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Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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Hi All,
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This email concerns where the invisible examples of Sha in
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[Cremona-Mazur] become visible. I've found a method to get further
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information, and analyzed each of the cases in Table 1 of
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[Cremona-Mazur]. I've talked to all of you about this problem at one
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point or another, so I thought you might find these computations
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interesting. The discussion of each case follows. The most
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interesting examples are 2849A and 5389A.
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2849A: Barry and Adam found the first known example of an _invisible_
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Shafarevich-Tate group. This was Sha(E), where E is the elliptic
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curve 2849A, which is given by the minimal Weierstrass equation
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y^2 + x*y + y = x^3 + x^2 - 53484*x - 4843180.
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I found an elliptic curve F of conductor 8547=2839*3 such that f_E =
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f_F (mod 3), where f_E and f_F are the newforms attached to E and F.
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Furthermore the Mordell-Weil group of F has rank TWO. The equation of
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F is
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y^2 + x*y + y = x^3 + x^2 - 154*x - 478
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Thus maybe Sha(E) becomes visible at level 8547. (This isn't
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completely obvious to me because the geometric, but not arithmetic,
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component group of E at 3 has order necessarily divisible by 3.)
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4343B: Again, according to Table 1 of [Cremona-Mazur], Sha(E) has
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order 9, but the modular degree prevents Sha(E) from being visible.
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At level 5*4343 there is an elliptic curve F of rank 1 that is
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congruent to E. Its equation is
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y^2 - x*y - y = x^3 - x^2 + 78*x - 256
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HOW I FOUND THESE CURVES: Let me explain the naive observation that
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allowed me find these curves of higher level, even though their
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conductors are out of the range of Cremona's tables. The elliptic
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curves labeled "NONE" in table 1 of [Cremona-Mazur] have unusually
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large height, as compared to their conductors. However, the
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explanatory factors often have unusually small height. So I just made
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a table of all elliptic curves [*, *, *, a4, a6] with * in {-1,0,1}
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and a4, a6 bounded by 1000 and of conductor < 50,000.
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3364C, 4229A, 5073d: I tried this method on the curves 3364C, 4229A,
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5073D each of which is labeled "none" (as opposed to NONE) in
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Cremona-Mazur. In none of these 3 cases was I able to find an
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explanatory factor at higher level.
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4194N: The curve 4914N is labeled "none" and has the remark "E has
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rational 3-torsion", even though there are invisible elements in Sha
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of order 3. Here I was able to find a congruent curve at level 24570.
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The congruent curve is
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F: y^2 - x*y = x^3 - x^2 - 15*x - 75.
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It has completely trivial Mordell-Weil group: F(Q) = {0}; there isn't
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even 3-torsion, so maybe Sha is not visible in E+F.
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5054C: The curve 5054C is labeled "none" and has invisible elements of
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Sha of order 3. I found a congruent curve of level 25270 of rank 1.
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The equation of the congruent curve is
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F: y^2 - x*y = x^3 + x^2 - 178*x + 882.
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5389A: The last curve labeled "NONE" in the table is curve 5389A. I
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do not know the equation for this curve, because the height is not
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small, John has not published it, and I have not bothered to compute
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it. However I found the eigenvalues a_p using modular symbols; they
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are
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[1, -2, 2, -2, 6, 2, 1, -4, 6, ...].
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We can raise the level at 3, since (-2)^2 = (3+1)^2 (mod 3). However,
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my table of curves of small height does not have any of conductor
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3*5389. However, (-2)^2 = (7+1)^2 (mod 3), so we can instead add 7
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into the level. My table does have a curve of conductor 37723, and
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*luckily* it is congruent mod 3 to 5389A! It's Weierstrass equations
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is
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y^2 - y = x^3 + x^2 + 34*x - 248.
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According to Cremona's mwrank program, this curve at level 37723 has
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rank 2.
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Please share your comments or suggestions with me.
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Best regards,
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William
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---------------------------------------
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Loic asked:
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> Thanks for the news. Is it known that the example in question does
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> not become visible in J_1(2849)?
