the degree of the
+ # extended field.
+ #
+ # This function uses the properties of resultants (I can provide a pdf
+ # explaining how it works).
+ #
+ # It is useful to compute, e.g., the fields obtained by adjoining the
+ # coordinates of the n-division points of a point (using the n-uplication
+ # formulas to get the required polynomials).
+ #
+ # When used to compute the 2-division fields, it gives the same output as
+ # E.division_field(2).
+
+
+ g = y^2 - x^3 - A*x - B
+ res = f.resultant(g,x)
+ res = res.subs(y=x)
+
+ #print "Splitting field of division pol:"
+ #print R1(f).splitting_field('r')
+ K.<b> = (R1(res*f)).splitting_field(degree_multiple=deg_mult)
+
+ return K
+
+
+A = -3
+B = 17/4
+
+E = EllipticCurve([0,0,0,A,B])
+print "*************************"
+print E
+print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens()
+print "CM:", E.has_cm()
+
+rep = E.galois_representation()
+print "mod 2 rep is surjective:", rep.is_surjective(2)
+
+if E.rank() == 0:
+ print "Stopping because rank 0"
+ exit()
+if len(E.gens()) == 0:
+ print "Stopping because no points of infinite order found"
+ exit()
+if E.has_cm():
+ print "Stopping because CM"
+ print ""
+ exit()
+if not rep.is_surjective(2):
+ print "Stopping because mod 2 rep is not surjective"
+ exit()
+
+K_2.<a> = E.division_field(2)
+print "2-division field:", K_2
+P = E(0)
+flag = False
+for P in E.gens():
+ if len(P.division_points(2)) == 0:
+ flag = True
+ break
+if not flag:
+ print "Stopping because the points found are 2-divisible"
+ exit()
+print "Taking the 2-division of P =", P
+
+# The following polynomial is derived from the duplication formula
+# (Silverman, p.54) using the x-coordinate of the 2-division point as
+# an indeterminate.x
+f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0])
+
+M = extended_field( f_P, A, B, 24 ) # The 2-division field of P
+print "2-division field of P:", M
+
+if M.degree() != 24:
+ print "Stopping because 2-division of P is too small"
+ exit()
+
+K_4 = extended_field( E.division_polynomial(4), A, B, 96 )
+print "4-division field:", K_4
+
+if K_4.degree() != 96:
+ print "Stopping because mod 4 representation not surjective"
+ exit()
+
+if len(f_P.roots(ring=K_4)) == 0:
+ print "Stopping because E is not contained in K_4"
+ exit()
+
+print "----------------"
+print "|Example Found!|"
+print "----------------"
+
+print "*************************"
+print ""
diff --git a/misc_code/2_division_withparisplit.sage b/misc_code/2_division_withparisplit.sage
@@ -0,0 +1,122 @@
+
+R1.<x> = PolynomialRing(QQ)
+R2.<x,y> = PolynomialRing(QQ)
+
+def extended_field( f, A, B, deg_mult ):
+ # f: a polynomial whose roots are the x-coordinates of some points of an
+ # elliptic curve E: y^2 = x^3 + Ax + B.
+ # return value: a field containing the x and y coordinates of those points
+ # deg_mult: a positive integer known to be a multiple of the degree of the
+ # extended field.
+ #
+ # This function uses the properties of resultants (I can provide a pdf
+ # explaining how it works).
+ #
+ # It is useful to compute, e.g., the fields obtained by adjoining the
+ # coordinates of the n-division points of a point (using the n-uplication
+ # formulas to get the required polynomials).
+ #
+ # When used to compute the 2-division fields, it gives the same output as
+ # E.division_field(2).
+
+
+ g = y^2 - x^3 - A*x - B
+ res = f.resultant(g,x)
+ res = res.subs(y=x)
+
+ #K.<b> = f.splitting_field()
+ #print aux
+ pol = pari(R1(res*f)).nfsplitting(deg_mult)
+ K.<b> = NumberField(g)
+
+ return K
+
+L = []
+
+for A in range(1,9):
+ for B in range(1,9):
+
+ print "Current list of examples:", len(L), "elements. List:"
+ print L
+
+ E = EllipticCurve([0,0,0,A,B])
+ print "*************************"
+ print E
+ print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens()
+ print "CM:", E.has_cm()
+
+ rep = E.galois_representation()
