We Know that from a conjecture by Goldfeld says that half of all elliptic curves have rank zero.
Are there any known infinite families of elliptic curves in form
$y^2=x^3+p^2x$ where p is prime with rank 0 ?

What do you mean by a family of curves of that type?
– Will SawinNov 11 '12 at 3:06

Also, duplicated at stackexchange.
– MattNov 11 '12 at 3:36

3

One should be able to perform a descent via 2-isogeny by following almost exactly the steps section X.6 of Silverman's Arithmetic of Elliptic Curves which deals with $y^2 = x^3 + px$. This should give an upper bound on the rank as a function of p modulo some power of 2, offhand I would guess 8, but my intuition is an artifact of working with the case of full 2-torsion. There may be some congruence classes of p for which this upper bound is 0, giving that the rank is exactly 0. I would be surprised if this did not happen.
– Jamie WeigandtNov 11 '12 at 5:06