For a given infinite set of primes, not too big, eg, satisfying Lang-Trotter conjecture, can we always find an E.C. with supersingular reduction (at least) at these primes? How about E.C. without CM?



The existence of elliptic curves with prescribed supersingular primes
No. In fact the set can be arbitrarily sparse, i.e. the $n$-th prime
in the set can be chosen to exceed $a\_n$ for any sequence $\{a\_n\}$.
This is because the rationals are countable. Fix an enumeration
$j\_1,j\_2,j\_3,\ldots$ of $\bf Q$. For each $n$ let $p\_n$ be the smallest
prime such that $p\_n > a\_n$ and $j\_n$ is not supersingular mod $p\_n$
(this is possible because the density of ordinary primes is positive,
namely $1/2$ if $j\_n$ is one of the thirteen CM $j$-invariants and $1$ otherwise).
Then no rational number is the $j$-invariant of an elliptic curve
with supersingular reduction at every $p\_n$.