+ print "mod 2 rep is surjective:", rep.is_surjective(2)
+
+ if E.rank() == 0:
+ print "Stopping because rank 0"
+ print ""
+ continue
+ if len(E.gens()) == 0:
+ print "Stopping because no points of infinite order found"
+ print ""
+ continue
+ if E.has_cm():
+ print "Stopping because CM"
+ print ""
+ print ""
+ continue
+ if not rep.is_surjective(2):
+ print "Stopping because mod 2 rep is not surjective"
+ print ""
+ continue
+
+ K_2.<a> = E.division_field(2)
+ print "2-division field:", K_2
+ P = E(0)
+ flag = False
+ for P in E.gens():
+ if len(P.division_points(2)) == 0:
+ flag = True
+ break
+ if not flag:
+ print "Stopping because the points found are 2-divisible"
+ print ""
+ continue
+ print "Taking the 2-division of P =", P
+
+ # The following polynomial is derived from the duplication formula
+ # (Silverman, p.54) using the x-coordinate of the 2-division point as
+ # an indeterminate.x
+ f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0])
+ #print f_P.roots()
+
+ M = extended_field( f_P, A, B, 24 ) # The 2-division field of P
+ print "2-division field of P:", M
+
+ if M.degree() != 24:
+ print "Stopping because 2-division of P is too small"
+ print ""
+ continue
+
+ div_pol_4 = E.division_polynomial(4)
+
+ #if div_pol_4.splitting_field('zz').degree() < 48:
+ # print "Stopping because splitting field of div_pol_4 < 48"
+ # exit()
+
+ K_4 = extended_field( div_pol_4, A, B, 96 )
+ print "4-division field:", K_4
+
+ if K_4.degree() != 96:
+ print "Stopping because mod 4 representation not surjective"
+ print ""
+ continue
+
+ if len(f_P.roots(ring=K_4)) == 0:
+ print "Stopping because M is not contained in K_4"
+ print ""
+ continue
+
+ print "----------------"
+ print "|Example Found!|"
+ print "----------------"
+ L.append((A,B))
+
+ print "*************************"
+ print ""
diff --git a/misc_code/Q49-49a1-49a2.gp b/misc_code/Q49-49a1-49a2.gp
@@ -0,0 +1,14 @@
+elltorsfield_deg( E, n, deg ) = {
+ my( f, eqn, res );
+ f = elldivpol( E, n );
+ eqn = y^2 + E[1]*x*y + E[3]*y - x^3 - E[2]*x^2 - E[4]*x - E[5];
+ res = substpol( polresultant( f, eqn ), y, x );
+ printf( "Computing %d-torsion field...\n", n );
+ return( nfinit( nfsplitting( f*res, deg ) ) );
+}
+
+print("Computing 49-torsion field for 49a1, expecting maximal degree 42*49\n");
+elltorsfield_deg( ellinit( "49a1" ), 49, 42*49 );
+
+print("Computing 49-torsion field for 49a1, expecting maximal degree 42*49\n");
+elltorsfield_deg( ellinit( "49a1" ), 49, 42*49 );
diff --git a/misc_code/adelic_failure_example b/misc_code/adelic_failure_example
@@ -0,0 +1,3 @@
+E = EllipticCurve([0, 1, 0, -44, -84])
+print "Rational torsion points:", E.torsion_points()
+print "Generator of the free part:", E.gens()
diff --git a/misc_code/adelic_failure_example.sage b/misc_code/adelic_failure_example.sage
@@ -0,0 +1,10 @@
+E = EllipticCurve([0, 1, 0, -44, -84])
+print "Rational torsion points:", E.torsion_points()
+print "Generators of the free part:", E.gens()
+P = E.gens()[0]
+
+K = CyclotomicField(3)
+EK = E.base_extend(K)
+PK = EK.point(P)
+
+print "2-division points over Q(zeta_3):", PK.division_points(2)
diff --git a/misc_code/ugly_computations/GL2_ZN.py b/misc_code/ugly_computations/GL2_ZN.py
@@ -0,0 +1,24 @@
+# Prints a list of matrices, the elements of GL_2(Z/nZ)
+
+def gcd( m, n ):
+ if n == 0:
+ return m
+ return gcd( n, m%n )
+
+def print_mat( a, b, c, d ):
+ print ""
+ print "|", a, " ", b, "|"
+ print "|", c, " ", d, "|"
+
+n = input("Choose n: ")
+
+count = 0
+for a in range(n):
+ for b in range(n):
+ for c in range(n):
+ for d in range(n):
+ if gcd( n, abs(a*d-b*c)) == 1:
+ print_mat(a,b,c,d)
+ count += 1
+
+print "Found", count, "elements"
diff --git a/misc_code/ugly_computations/cm/cm-rank+-all.sage b/misc_code/ugly_computations/cm/cm-rank+-all.sage
@@ -0,0 +1,3361 @@
+
+# Elliptic curves downloaded from the LMFDB downloaded on 23 October 2019.
