A caveat: the following questions are so obvious that they have surely been asked elsewhere, and possibly given much more convincing answers. References welcome!

The Sato-Tate conjecture implies that the normalized trace of Frobenius for a non-CM elliptic curve is equidistributed with respect to the pushforward of the Haar measure of SU(2) under the trace map. This gives a perfectly good account of the behavior of the unnormalized over regions which have positive measure, namely, intervals of the form for distinct multiples of

If one tries to make global conjectures on a finer scale, however, one quickly runs into difficult conjectures of Lang-Trotter type. For example, given a non-CM elliptic curve E over if you want to count the number of primes p < X such that (say), an extremely generous interpretation of Sato-Tate would suggest that probability that would be

and hence the number of such primes < X should be something like:

except one *also* has to account for the fact that there are congruence obstructions/issues, so one should multiply this factor by a (possibly zero) constant depending one adelic image of the Galois representation. So maybe this does give something like Lang-Trotter.

But what happens at the other extreme end of the scale? Around the boundaries of the interval [-2,2], the Sato-Tate measure converges to zero with exponent one half. There is a trivial bound where is the largest square less than 4p. How often does one have an equality Again, being very rough and ready, the generous conjecture would suggest that this happens with probability very roughly equal to

and hence the number of such primes < X should be something like:

Is it at all reasonable to expect primes of this form? If one takes the elliptic curve one finds to be as big as possible for the following primes:

but no more from the first 500,000 primes. That's not completely out of line for the formula above!

Possibly a more sensible thing to do is to simply ignore the Sato-Tate measure completely, and model by simply choosing a randomly chosen elliptic curve over Now one can ask in this setting for the probability that is as large as possible. Very roughly, the number of elliptic curves modulo up to isomorphism is of order and the number with is going to be approximately the class number of where perhaps it is even exactly equal to the class number for some appropriate definition of the class number. Now the behaviour of this quantity is going to depend on how close is to a square. If is very slightly — say — more than a square, then is pretty much a constant, and the expected probability going to be around in On the other hand, for a generic value of the smallest value of will have order and then the class group will have approximate size and so one (more or less) ends up with a heuristic fairly close to the prediction above (at least in the sense of the main term being around

But why stop there? Let's push things even closer to the boundary. How small can get relative to For example, let us restrict to the set of prime numbers p such that

For such primes, the relative probability that is approximately So the expected number of primes with this property will be infinite providing that

is infinite, or, in other words, when So this leads to the following guess (don't call it a conjecture!):

** Guess:** Let be an elliptic curve without CM. Is

Of course, one can go crazy with even more outrageous guesses, but let me stop here before saying anything more stupid.

This is a long long way from the

Guess, but note that (at least) one can prove, for a non CM-curve E, that . If then the eigenvalue of Frobenius generates the order But, for fixed there are only finitely many j-invariants in with CM by this order, and there cannot be a congruence for infinitely many primes unless there is an equality, which would contradict the assumption that E has CM.I think the same method can be used to give some explicit bound like by showing that the norm of is at most exponential in by looking explicitly at how the j invariants are distributed on the modular curve.

Good point! (I didn’t check the precise bound, but I your main point is that one *can* give such explicit bounds).