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Cyclotomic points on modular curves

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  $\mathbf{Q}(\mu_p)$. For which primes p does there exist an elliptic curve E over $\mathbf{Q}(\mu_p)$with all of its p-torsion rational over $\mathbf{Q}(\mu_p)$? 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 $p \equiv 3\pmod{4}$ be a prime satisfying $7 \leq p < 1000$. There are no elliptic curves over $\mathbf{Q}(\mu_p)$ all of whose p-torsion is rational over $\mathbf{Q}(\mu_p)$.

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