+# Below is a list called data. Each entry has the form:
+# [a1,a2,a3,a4,a6] (Weierstrass Coefficients)
+
+
+data = [\
+[0, -1, 1, -887, -10143],\
+[0, -1, 1, -7, 10],\
+[0, 0, 1, 0, -34],\
+[0, 0, 1, 0, 1],\
+[0, 0, 1, 0, -1],\
+[0, 0, 1, 0, 20],\
+[0, 1, 0, -13, -21],\
+[0, 1, 0, -3, 1],\
+[0, 0, 0, -2, 0],\
+[0, 0, 0, 8, 0],\
+[0, 0, 0, -12, 0],\
+[0, 0, 0, 3, 0],\
+[0, 0, 1, -13718, -619025],\
+[0, 0, 1, -38, 90],\
+[0, 0, 0, 0, -4],\
+[0, 0, 0, 0, 108],\
+[1, -1, 1, -16400, -804212],\
+[1, -1, 1, -965, -13940],\
+[1, -1, 1, -335, 2440],\
+[1, -1, 1, -20, 46],\
+[0, 0, 1, 0, -331],\
+[0, 0, 1, 0, 12],\
+[0, 0, 0, -396, -3024],\
+[0, 0, 0, -396, 3024],\
+[0, 0, 0, -36, 0],\
+[0, 0, 0, 9, 0],\
+[0, 0, 0, -540, -4752],\
+[0, 0, 0, -60, 176],\
+[0, 0, 0, 0, -216],\
+[0, 0, 0, 0, 8],\
+[0, 0, 1, -6750, -213469],\
+[0, 0, 1, -750, 7906],\
+[0, 0, 1, 0, -844],\
+[0, 0, 1, 0, 31],\
+[0, 0, 0, -29155, -1915998],\
+[0, 0, 0, -1715, -33614],\
+[0, 0, 0, -595, 5586],\
+[0, 0, 0, -35, 98],\
+[0, 0, 0, -275, -1750],\
+[0, 0, 0, -275, 1750],\
+[0, 0, 0, -25, 0],\
+[0, 0, 0, 100, 0],\
+[0, 0, 0, -5, 0],\
+[0, 0, 0, 20, 0],\
+[0, 0, 0, 0, -2700],\
+[0, 0, 0, 0, 100],\
+[0, 0, 0, 0, -972],\
+[0, 0, 0, 0, 36],\
+[0, 0, 0, 0, -243],\
+[0, 0, 0, 0, 9],\
+[0, 0, 1, 0, -2],\
+[0, 0, 1, 0, 47],\
+[0, 0, 1, 0, -16207],\
+[0, 0, 1, 0, 600],\
+[0, 0, 1, 0, -192787],\
+[0, 0, 1, 0, 7140],\
+[0, 0, 1, 0, -88],\
+[0, 0, 1, 0, 3],\
+[0, 0, 0, -539, -4802],\
+[0, 0, 0, -539, 4802],\
+[0, 0, 0, -49, 0],\
+[0, 0, 0, 196, 0],\
+[0, 0, 0, -500, 0],\
+[0, 0, 0, 125, 0],\
+[0, 0, 0, -20, 0],\
+[0, 0, 0, 5, 0],\
+[0, 0, 0, -1080, -13662],\
+[0, 0, 0, -120, 506],\
+[0, 0, 0, 0, -54],\
+[0, 0, 0, 0, 2],\
+[0, 0, 0, -1080, 13662],\
+[0, 0, 0, -120, -506],\
+[0, 0, 0, 0, -2],\
+[0, 0, 0, 0, 54],\
+[0, 0, 0, 0, -28],\
+[0, 0, 0, 0, 756],\
+[0, 0, 0, -6615, 203742],\
+[0, 0, 0, -735, -7546],\
+[0, 0, 0, 0, -343],\
+[0, 0, 0, 0, 9261],\
+[0, 0, 1, -1590140, -771794326],\
+[0, 0, 1, -860, 9707],\
+[0, 0, 0, -72, 0],\
+[0, 0, 0, 18, 0],\
+[0, 0, 0, -6, 0],\
+[0, 0, 0, 24, 0],\
+[0, 0, 0, -120, -448],\
+[0, 0, 0, -30, 56],\
+[0, 0, 0, -54, 0],\
+[0, 0, 0, 216, 0],\
+[0, 0, 0, 0, -13500],\
+[0, 0, 0, 0, 500],\
+[0, 0, 0, 0, -16875],\
+[0, 0, 0, 0, 625],\
+[0, 0, 0, 0, -135],\
+[0, 0, 0, 0, 5],\
+[0, 1, 1, -22183, -1312206],\
+[0, 1, 1, -183, 919],\
+[0, 0, 0, -7, 0],\
+[0, 0, 0, 28, 0],\
+[0, 0, 0, -2156, -38416],\
+[0, 0, 0, -2156, 38416],\
+[0, 0, 0, -196, 0],\
+[0, 0, 0, 49, 0],\
+[0, 0, 0, -116620, -15327984],\
+[0, 0, 0, -6860, -268912],\
+[0, 0, 0, -2380, 44688],\
+[0, 0, 0, -140, 784],\
+[0, 0, 0, -343, 0],\
+[0, 0, 0, 1372, 0],\
+[0, 0, 1, -32670, 2273015],\
+[0, 0, 1, -3630, -84186],\
+[0, 0, 1, 0, -333],\
+[0, 0, 1, 0, 8984],\
+[0, 0, 0, 0, -10000],\
+[0, 0, 0, 0, 270000],\
+[0, 0, 0, 0, -100],\
+[0, 0, 0, 0, 2700],\
+[0, 0, 0, -5324, 0],\
+[0, 0, 0, 1331, 0],\
+[0, 0, 0, 0, -81],\
+[0, 0, 0, 0, 3],\
+[0, 0, 0, 0, -324],\
+[0, 0, 0, 0, 12],\
+[0, 0, 0, 0, -1296],\
+[0, 0, 0, 0, 48],\
