Local conductor bounds for modular abelian varieties. (English) Zbl 07835523
In this paper, the author proves a bound on local conductor exponents for an abelian variety \(A/\mathbb{Q}\) (in terms of the dimension of \(A\) and the localization prime \(p\)) in the case that \(A\) has maximal real multiplication. This bound is an improvement of the bounds in an earlier work of Brumer and Kramer without assuming \(A\) has maximal real multiplication.
More specifically, let \(A/\mathbb{Q}\) be a \(d\)-dimensional abelian variety of conductor \(N_A\). Brumer and Kramer gave upper bounds \(\nu_p(N_A) \leq B(p,d)\) and the bounds are sharp in the sense that, for any \(p,d\), there exists an abelian variety \(A\) with \(\nu_p(N_A) = B(p,d)\).
The abelian variety \(A\) is said to have real multiplication if the endomorphism algebra \(\textrm{End}(A) \otimes \mathbb{Q}\) contains a totally real number \(K \supsetneq \mathbb{Q}\). We say \(A\) has maximal real multiplication if \(K\) has degree \(d = \textrm{dim}(A)\). If \(A\) has maximal real multiplication, then it is of \(\textrm{GL}(2)\)-type. By a theorem of Ribet, it is simple over \(\mathbb{Q}\) if and only if \(\textrm{End}(A) \otimes \mathbb{Q} \simeq K\) is a totally real field of degree \(d\). Up to isogeny, the simple abelian varieties with maximal real multiplication are the simple factors of the Jacobians \(J_0(N) = \textrm{Jac}(X_0(N))\) of the modular curves \(X_0(N)\). If \(N\) is minimal with \(A\) isogenous to a simple factor of \(J_0(N)\), then \(N_A = N^d\).
Define \[ B_p(p,d) := \begin{cases} 8+2\nu_2(d) & \text{if \(p=2\)} \\ 5+2\nu_3(d) & \text{if \(p=3\)} \\ 4+2\nu_p(d) & \text{if \(p \geq 5\) and \((p-1) \mid 2d\)}\\ 2 & \text{otherwise}. \end{cases} \] The main theorem of the paper is as follows: Let \(A/\mathbb{Q}\) be a \(d\)-dimensional simple abelian variety with maximal real multiplication and conductor \(N^d\).
More specifically, let \(A/\mathbb{Q}\) be a \(d\)-dimensional abelian variety of conductor \(N_A\). Brumer and Kramer gave upper bounds \(\nu_p(N_A) \leq B(p,d)\) and the bounds are sharp in the sense that, for any \(p,d\), there exists an abelian variety \(A\) with \(\nu_p(N_A) = B(p,d)\).
The abelian variety \(A\) is said to have real multiplication if the endomorphism algebra \(\textrm{End}(A) \otimes \mathbb{Q}\) contains a totally real number \(K \supsetneq \mathbb{Q}\). We say \(A\) has maximal real multiplication if \(K\) has degree \(d = \textrm{dim}(A)\). If \(A\) has maximal real multiplication, then it is of \(\textrm{GL}(2)\)-type. By a theorem of Ribet, it is simple over \(\mathbb{Q}\) if and only if \(\textrm{End}(A) \otimes \mathbb{Q} \simeq K\) is a totally real field of degree \(d\). Up to isogeny, the simple abelian varieties with maximal real multiplication are the simple factors of the Jacobians \(J_0(N) = \textrm{Jac}(X_0(N))\) of the modular curves \(X_0(N)\). If \(N\) is minimal with \(A\) isogenous to a simple factor of \(J_0(N)\), then \(N_A = N^d\).
Define \[ B_p(p,d) := \begin{cases} 8+2\nu_2(d) & \text{if \(p=2\)} \\ 5+2\nu_3(d) & \text{if \(p=3\)} \\ 4+2\nu_p(d) & \text{if \(p \geq 5\) and \((p-1) \mid 2d\)}\\ 2 & \text{otherwise}. \end{cases} \] The main theorem of the paper is as follows: Let \(A/\mathbb{Q}\) be a \(d\)-dimensional simple abelian variety with maximal real multiplication and conductor \(N^d\).
- 1.
- We have \(\nu_p(N) \leq B_0(p,d)\), that is, \(\nu_p(N_A) \leq dB_0(p,d)\).
- 2.
- For all \(p,d\), we have \(B_p(p,d) \leq \lfloor B(p,d)/d \rfloor\). This is a strict inequality if either (a) \(5 \leq p < 2d+1\) and \((p-1) \nmid 2d\), or (b) \(p \leq 3\), \(d > 3\), and \(p \nmid d\). It is an equality when \(p \geq 2d+1\). In other cases, this is sometimes an equality and sometimes not.
- 3.
- We have \(\nu_p(N) = B_0(p,d)\) for some \(A\), for all \((p,d)\) with \(d \leq 10\), with the possible exclusion of the following five cases: \(B_0(5,10) = 6\), \(B_0(11,10)=4\), and \(B_0(2d+1, d) = 4\) for \(d = 6,7,8\).
Reviewer: Xiao Xiao (Utica)
MSC:
| 11G10 | Abelian varieties of dimension \(> 1\) |
| 11F11 | Holomorphic modular forms of integral weight |
| 11F30 | Fourier coefficients of automorphic forms |
Keywords:
abelian varieties of \(\mathrm{GL}(2)\)-type; endomorphism algebras; conductors; rationality fields of modular formsReferences:
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