Mod p representations on elliptic curves
  Modular Galois representations into GL_2(F_p) with cyclotomic determinant arise from elliptic curves for p = 2,3,5. We show (by constructing explicit examples) that such elliptic curves cannot be chosen to have conductor as small as possible at all primes other than p. Our proof involves finding all elliptic curves of conductor 85779, a custom computation carried out for us by Cremona. This leads to a counterexample to a conjecture of Lario and Rio. For p > 5, we construct irreducible representations with cyclotomic determinant that do not arise from any elliptic curve over Q. 
Frank Calegari