+[0, 0, 0, 0, -1581228],\
+[0, 0, 0, 0, 58564],\
+[0, 0, 0, 0, -44],\
+[0, 0, 0, 0, 1188],\
+[0, 0, 0, -16335, 790614],\
+[0, 0, 0, -1815, -29282],\
+[0, 0, 0, 0, -1331],\
+[0, 0, 0, 0, 35937],\
+[0, 0, 1, -33083930, -73244287055],\
+[0, 0, 1, -7370, 243528],\
+[0, 0, 1, 0, -1141],\
+[0, 0, 1, 0, 42],\
+[0, 0, 1, 0, -2506228],\
+[0, 0, 1, 0, 92823],\
+[0, 0, 1, 0, -6],\
+[0, 0, 1, 0, 155],\
+[0, 0, 0, 0, -64827],\
+[0, 0, 0, 0, 2401],\
+[0, 0, 0, 0, -67228],\
+[0, 0, 0, 0, 1815156],\
+[0, 0, 0, 0, -5292],\
+[0, 0, 0, 0, 196],\
+[0, 0, 0, 0, -7],\
+[0, 0, 0, 0, 189],\
+[0, 0, 0, -2197, 0],\
+[0, 0, 0, 8788, 0],\
+[0, 0, 0, -1859, -30758],\
+[0, 0, 0, -1859, 30758],\
+[0, 0, 0, -169, 0],\
+[0, 0, 0, 676, 0],\
+[1, -1, 1, -220485, -39791650],\
+[1, -1, 1, -12970, -695824],\
+[1, -1, 1, -4500, 117296],\
+[1, -1, 1, -265, 2104],\
+[0, 0, 1, 0, -7594],\
+[0, 0, 1, 0, 281],\
+[0, 0, 1, 0, -37969],\
+[0, 0, 1, 0, 1406],\
+[0, 0, 1, 0, -1519],\
+[0, 0, 1, 0, 56],\
+[0, 0, 1, 0, -189844],\
+[0, 0, 1, 0, 7031],\
+[0, 0, 1, 0, -304],\
+[0, 0, 1, 0, 11],\
+[0, 0, 0, -10, 0],\
+[0, 0, 0, 40, 0],\
+[0, 0, 0, -50, 0],\
+[0, 0, 0, 200, 0],\
+[0, 0, 0, -250, 0],\
+[0, 0, 0, 1000, 0],\
+[0, -1, 0, -333, -1963],\
+[0, -1, 0, -83, 287],\
+[0, 0, 0, 0, -756],\
+[0, 0, 0, 0, 28],\
+[0, 0, 0, -6615, -203742],\
+[0, 0, 0, -735, 7546],\
+[0, 0, 0, 0, -9261],\
+[0, 0, 0, 0, 343],\
+[0, 0, 0, 0, -7260624],\
+[0, 0, 0, 0, 268912],\
+[0, 0, 0, -2475, -47250],\
+[0, 0, 0, -2475, 47250],\
+[0, 0, 0, -225, 0],\
+[0, 0, 0, 900, 0],\
+[0, 0, 0, -13500, 0],\
+[0, 0, 0, 3375, 0],\
+[0, 0, 0, -45, 0],\
+[0, 0, 0, 180, 0],\
+[0, 0, 0, -60, 0],\
+[0, 0, 0, 15, 0],\
+[0, 0, 0, -2700, 0],\
+[0, 0, 0, 675, 0],\
+[0, 0, 1, 0, -138450256],\
+[0, 0, 1, 0, 5127787],\
+[0, 0, 0, -5324, -149072],\
+[0, 0, 0, -5324, 149072],\
+[0, 0, 0, -484, 0],\
+[0, 0, 0, 121, 0],\
+[0, 1, 0, -3549, -84691],\
+[0, 1, 0, -29, 53],\
+[0, 0, 1, 0, -9584035],\
+[0, 0, 1, 0, 354964],\
+[0, 0, 1, 0, -563767],\
+[0, 0, 1, 0, 20880],\
+[0, 0, 1, -78030, -8390176],\
+[0, 0, 1, -8670, 310747],\
+[0, 0, 1, 0, -33163],\
+[0, 0, 1, 0, 1228],\
+[0, 0, 1, 0, -115],\
+[0, 0, 1, 0, 4],\
+[0, 0, 1, 0, -1951],\
+[0, 0, 1, 0, 72],\
+[0, 0, 1, 0, -8],\
+[0, 0, 1, 0, 209],\
+[0, 0, 1, 0, -6233767],\
+[0, 0, 1, 0, 230880],\
+[0, 0, 1, -342950, -77378094],\
+[0, 0, 1, -950, 11281],\
+[0, 0, 0, -17, 0],\
+[0, 0, 0, 68, 0],\
+[0, 0, 1, 0, -2437],\
+[0, 0, 1, 0, 90],\
+[0, 0, 0, 0, -459],\
+[0, 0, 0, 0, 17],\
+[0, 0, 0, 0, -7803],\
+[0, 0, 0, 0, 289],\
+[0, 0, 0, 0, -2255067],\
+[0, 0, 0, 0, 83521],\
+[0, 0, 0, 0, -25],\
+[0, 0, 0, 0, 675],\
+[0, 0, 0, 0, -20],\
+[0, 0, 0, 0, 540],\
+[0, 0, 0, -108000, 13662000],\
+[0, 0, 0, -12000, -506000],\
+[0, 0, 0, 0, -2000],\
+[0, 0, 0, 0, 54000],\
+[0, 0, 0, 0, -2500],\
+[0, 0, 0, 0, 67500],\
+[0, 0, 0, 0, -3125],\
+[0, 0, 0, 0, 84375],\
+[0, 0, 0, -52, 0],\
+[0, 0, 0, 13, 0],\
+[0, 0, 0, -8788, 0],\
+[0, 0, 0, 2197, 0],\
+[0, 0, 1, 0, -53594],\
+[0, 0, 1, 0, 1447031],\
+[0, 0, 1, 0, -44],\
