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arXiv 1508.01293math.NTmath.AG
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具有平凡自同态环的阿贝尔簇的显式开像定理

Explicit open image theorems for abelian varieties with trivial endomorphism ring

Matthew Bisatt, Davide Lombardo

备注：
v3: added new author, streamlined exposition, several new results. v2: the result now applies to abelian varieties of arbitrary dimension
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AI总结：

针对数域上具有平凡自同态环的阿贝尔簇，在伽罗瓦表示像满足特定条件时给出半有效界，并基于Faltings高度等给出界的表达式，还描述了$\text{ℚ}$上阿贝尔三维簇的优化算法方法。

摘要：

设$K$为数域，$A/K$为$g$维阿贝尔簇。假设对所有充分大的素数$\boldsymbol{\text{ℓ}}$， Tate模$T_\text{ℓ}(A)$所附自然伽罗瓦表示的像$G_{\text{ℓ}^\text{∞}}$为$\boldsymbol{\text{GSp}_{2g}(\text{ℤ}_\text{ℓ})}$，我们给出半有效界$\text{ℓ}_0(A/K)$，使得对所有素数$\text{ℓ} > \text{ℓ}_0(A/K)$，有$G_{\text{ℓ}^\text{∞}}=\text{GSp}_{2g}(\text{ℤ}_\text{ℓ})$。该界由$A$的Faltings高度及$K$的适当一般位的剩余域的基数给出。我们还描述了一种算法方法，以获得$\text{ℚ}$上阿贝尔三维簇的更好界。

英文摘要：

Let $K$ be a number field and $A/K$ be an abelian variety of dimension $g$. Assuming that the image $G_{\ell^\infty}$ of the natural Galois representation attached to the Tate module $T_\ell(A)$ is $\operatorname{GSp}_{2g}(\mathbb{Z}_\ell)$ for all sufficiently large primes $\ell$, we provide a semi-effective bound $\ell_0(A/K)$ such that $G_{\ell^\infty}=\operatorname{GSp}_{2g}(\mathbb{Z}_\ell)$ for all primes $\ell > \ell_0(A/K)$. The bound is given in terms of the Faltings height of $A$ and of the cardinality of the residue field at a suitably generic place of $K$. We also describe an algorithmic approach to obtain better bounds for abelian threefolds over $\mathbb{Q}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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