Local conductor bounds for modular abelian varieties
Kimball Martin
TL;DR
This work refines local conductor bounds for abelian varieties over $\mathbb{Q}$ with maximal real multiplication, exploiting the rationality fields of modular forms. The key technical tool shows that for large $p$-power level divisors, the real cyclotomic field $\mathbb{Q}(\zeta_{p^r})^+$ embeds in the rationality field $K_f$ of the associated newform, forcing explicit bounds $v_p(N) \le B_0(p,d)$ and hence $v_p(N_A) \le d\,B_0(p,d)$. It also develops RM-field restrictions and GL(2)-type considerations, demonstrating sharpness in many small-dimension cases and providing cross-prime interactions that constrain endomorphism algebras. The results sharpen Brumer–Kramer bounds in numerous regimes and have computational verification for dimensions up to $d\le 10$, with implications for endomorphism algebras and RM-field determination from conductor data.
Abstract
Brumer and Kramer gave bounds on local conductor exponents for an abelian variety $A/\mathbb Q$ 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 $Γ_0(N)$, and thus the endomorphism algebra of $A$, contains $\mathbb Q(ζ_{p^r})^+$ 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.