+[0, 0, 1, 0, 1181],\
+[0, 0, 1, 0, -50646094],\
+[0, 0, 1, 0, 1875781],\
+[0, 0, 1, 0, -41344],\
+[0, 0, 1, 0, 1531],\
+[1, -1, 0, -409992, -100936459],\
+[1, -1, 0, -24117, -1766584],\
+[1, -1, 0, -8367, 296666],\
+[1, -1, 0, -492, 5291],\
+[0, 0, 0, -76, 0],\
+[0, 0, 0, 19, 0],\
+[0, 0, 1, 0, -142],\
+[0, 0, 1, 0, 5],\
+[0, 0, 1, 0, -340342],\
+[0, 0, 1, 0, 12605],\
+[0, 0, 1, 0, -37],\
+[0, 0, 1, 0, 992],\
+[0, 0, 1, 0, -1801],\
+[0, 0, 1, 0, 48620],\
+[0, 0, 1, 0, -6946],\
+[0, 0, 1, 0, 257],\
+[0, 1, 0, -653, -5909],\
+[0, 1, 0, -163, 657],\
+[0, 0, 0, -392, 0],\
+[0, 0, 0, 98, 0],\
+[0, 0, 0, -56, 0],\
+[0, 0, 0, 14, 0],\
+[0, 0, 0, -686, 0],\
+[0, 0, 0, 2744, 0],\
+[0, 0, 0, -14, 0],\
+[0, 0, 0, 56, 0],\
+[0, 0, 0, -48735, 4074246],\
+[0, 0, 0, -5415, -150898],\
+[0, 0, 0, 0, -6859],\
+[0, 0, 0, 0, 185193],\
+[0, 0, 0, 0, -19],\
+[0, 0, 0, 0, 513],\
+[0, 0, 0, 0, -2476099],\
+[0, 0, 0, 0, 66854673],\
+[0, 0, 0, 0, -11],\
+[0, 0, 0, 0, 297],\
+[0, 0, 0, 0, -5324],\
+[0, 0, 0, 0, 143748],\
+[0, 0, 0, 0, -3267],\
+[0, 0, 0, 0, 121],\
+[0, 0, 0, 0, -161051],\
+[0, 0, 0, 0, 4348377],\
+[0, 0, 0, -252, 0],\
+[0, 0, 0, 63, 0],\
+[0, 0, 0, -5292, 0],\
+[0, 0, 0, 1323, 0],\
+[0, 0, 0, -4851, -129654],\
+[0, 0, 0, -4851, 129654],\
+[0, 0, 0, -441, 0],\
+[0, 0, 0, 1764, 0],\
+[0, 0, 0, -21, 0],\
+[0, 0, 0, 84, 0],\
+[0, 0, 0, -1029, 0],\
+[0, 0, 0, 4116, 0],\
+[0, 0, 1, -142830, 20778194],\
+[0, 0, 1, -15870, -769563],\
+[0, 0, 1, 0, -3042],\
+[0, 0, 1, 0, 82127],\
+[0, 0, 0, 0, -1250],\
+[0, 0, 0, 0, 33750],\
+[0, 0, 0, 0, -270],\
+[0, 0, 0, 0, 10],\
+[0, 0, 0, -15, 0],\
+[0, 0, 0, 60, 0],\
+[0, 0, 0, -135, 0],\
+[0, 0, 0, 540, 0],\
+[0, 0, 0, -3375, 0],\
+[0, 0, 0, 13500, 0],\
+[0, 0, 0, -375, 0],\
+[0, 0, 0, 1500, 0],\
+[0, 0, 0, -9900, -378000],\
+[0, 0, 0, -9900, 378000],\
+[0, 0, 0, -900, 0],\
+[0, 0, 0, 225, 0],\
+[0, 0, 0, 0, -2700000],\
+[0, 0, 0, 0, 100000],\
+[0, 0, 0, 0, -800],\
+[0, 0, 0, 0, 21600],\
+[0, 0, 0, -13500, -594000],\
+[0, 0, 0, -1500, 22000],\
+[0, 0, 0, 0, -27000],\
+[0, 0, 0, 0, 1000],\
+[0, 0, 1, 0, -277],\
+[0, 0, 1, 0, 10],\
+[0, 0, 0, 0, -648],\
+[0, 0, 0, 0, 24],\
+[0, 0, 0, 0, -18],\
+[0, 0, 0, 0, 486],\
+[0, 0, 0, 0, -7776],\
+[0, 0, 0, 0, 288],\
+[0, 0, 0, 0, -1944],\
+[0, 0, 0, 0, 72],\
+[0, 0, 0, 0, -72],\
+[0, 0, 0, 0, 1944],\
+[0, 0, 0, 0, -288],\
+[0, 0, 0, 0, 7776],\
+[0, 0, 0, 0, -486],\
+[0, 0, 0, 0, 18],\
+[0, 0, 1, 0, -12481],\
+[0, 0, 1, 0, 462],\
+[0, 0, 0, -5819, -170338],\
+[0, 0, 0, -5819, 170338],\
+[0, 0, 0, -529, 0],\
+[0, 0, 0, 2116, 0],\
+[0, 0, 0, 0, -1936],\
+[0, 0, 0, 0, 52272],\
+[0, 0, 0, -16335, -790614],\
+[0, 0, 0, -1815, 29282],\
+[0, 0, 0, 0, -35937],\
+[0, 0, 0, 0, 1331],\
+[0, 0, 0, 0, -1188],\
+[0, 0, 0, 0, 44],\
+[0, 0, 0, 0, -58564],\
+[0, 0, 0, 0, 1581228],\
+[0, 0, 0, 0, -69574032],\
+[0, 0, 0, 0, 2576816],\
+[0, 0, 0, -127776, -18037712],\
+[0, 0, 0, -1056, 13552],\
+[1, -1, 1, -657810, -205176646],\
+[1, -1, 1, -38695, -3592802],\
+[1, -1, 1, -13425, 602018],\
+[1, -1, 1, -790, 10700],\
