1Hi All,
2
3This email concerns where the invisible examples of Sha in
4[Cremona-Mazur] become visible.  I've found a method to get further
5information, and analyzed each of the cases in Table 1 of
7point or another, so I thought you might find these computations
8interesting.  The discussion of each case follows.  The most
9interesting examples are 2849A and 5389A.
10
112849A: Barry and Adam found the first known example of an _invisible_
12Shafarevich-Tate group.  This was Sha(E), where E is the elliptic
13curve 2849A, which is given by the minimal Weierstrass equation
14
15       y^2 + x*y + y = x^3 + x^2 - 53484*x - 4843180.
16
17I found an elliptic curve F of conductor 8547=2839*3 such that f_E =
18f_F (mod 3), where f_E and f_F are the newforms attached to E and F.
19Furthermore the Mordell-Weil group of F has rank TWO.  The equation of
20F is
21
22       y^2 + x*y + y = x^3 + x^2 - 154*x - 478
23
24Thus maybe Sha(E) becomes visible at level 8547.  (This isn't
25completely obvious to me because the geometric, but not arithmetic,
26component group of E at 3 has order necessarily divisible by 3.)
27
284343B: Again, according to Table 1 of [Cremona-Mazur], Sha(E) has
29order 9, but the modular degree prevents Sha(E) from being visible.
30At level 5*4343 there is an elliptic curve F of rank 1 that is
31congruent to E.  Its equation is
32
33                    y^2 - x*y - y = x^3 - x^2 + 78*x - 256
34
35
36HOW I FOUND THESE CURVES:  Let me explain the naive observation that
37allowed me find these curves of higher level, even though their
38conductors are out of the range of Cremona's tables.  The elliptic
39curves labeled "NONE" in table 1 of [Cremona-Mazur] have unusually
40large height, as compared to their conductors.  However, the
41explanatory factors often have unusually small height.  So I just made
42a table of all elliptic curves [*, *, *, a4, a6] with * in {-1,0,1}
43and a4, a6 bounded by 1000 and of conductor < 50,000.
44
453364C, 4229A, 5073d:  I tried this method on the curves 3364C, 4229A,
465073D each of which is labeled "none" (as opposed to NONE) in
47Cremona-Mazur.  In none of these 3 cases was I able to find an
48explanatory factor at higher level.
49
504194N: The curve 4914N is labeled "none" and has the remark "E has
51rational 3-torsion", even though there are invisible elements in Sha
52of order 3.  Here I was able to find a congruent curve at level 24570.
53The congruent curve is
54
55         F: y^2 - x*y = x^3 - x^2 - 15*x - 75.
56
57It has completely trivial Mordell-Weil group: F(Q) = {0}; there isn't
58even 3-torsion, so maybe Sha is not visible in E+F.
59
605054C: The curve 5054C is labeled "none" and has invisible elements of
61Sha of order 3.  I found a congruent curve of level 25270 of rank 1.
62The equation of the congruent curve is
63
64         F: y^2 - x*y = x^3 + x^2 - 178*x + 882.
65
665389A: The last curve labeled "NONE" in the table is curve 5389A.  I
67do not know the equation for this curve, because the height is not
68small, John has not published it, and I have not bothered to compute
69it.  However I found the eigenvalues a_p using modular symbols; they
70are
71
72            [1, -2, 2, -2, 6, 2, 1, -4, 6, ...].
73
74We can raise the level at 3, since (-2)^2 = (3+1)^2 (mod 3).  However,
75my table of curves of small height does not have any of conductor
763*5389.  However, (-2)^2 = (7+1)^2 (mod 3), so we can instead add 7
77into the level.  My table does have a curve of conductor 37723, and
78*luckily* it is congruent mod 3 to 5389A!  It's Weierstrass equations
79is
80
81            y^2 - y = x^3 + x^2 + 34*x - 248.
82
83According to Cremona's mwrank program, this curve at level 37723 has
84rank 2.
85
87
88Best regards,
89  William
90
91
92---------------------------------------
94
95> Thanks for the news. Is it known that the example in question does
96> not become visible in J_1(2849)?
97
98To the best of my knowledge this still isn't known.  I haven't asked
99Mazur in a while, though.
