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On the μ-invariants of abelian varieties over function fields of positive characteristic
Algebra & Number Theory ( IF 1.3 ) Pub Date : 2021-05-29 , DOI: 10.2140/ant.2021.15.863
King-Fai Lai , Ignazio Longhi , Takashi Suzuki , Ki-Seng Tan , Fabien Trihan

Let A be an abelian variety over a global function field K of characteristic p. We study the μ-invariant appearing in the Iwasawa theory of A over the unramified p-extension of K. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate–Shafarevich group of A and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate–Shafarevich group (which is now the μ-invariant) in terms of other quantities including the Faltings height of A and Frobenius slopes of the numerator of the Hasse–Weil L-function of AK assuming the conjectural Birch–Swinnerton-Dyer formula. Our next result is to prove this μ-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the “μ = 0” locus of the moduli of isomorphism classes of minimal elliptic surfaces endowed with a section and with fixed large enough Euler characteristic is a dense open subset.



中文翻译:

正特征函数域上阿贝尔簇的μ-不变量

一种 是全局函数域上的阿贝尔变体 有特色的 . 我们研究μ-不变量出现在 Iwasawa 理论中 一种 在无分支的 -的扩展 . Ulmer 认为这个不变量等于他所说的 Tate-Shafarevich 群的维数一种并且它确实是一些规范定义的组方案的维度。我们的第一个结果是验证他的建议。他还给出了 Tate-Shafarevich 群(现在是μ-不变量)在其他数量方面,包括法尔廷斯高度 一种 和 Hasse-Weil 分子的 Frobenius 斜率 - 功能 一种假设猜想的 Birch-Swinnerton-Dyer 公式。我们的下一个结果是证明这一点μ- 无条件的雅可比矩阵和半稳定阿贝尔变体的不变公式。最后,我们证明“μ = 0” 具有截面和固定足够大的欧拉特性的最小椭圆表面同构类的模的轨迹是一个密集的开子集。

更新日期:2021-05-30
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