+[0, 0, 0, 0, -1404],\
+[0, 0, 0, 0, 52],\
+[0, 0, 0, 0, -4563],\
+[0, 0, 0, 0, 169],\
+[0, 0, 0, 0, -10024911],\
+[0, 0, 0, 0, 371293],\
+[0, 0, 0, 0, -3084588],\
+[0, 0, 0, 0, 114244],\
+[0, 0, 0, -12716, -550256],\
+[0, 0, 0, -12716, 550256],\
+[0, 0, 0, -1156, 0],\
+[0, 0, 0, 289, 0],\
+[0, 0, 0, 0, -57132],\
+[0, 0, 0, 0, 2116],\
+[0, 0, 0, 0, -25745372],\
+[0, 0, 0, 0, 695125044],\
+[0, 0, 0, -71415, 7227198],\
+[0, 0, 0, -7935, -267674],\
+[0, 0, 0, 0, -12167],\
+[0, 0, 0, 0, 328509],\
+[0, 0, 0, -728875, -239499750],\
+[0, 0, 0, -42875, -4201750],\
+[0, 0, 0, -14875, 698250],\
+[0, 0, 0, -875, 12250],\
+[0, -1, 1, -149959, -22883378],\
+[0, -1, 1, -1239, 17643],\
+[0, 0, 0, 0, -453789],\
+[0, 0, 0, 0, 16807],\
+[0, 0, 0, 0, -3024],\
+[0, 0, 0, 0, 112],\
+[0, 0, 0, 0, -38416],\
+[0, 0, 0, 0, 1037232],\
+[0, 0, 0, 0, -49],\
+[0, 0, 0, 0, 1323],\
+[0, 0, 0, 0, -37044],\
+[0, 0, 0, 0, 1372],\
+[0, 0, 1, 0, -196],\
+[0, 0, 1, 0, 7],\
+[0, 0, 1, -227070, -41650315],\
+[0, 0, 1, -25230, 1542604],\
+[0, 0, 1, 0, -164626],\
+[0, 0, 1, 0, 6097],\
+[0, 0, 0, -54872, 4952198],\
+[0, 0, 0, -152, -722],\
+[0, 0, 0, -54872, -4952198],\
+[0, 0, 0, -152, 722],\
+[0, 0, 0, -15884, -768208],\
+[0, 0, 0, -15884, 768208],\
+[0, 0, 0, -1444, 0],\
+[0, 0, 0, 361, 0],\
+[0, 0, 0, 0, -375],\
+[0, 0, 0, 0, 10125],\
+[0, 0, 0, 0, -60],\
+[0, 0, 0, 0, 1620],\
+[0, 0, 0, 0, -6075],\
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diff --git a/misc_code/ugly_computations/cm/test-cm-rank.sage b/misc_code/ugly_computations/cm/test-cm-rank.sage
@@ -0,0 +1,16 @@
+def test():
+ attach("cm-rank+-all.sage")
+ no = 0
+ unk = 0
+ for cf in data:
+ E = EllipticCurve(cf)
+ E_K = E.base_extend(NumberField(x^2-E.cm_discriminant(),'a'))
+ try:
+ if E_K.rank() != 2* E.rank():
+ print "Non doubling:", cf, E.rank(), E_K.rank()
+ no += 1
+ except:
+ print "Can't compute the rank for", cf
+ unk += 1
+ print "Total ok curves:", (len(data)-(no+unk)), "/", len(data), \
+ "( unknown:", unk, ")"
diff --git a/misc_code/ugly_computations/order_GLn.pdf b/misc_code/ugly_computations/order_GLn.pdf
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diff --git a/misc_code/ugly_computations/quadratic_class_number.sage b/misc_code/ugly_computations/quadratic_class_number.sage
@@ -0,0 +1,12 @@
+# Determines the proportion of real quadratic fields with odd class number.
+def f(B):
+ sqf = [ b for b in range(2,B) if b%4==1 and is_squarefree(b) ]
+ total_odd = 0
+
+ R.<x> = PolynomialRing(QQ)
+ for a in sqf:
+ if NumberField( x^2 - a, 'c' ).class_number() % 2 == 1:
+ total_odd += 1
+
+ print "=1 mod 4 squarefree integers up to", B, ": ", len(sqf)
+ print "With odd class number:", total_odd
diff --git a/necessity_52b/2_division.sage b/necessity_52b/2_division.sage
@@ -0,0 +1,126 @@
+
+R1.<x> = PolynomialRing(QQ)
+R2.<x,y> = PolynomialRing(QQ)
+
+def extended_field( f, A, B, deg_mult ):
+ # f: a polynomial whose roots are the x-coordinates of some points of an
+ # elliptic curve E: y^2 = x^3 + Ax + B.
