?? solutions over finite fields to y^2 = x^3 - x (for example ...) ... ?? various possible reasons for being interested in such ... ?? relationship among such reasons ... ?? ....

?? so .... ??? let's imagine for a moment that we have some idea of what the finite abelian group of points on the lemniscate is over some or maybe all finite fields ... ??? .... ?? then what might we try to do with this info ??? ....

?? so ... ?? might try to "turn it around", in sense of ... ??? for given p (say ... ?? ...), consider all those finite fields for which that finite abelian group includes a factor of z/p .... ???? .... "case where the point ab gp has p-torsion" .... ???? ...

?? meanwhile .... ???? try applying "quadratic reciprocity" over function field over finite field .... ???? .... ..... ?????? ......