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Visibility of Shafarevich-Tate groups

A few years ago Mazur started lecturing about visibility of Shafarevich-Tate groups of elliptic curves. Assuming the BSD conjecture, Adam Logan exhibited an elliptic curve whose Shafarevich-Tate group is invisible in J0(2849), and other such examples followed. For a while it was believed out of reach to numerically find specific levels Np in which invisible Shafarevich-Tate groups becomes visible, because they are too large. I recently made a naive observation which has changed this view.

History
Temptation
Examples

History

  1. Mazur and Logan considered

    2849A        E:       y2 + xy + y = x3 + x2 - 53484x - 4843180

    BSD ==> #Sha(E) = 9

    They proved that

    Sha(E) ------> Sha(J0(2849))

    is injective (assuming BSD).

    "Sha(E) is invisible in J0(2849)."

    Likewise, for 5389A and a handful of other examples.

Temptation

  1. Murphy (of "Murphy's Law"):
    Does Sha(E) inject into Sha(J0(2849p)) for all primes p with p+1 not divisible by 3?
    (We assume that p+1 is not divisible by 3, so E[3] is not in the kernel of the map E-->J0(2849p).)

Examples

Observation: Explanatory rank 2 curves often have small height. Table of curves of small height yields possible counterexamples to the temptation.

  1. Conjectural counterexample to Murphy

    p=3,      2849p = 8547 = 3*7*11*37

    There is an elliptic curve


    F:        y2 + xy + y = x3 + x2 - 154x - 478

    of conductor 8547, such that

    fE = fF (mod 3) and F has rank 2.

    Maybe Sel3(E) = Sel3(F), which would imply Sha(E) is "explained by a jump in the rank of the Mordell-Weil"; equivalently, Sha(E) is visible in J0(8547).
    This is only a conjecture.

  2. Counterexample to Murphy
    5389A is

    E:        y2 + xy + y = x3 - 35590x - 2587197

    BSD ==> #Sha(E) = 9.

    At level 37723 = 7*17*317 there is a rank 2 elliptic curve

    F:        y2 - y = x3 + x2 + 34x - 248

    with F(Q) = Z*(5, 9) + Z*(8, 25)      

    such that

    ap(E) = ap(F) (mod 3)        for all p =/= 7,17,317.

    I can use F to prove that Sha(E) is visible in J0(37723).
    Details: We are lucky in this particular example, and I think we can get around every single problem that comes up in following the visibility proof in my thesis through. For example, the 7-component group of the explanatory factor F has order 3. However, the Galois action on this component group is nontrivial so H1(F7,PhiF,7) = 0. Also, we verify by a direct computation with 3-division polynomials that if P, Q are generators of the Mordell-Weil group of F, then there exists R, S such that 3R = P and 3S = Q, and R, S are defined over extensions which are unramified at 7.
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Last modified: May 16, 2000