+ # return value: a field containing the x and y coordinates of those points
+ # deg_mult: a positive integer known to be a multiple of the degree of the
+ # extended field.
+ #
+ # This function uses the properties of resultants (I can provide a pdf
+ # explaining how it works).
+ #
+ # It is useful to compute, e.g., the fields obtained by adjoining the
+ # coordinates of the n-division points of a point (using the n-uplication
+ # formulas to get the required polynomials).
+ #
+ # When used to compute the 2-division fields, it gives the same output as
+ # E.division_field(2).
+
+
+ g = y^2 - x^3 - A*x - B
+ res = f.resultant(g,x)
+ res = res.subs(y=x)
+
+ #K.<b> = f.splitting_field()
+ #print aux
+ print "+++ Computing splitting field of the following: +++"
+
+ pol = R1(res*f)
+ print pol
+
+ K.<b> = NumberField( R( pari(pol).nfsplitting(deg_mult) ) )
+
+ return K
+
+L = []
+
+for A in range(1,9):
+ for B in range(1,9):
+
+ print "Current list of examples:", len(L), "elements. List:"
+ print L
+
+ E = EllipticCurve([0,0,0,A,B])
+ print "*************************"
+ print E
+ print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens()
+ print "CM:", E.has_cm()
+
+ rep = E.galois_representation()
+ print "mod 2 rep is surjective:", rep.is_surjective(2)
+
+ if E.rank() == 0:
+ print "Stopping because rank 0"
+ print ""
+ continue
+ if len(E.gens()) == 0:
+ print "Stopping because no points of infinite order found"
+ print ""
+ continue
+ if E.has_cm():
+ print "Stopping because CM"
+ print ""
+ print ""
+ continue
+ if not rep.is_surjective(2):
+ print "Stopping because mod 2 rep is not surjective"
+ print ""
+ continue
+
+ K_2.<a> = E.division_field(2)
+ print "2-division field:", K_2
+ P = E(0)
+ flag = False
+ for P in E.gens():
+ if len(P.division_points(2)) == 0:
+ flag = True
+ break
+ if not flag:
+ print "Stopping because the points found are 2-divisible"
+ print ""
+ continue
+ print "Taking the 2-division of P =", P
+
+ # The following polynomial is derived from the duplication formula
+ # (Silverman, p.54) using the x-coordinate of the 2-division point as
+ # an indeterminate.x
+ f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0])
+ #print f_P.roots()
+
+ M = extended_field( f_P, A, B, 24 ) # The 2-division field of P
+ print "2-division field of P:", M
+
+ if M.degree() != 24:
+ print "Stopping because 2-division of P is too small"
+ print ""
+ continue
+
+ div_pol_4 = E.division_polynomial(4)
+
+ #if div_pol_4.splitting_field('zz').degree() < 48:
+ # print "Stopping because splitting field of div_pol_4 < 48"
+ # exit()
+
+ K_4 = extended_field( div_pol_4, A, B, 96 )
+ print "4-division field:", K_4
+
+ if K_4.degree() != 96:
+ print "Stopping because mod 4 representation not surjective"
+ print ""
+ continue
+
+ if len(f_P.roots(ring=K_4)) == 0:
+ print "Stopping because M is not contained in K_4"
+ print ""
+ continue
+
+ print "----------------"
+ print "|Example Found!|"
+ print "----------------"
+ L.append((A,B))
+
+ print "*************************"
+ print ""
diff --git a/necessity_52b/2_division_counterex.sage b/necessity_52b/2_division_counterex.sage
@@ -0,0 +1,59 @@
+
+R1.<x> = PolynomialRing(QQ)
+R2.<x,y> = PolynomialRing(QQ)
+
+def extended_field( f, A, B, deg_mult ):
+ # f: a polynomial whose roots are the x-coordinates of some points of an
+ # elliptic curve E: y^2 = x^3 + Ax + B.
+ # return value: a field containing the x and y coordinates of those points
+ # deg_mult: a positive integer known to be a multiple of the degree of the
+ # extended field.
+ #
+ # This function uses the properties of resultants (I can provide a pdf
+ # explaining how it works).
+ #
+ # It is useful to compute, e.g., the fields obtained by adjoining the
+ # coordinates of the n-division points of a point (using the n-uplication
+ # formulas to get the required polynomials).
+ #
+ # When used to compute the 2-division fields, it gives the same output as
+ # E.division_field(2).