100
101I just tried a computation which *might* show that the element does not
102become visible in J_1(2849).  However, I haven't thought hard enough
103about it to decide whether or not the implication is justified.  Here's
104what I did.   Let M_2(2849,F_3)(eps) denote the space of modular symbols
105computed modulo 3 using the relations from your SLNM article.   There
106are 8 Dirichlet character eps:(Z/2849Z)^* ---> F_3^*.   Four of them are
107odd, and for these the corresponding space of modular symbols is
108trivial.  The other four are even, and the corresponding spaces of
109modular symbols are F_3-vector spaces of dimension just over 600.
110I searched in each of these spaces and found *exactly one* eigenvector
111with the same Hecke eigenvalues as that of the elliptic curve 2849A.
112This one form is the mod-3 reduction reduction of eigenvector
113corresponding to 2849A.
114
115What do you think?
116
117Regards,
118 William
119
120
121> EulerPhi(2849);
1222160
123> G:=DirichletGroup(2849,GF(3));
124> e:=Elements(G);
125> SetVerbose("ModularForm",2);
127> E;
128Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 53484*x - 4843180 over Rational Field
129> factor(2849);
130[ <7, 1>, <11, 1>, <37, 1> ]
131> ellap(E,2);
132-1
133> ellap(E,3);
1342
135> ellap(E,5);
136-2
137> R<x>:=PolynomialRing(GF(3));
138> I:=[<2,x-ellap(E,2)>,<3,x-ellap(E,3)>,<5,x-ellap(E,5)>];
139> time M := [MS(e[i],2,+1) : i in [1..#e]];
140Time: 63.620
141> Mnz:=[m : m in M | Dimension(m) ne 0];
142> Mnz;
143[
144Full Modular symbols space of level 2849, weight 2, and dimension 308,
145Full Modular symbols space of level 2849, weight 2, character $.1*$.2, and dimension 304,
146Full Modular symbols space of level 2849, weight 2, character $.3, and dimension 308, 147Full Modular symbols space of level 2849, weight 2, character$.1*$.2*$.3, and dimension 304
148]
149> time K1:=Kernel(I,Mnz[1]);
150Time: 1.700
151> K1;
152Modular symbols space of level 2849, weight 2, and dimension 1
153> SetVerbose("ModularForm",0);
154> time K2:=Kernel(I,Mnz[2]);
155Time: 3.580
156> K2;
157Modular symbols space of level 2849, weight 2, character $.1*$.2, and dimension 2
158> DirichletCharacter(Mnz[2]);
159$.1*$.2
160> qEigenform(K1,40);
161q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6 + q^7 + q^9 + 2*q^10 + 2*q^11 + q^12 + q^13 + 2*q^14 + 2*q^15 + 2*q^16 + q^17 + 2*q^18 + 2*q^20 + 2*q^21 + q^22 + 2*q^25 + 2*q^26 + 2*q^27 + 2*q^28 + q^30 + q^31 + q^32 + q^33 + 2*q^34 + q^35 + 2*q^36 + O(q^37)
162> DualHeckeOperator(K2,13);DualHeckeOperator(K2,13);
163[1 0]
164[0 1]
165> DualHeckeOperator(K2,31);
166[1 1]
167[0 1]
168> qEigenform(K1,40);
169q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6 + q^7 + q^9 + 2*q^10 + 2*q^11 + q^12 + q^13 + 2*q^14 + 2*q^15 + 2*q^16 + q^17 + 2*q^18 + 2*q^20 + 2*q^21 + q^22 + 2*q^25 + 2*q^26 + 2*q^27 + 2*q^28 + q^30 + q^31 + q^32 + q^33 + 2*q^34 + q^35 + 2*q^36 + 2*q^37 + 2*q^39 + O(q^40)
170> DualHeckeOperator(K2,31);
171[0 0]
172[0 0]
173> I:=[<p,x-ellap(E,p)> : p in [2,3,5,13,17,19,23,29,31]];
174> time K1:=Kernel(I,Mnz[1]);
175Time: 1.369
176> K1;
177Modular symbols space of level 2849, weight 2, and dimension 1
178> time K2:=Kernel(I,Mnz[2]);
179Time: 1.679
180> K2;
181Modular symbols space of level 2849, weight 2, character $.1*$.2, and dimension 0
182> time K3:=Kernel(I,Mnz[3]);
183Time: 3.890
184> K3;
185Modular symbols space of level 2849, weight 2, character $.3, and dimension 0 186> time K4:=Kernel(I,Mnz[4]); 187Time: 2.800 188> K4; 189Modular symbols space of level 2849, weight 2, character$.1*$.2*$.3, and dimension 0
190> #Mnz;
1914
192> Mnz[1];
193Full Modular symbols space of level 2849, weight 2, and dimension 308
194>