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For $d$ a non-zero integer, let $E_d$ be the elliptic curve $$ E_d : y^2 = x^3+dx. $$ When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD, $$ \# Ш(E_p) = 2^{2k-2}. $$ Together with the fact that $\operatorname{Sel}^{(2)}(E_p) \cong (\mathbb{Z}/2\mathbb{Z})^2$ if $k = 1$ and $\operatorname{Sel}^{(2)}(E_p) \cong (\mathbb{Z}/2\mathbb{Z})^3$ if $k \geq 2$ (this can be found in X.6 of Silverman's book) the above implies, still assuming BSD, $$ Ш(E_p) \cong (\mathbb{Z}/2^{k-1}\mathbb{Z})^2. $$

Question

Restating the above in a more explicit format, assuming BSD we have: \begin{align} Ш(E_5) & = 0 \\\ Ш(E_{17}) & \cong (\mathbb{Z}/2\mathbb{Z})^2 \\\ Ш(E_{257}) & \cong (\mathbb{Z}/4\mathbb{Z})^2 \\\ Ш(E_{65537}) & \cong (\mathbb{Z}/8\mathbb{Z})^2 \end{align} Is there any "reason" why this should be true? I realize that this is a "soft" question, but the above pattern seems so remarkable that I feel compelled to look for an explanation at some level.

An alternative question would run as follows: if $k \geq 1$ is an integer and $p=2^{2^k}+1$ is prime, do we always have $$ Ш(E_p)\cong(\mathbb{Z}/2^{k-1}\mathbb{Z})^2? $$

A weaker question could be: if $k \geq 1$ is an integer and $p=2^{2^k}+1$ is prime, do we always have that the exponent of $Ш(E_p)$ is a multiple of $2^{k-1}$? (And what if we do not insist that $2^{2^k}+1$ be prime?) I have no idea how to prove lower bounds for the exponent of Ш of elliptic curves, other than running an explicit descent, but that doesn't seem the right method to attack the questions above.

How this came up: I was using sage to search for elliptic curves $E_d$ of the form $$ E_d : y^2 = x^3 + dx $$ with the additional property that $$ Ш(E_d)[2] = 0, $$ the underlying motivation being simply that for the course I am teaching I wanted to compile a list of elliptic curves on which a partial $2$-descent is possible without extending the ground field, and gives a sharp bound on the rank. I noticed then that the smallest $d$ with $\# Ш(E_d)[2^{\infty}] > 1$ is $d=17$, and the smallest $d$ with $\# Ш(E_d)[2^{\infty}] > 4$ is $d=257$.

While $2^{32}+1$ is of course not prime, out of curiosity I did try to compute the analytic rank and conjectural order of Ш of $E_d$ for $d=2^{32}+1$ (both with sage and magma), but the computations seem to be too lengthy to get an answer within a reasonable time.

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    $\begingroup$ There are similar phenomena in connection with class groups of quadratic fields whose discriminants are Mersenne or Fermat primes. These things are not trivial. $\endgroup$
    – Franz Lemmermeyer
    Commented Dec 10, 2013 at 17:21
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    $\begingroup$ Indeed I was wondering about that. Is there a particular reference for this that you would recommend? $\endgroup$
    – R.P.
    Commented Dec 10, 2013 at 17:23
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    $\begingroup$ The value of the $L$-function at $s=1$ for (quartic) twists of $y^2 = x^3+x$ could help. Or maybe the $2$-adic $L$-function. But I don't know enough about CM-curves to help you. $\endgroup$
    – Chris Wuthrich
    Commented Dec 10, 2013 at 22:22
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    $\begingroup$ I think Chris is right: your statement is equivalent to $L(E_d,1)/\Omega=2,8,32$ for $d=17,257,65537$, and this quotient could be perhaps computed with modular symbols? (Or at least proving the congruence $L(E_p,1)/\Omega\cong 0$ mod $2^{...}$ using that $p=2^{2^k}+1$?) $\endgroup$
    – Tim Dokchitser
    Commented Dec 12, 2013 at 17:56
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    $\begingroup$ Magma returns some information about 2^32+1. Try the commands: d:=2^(2^5)+1;E := EllipticCurve([0, 0, 0, d, 0]);MordellWeilShaInformation(E);. Returns (Z/2)^4 <= Sha(E)[4] <= (Z/2)^4 and <4, [ 0, 0 ]> $\endgroup$
    – joro
    Commented Nov 3, 2014 at 13:20

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