Local conductor bounds for modular abelian varieties

Brumer and Kramer gave bounds on local conductor exponents for an abelian variety $A\mathrm{/}\mathbb{Q}$ 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 $X_0(N)$. In many cases, these bounds are sharp. The proof relies on showing that the rationality field of a newform for ${\mathrm{\Gamma }}_0(N)$, and thus the endomorphism algebra of $A$, contains $\mathbb{Q}({\zeta }_{p^r}{)}^{+}$ 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.