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Interesting question.<br>
<br>
Let E : y^2 = x^3 + a*x + b be an elliptic curve and E' : Y^2 = X^3
+ d^2*a*x + d^3*b be its quadratic twist. The primality of E( F_q )
and E'( F_q ) are not independent events!! Indeed, far from it. <br>
<br>
Look at this magma run:<br>
<br>
> p := NextPrime( Random(500)); for X := 1 to 20 do print
IsPrime( p + 1 + X), " ", IsPrime( p + 1 - X ); end for;<br>
false true<br>
false false<br>
false false<br>
false false<br>
true true<br>
false false<br>
false true<br>
false false<br>
false false<br>
false false<br>
true true<br>
false false<br>
true false<br>
false false<br>
false false<br>
false false<br>
false true<br>
false false<br>
true true<br>
false false<br>
<br>
Try it yourself. Use sage if you prefer. Be amazed at the magic
of mathematics.<br>
<br>
Some sort of vague explanation is given in the paper:<br>
<br>
S. D. Galbraith, J. F. McKee, The probability that the number of
points on an elliptic curve over a finite field is prime, Journal of
the London Mathematical Society, 62, no. 3, p. 671-684 (2000) <br>
<br>
pdf is on my webpage.<br>
<br>
This does not fully answer your question. But somehow the answer
should vaguely be that the distribution for "nearly prime order of
E" and "nearly prime order of both E and its twist" are not so
different.<br>
<br>
Steven<br>
<br>
<br>
<br>
<div class="moz-cite-prefix">On 13/11/14 06:22, David Leon Gil
wrote:<br>
</div>
<blockquote
cite="mid:CAA7UWsVB6+i7wZFYVQvmBXgBdH1+uSDJP3WA54+SfPL0Bna60g@mail.gmail.com"
type="cite">
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Question:
<div><br>
</div>
<div>What is the distribution induced on the trace of
Frobenius by choosing a curve such that its twist has nearly
prime order?<br>
<div><br>
</div>
<div>(I.e., is it any different from the distribution induced by
choosing a curve parameter at random, which -- as I understand
it -- is the Sato-Tate distribution?)</div>
</div>
<br>
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<br>
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