非分裂Cartan曲线与伪椭圆曲线的一个刻画
Non-split Cartan curves and a characterization of fake elliptic curves
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中文总结 AI 辅助
该文针对虚二次域上的无潜在CM且L-函数为平方的阿贝尔曲面，结合非分裂Cartan模曲线的二次点研究，给出了其具有四元数乘法的刻画条件，部分情形为无条件结论。
中文摘要 AI 辅助
带有四元数乘法（QM）的阿贝尔曲面的ℓ进Tate模分解为两个二维ℚ有理表示的拷贝。至今尚无判别这些表示与椭圆曲线所附表示的准则，因此QM阿贝尔曲面通常被称为伪椭圆曲线。本文刻画了定义在虚二次域K上的QM阿贝尔曲面：考虑一个无潜在复乘（CM）且其L-函数为平方的阿贝尔曲面A/K，在一个合理的猜想下，证明A具有QM当且仅当它在至少两个素数模下的剩余像包含于非分裂Cartan群中；对于所有不超过33的不定四元数判别式，该刻画是无条件的。证明基于Siksek与Michaud-Jacobs关于非分裂Cartan模曲线二次点的前期工作，具体计算了曲线X_{ns}^+(15)的有理点，并证明X_{ns}(6)、X_{ns}(10)和X_{ns}(15)上的所有二次点均为非例外点。
英文摘要
The $\ell$-adic Tate module of an abelian surface with quaternionic multiplication (QM) decomposes as two copies of a two-dimensional $\mathbb{Q}$-rational representation. To this day, there is no criterion to distinguish these representations from those attached to elliptic curves. For this reason, QM abelian surfaces are usually called fake elliptic curves. In this paper we characterize QM abelian surfaces defined over an imaginary quadratic field $K$. Namely, we consider an abelian surface $A/K$ without potential CM and whose $L$-function is a square. Under a reasonable conjecture, we show that $A$ has QM if and only if it has residual image contained in a non-split Cartan group modulo at least two primes. The characterization is unconditional for all indefinite quaternion discriminants up to 33. The proof is based on previous work of Siksek and Michaud-Jacobs on quadratic points on non-split Cartan modular curves. In particular, we prove that all quadratic points on $X_{ns}(6)$, $X_{ns}(10)$ and $X_{ns}(15)$ are non-exceptional.
发表机构
- Universitat Oberta de Catalunya（加泰罗尼亚开放大学）
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