论文标题
固定的阿伯利亚品种和不同数字的扭转边界
Torsion bounds for a fixed abelian variety and varying number field
论文作者
论文摘要
让$ a $是在数字字段$ k $上定义的阿贝里安品种。对于有限扩展$ l/k $，组$ a（l）_ {\ operatotorname {tors}} $的基数可以根据度$ $ [l：k] $的程度限制。我们研究了最小的实际数字$β_A$，以便对于任何有限扩展名$ l/k $和$ \ varepsilon> 0 $，我们有$ | a（l）_ {\ operatorAtorName {tors}} | \ leq c \ cdot [l：k]^{β_a+\ varepsilon} $，其中常数$ c $仅取决于$ a $ a $ and $ \ varepsilon $（而不是$ l $）。假设Mumford-Tate的猜想是$ a $，我们将表明$β_A$与Hindry和Ratazzi的猜想价值一致。我们还为$ a（l）$的扭转点的最大顺序提供了类似的限制。
Let $A$ be an abelian variety defined over a number field $K$. For a finite extension $L/K$, the cardinality of the group $A(L)_{\operatorname{tors}}$ of torsion points in $A(L)$ can be bounded in terms of the degree $[L:K]$. We study the smallest real number $β_A$ such that for any finite extension $L/K$ and $\varepsilon>0$, we have $|A(L)_{\operatorname{tors}}| \leq C \cdot [L:K]^{β_A+\varepsilon}$, where the constant $C$ depends only on $A$ and $\varepsilon$ (and not $L$). Assuming the Mumford--Tate conjecture for $A$, we will show that $β_A$ agrees with the conjectured value of Hindry and Ratazzi. We also give a similar bound for the maximal order of a torsion point in $A(L)$.