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To the best of my knowledge this still isn't known. I haven't asked
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Mazur in a while, though.
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I just tried a computation which *might* show that the element does not
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become visible in J_1(2849). However, I haven't thought hard enough
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about it to decide whether or not the implication is justified. Here's
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what I did. Let M_2(2849,F_3)(eps) denote the space of modular symbols
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computed modulo 3 using the relations from your SLNM article. There
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are 8 Dirichlet character eps:(Z/2849Z)^* ---> F_3^*. Four of them are
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odd, and for these the corresponding space of modular symbols is
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trivial. The other four are even, and the corresponding spaces of
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modular symbols are F_3-vector spaces of dimension just over 600.
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I searched in each of these spaces and found *exactly one* eigenvector
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with the same Hecke eigenvalues as that of the elliptic curve 2849A.
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This one form is the mod-3 reduction reduction of eigenvector
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corresponding to 2849A.
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What do you think?
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Regards,
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William
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> EulerPhi(2849);
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2160
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> G:=DirichletGroup(2849,GF(3));
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> e:=Elements(G);
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> SetVerbose("ModularForm",2);
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> E:=EllipticCurve(CremonaDatabase(),"2849A");
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> E;
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Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 53484*x - 4843180 over Rational Field
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> factor(2849);
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[ <7, 1>, <11, 1>, <37, 1> ]
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> ellap(E,2);
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-1
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> ellap(E,3);
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2
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> ellap(E,5);
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-2
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> R<x>:=PolynomialRing(GF(3));
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> I:=[<2,x-ellap(E,2)>,<3,x-ellap(E,3)>,<5,x-ellap(E,5)>];
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> time M := [MS(e[i],2,+1) : i in [1..#e]];
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Time: 63.620
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> Mnz:=[m : m in M | Dimension(m) ne 0];
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> Mnz;
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[
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Full Modular symbols space of level 2849, weight 2, and dimension 308,
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Full Modular symbols space of level 2849, weight 2, character $.1*$.2, and dimension 304,
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Full Modular symbols space of level 2849, weight 2, character $.3, and dimension 308,
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Full Modular symbols space of level 2849, weight 2, character $.1*$.2*$.3, and dimension 304
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]
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> time K1:=Kernel(I,Mnz[1]);
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Time: 1.700
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> K1;
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Modular symbols space of level 2849, weight 2, and dimension 1
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> SetVerbose("ModularForm",0);
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> time K2:=Kernel(I,Mnz[2]);
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Time: 3.580
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> K2;
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Modular symbols space of level 2849, weight 2, character $.1*$.2, and dimension 2
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> DirichletCharacter(Mnz[2]);
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$.1*$.2
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> qEigenform(K1,40);
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q + 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)
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> DualHeckeOperator(K2,13);DualHeckeOperator(K2,13);
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[1 0]
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[0 1]
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> DualHeckeOperator(K2,31);
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[1 1]
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[0 1]
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> qEigenform(K1,40);
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q + 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)
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> DualHeckeOperator(K2,31);
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[0 0]
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[0 0]
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> I:=[<p,x-ellap(E,p)> : p in [2,3,5,13,17,19,23,29,31]];
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> time K1:=Kernel(I,Mnz[1]);
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Time: 1.369
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> K1;
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Modular symbols space of level 2849, weight 2, and dimension 1
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> time K2:=Kernel(I,Mnz[2]);
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Time: 1.679
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> K2;
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Modular symbols space of level 2849, weight 2, character $.1*$.2, and dimension 0
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> time K3:=Kernel(I,Mnz[3]);
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Time: 3.890
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> K3;
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Modular symbols space of level 2849, weight 2, character $.3, and dimension 0
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> time K4:=Kernel(I,Mnz[4]);
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Time: 2.800
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> K4;
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Modular symbols space of level 2849, weight 2, character $.1*$.2*$.3, and dimension 0
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> #Mnz;
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4
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> Mnz[1];
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Full Modular symbols space of level 2849, weight 2, and dimension 308
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>