Local conductor bounds for modular abelian varieties

Brumer and Kramer gave bounds on local conductor exponents for an abelian variety A/Q in terms of the dimension of A and the localization prime p. Here we give improved bounds in the case that A has maximal real multiplication, i.e., A is isogenous to a factor of the Jacobian of a modular curve X0(N). In many cases, these bounds are sharp. The proof relies on showing that the rationality field of a newform for Gamma 0(N), and thus the endomorphism algebra of A, contains Q(zeta pr )+ when p divides N to a sufficiently high power. We also deduce that certain divisibility conditions on N determine the endomorphism algebra when A is simple.