If E is an elliptic curve over
Qand p is an odd prime, then the p-torsion
on E can not all lie in
Q; because of
the Weil pairing the p-torsion generates a field
that contains
.
For which primes p does
there exist an elliptic curve E over
with all of its p-torsion rational over
?
When p=2,3,5 the corresponding moduli space has genus zero
and infinitely many examples exist. Recent work of L. Merel,
combined with computations he enlisted me to do,
suggest that these are the only primes p for
which such elliptic curves exist.
In [19], Merel exploits
cyclotomic analogues of the techniques used
in his proof of the uniform boundedness conjecture
to obtain an explicit criterion that can
be used to answer the above question for many primes p,
on a case-by-case basis.
Theoretical work of Merel, combined with my computations of
twisted L-values and character groups of tori,
give the following result (see [19, §3.2]):

Theorem 7
Let
be a prime satisfying
.
There are no elliptic curves
over
all of whose p-torsion is rational
over
.

The case in which p is congruent to 1 modulo 4 presents
additional difficulties that involve showing that Y(p) has no
-rational points. Merel and I hope to tackle these
difficulties in the near future.