+
+
+ g = y^2 - x^3 - A*x - B
+ res = f.resultant(g,x)
+ res = res.subs(y=x)
+
+ #print "Splitting field of division pol:"
+ #print R1(f).splitting_field('r')
+ K.<b> = (R1(res*f)).splitting_field(degree_multiple=deg_mult)
+
+ return K
+
+
+A = -3
+B = 17/4
+
+E = EllipticCurve([0,0,0,A,B])
+print E
+print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens()
+print "CM:", E.has_cm()
+
+rep = E.galois_representation()
+print "mod 2 rep is surjective:", rep.is_surjective(2)
+
+K_2.<a> = E.division_field(2)
+print "2-division field:", K_2
+P = [4,15/2]
+
+# The following polynomial is derived from the duplication formula
+# (Silverman, p.54) using the x-coordinate of the 2-division point as
+# an indeterminate.x
+f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0])
+
+M = extended_field( f_P, A, B, 24 ) # The 2-division field of P
+print "2-division field of P:", M
+
+if M.degree() != 24:
+ print "Stopping because 2-division of P is too small"
+ exit()
diff --git a/necessity_52b/necessity_52b.pdf b/necessity_52b/necessity_52b.pdf
Binary files differ.
diff --git a/necessity_52b/necessity_52b.tex b/necessity_52b/necessity_52b.tex
@@ -0,0 +1,96 @@
+\documentclass[10pt,a4paper]{article}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+\usepackage{amsthm}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage[a4paper, top=3cm, bottom=3cm, left=2.5cm, right=2.5cm]{geometry}
+
+\DeclareMathOperator{\alg}{alg}
+\DeclareMathOperator{\obj}{Obj}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\hol}{Hol}
+\DeclareMathOperator{\aut}{Aut}
+\DeclareMathOperator{\gal}{Gal}
+\DeclareMathOperator{\id}{id}
+\DeclareMathOperator{\res}{res}
+\DeclareMathOperator{\im}{Im}
+\DeclareMathOperator{\Id}{Id}
+\DeclareMathOperator{\fib}{Fib}
+\DeclareMathOperator{\spec}{Spec}
+\DeclareMathOperator{\proj}{Proj}
+\DeclareMathOperator{\trdeg}{trdeg}
+\DeclareMathOperator{\car}{char}
+\DeclareMathOperator{\Frac}{Frac}
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+\DeclareMathOperator{\den}{den}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\lcm}{lcm}
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+\DeclareMathOperator{\op}{op}
+\DeclareMathOperator{\sep}{sep}
+\DeclareMathOperator{\unr}{unr}
+\DeclareMathOperator{\et}{\acute et}
+\DeclareMathOperator{\ev}{ev}
+\DeclareMathOperator{\gl}{GL}
+\DeclareMathOperator{\SL}{SL}
+
+\newcommand{\grp}{\textsc{Grp}}
+\newcommand{\set}{\textsc{Set}}
+\newcommand{\x}{\mathbf{x}}
+\newcommand{\naturalto}{\overset{.}{\to}}
+\newcommand{\qbar}{\overline{\mathbb{Q}}}
+\newcommand{\zbar}{\overline{\mathbb{Z}}}
+
+\newcommand{\pro}{\mathbb{P}}
+\newcommand{\aff}{\mathbb{A}}
+\newcommand{\quat}{\mathbb{H}}
+\newcommand{\rea}{\mathbb{R}}
+\newcommand{\kiu}{\mathbb{Q}}
+\newcommand{\F}{\mathbb{F}}
+\newcommand{\zee}{\mathbb{Z}}
+\newcommand{\ow}{\mathcal{O}}
+\newcommand{\mcx}{\mathcal{X}}
+\newcommand{\mcy}{\mathcal{Y}}
+\newcommand{\mcs}{\mathcal{S}}
+\newcommand{\mca}{\mathcal{A}}
+\newcommand{\mcb}{\mathcal{B}}
+\newcommand{\mcf}{\mathcal{F}}
+\newcommand{\mcg}{\mathcal{G}}
+\newcommand{\mct}{\mathcal{T}}
+\newcommand{\mcq}{\mathcal{Q}}
+\newcommand{\mcr}{\mathcal{R}}
+\newcommand{\adl}{\mathbf{A}}
+\newcommand{\mbk}{\mathbf{k}}
+\newcommand{\m}{\mathfrak{m}}
+\newcommand{\p}{\mathfrak{p}}
+
+\newcommand{\kbar}{\overline{K}}
+\newtheorem{lemma}{Lemma}
+
+\author{Sebastiano Tronto}
+\title{On the Second Condition of Theorem 5.2 ($\ell=2$)}
+
+\begin{document}
+
+\maketitle
+
+\textbf{Question:} is Lemma 1 of Lang's \emph{Elliptic Curves Diophantine Analysis}, Chapter 5 Section 5, page 117 (References/Kummer theory/LANG-book-EC.pdf) consistent with Jones-Rouse's Theorem 5.2, i.e. can we deduce that $F(\beta_1)$ is partially contained in $F(A[2])$ already?
+
+I believe the answer is no. Lang is working over a base field that contains all the $\ell^\infty$ torsion of $A$, so it doesn't apply in our case (J\&R assume surjectivity of the torsion part).
+
+A counterexample is actually given by J\&R right after the proof of the theorem (``Remark''). They don't say explicitly that $\kiu(\beta_1)\cap \kiu(A[2])=\kiu$ in this case, but I have tested this with sage (see the file \texttt{2-division-counterex.sage}): $\kiu(\beta_1)$ has degree $24$ over $\kiu$ and $\kiu(A[2])$ has degree $2$, which together imply that $[\kiu(A[2],\beta_1):\kiu(A[2])]\geq 4$, so it is maximal.
+
+The file \texttt{2-division.sage} contains some code that looks for other counterexamples (varying the parameters of a short Weierstrass equation), but it is quite slow (about 3 minutes on my pc for each elliptic curve of which it computes the 4-torsion field).
+
+\end{document}
+\ No newline at end of file
