Elementary abelian groups are finite groups in the form of A = ( ℤ / p ⁢ ℤ ) r {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p . For every integer ℓ > 1 {\ell>1} and r > 1 {r>1} , we prove a non-trivial upper bound on the ℓ {\ell} -torsion in class groups of every A -extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G , the ℓ {\ell} -torsion in class groups are bounded non-trivially for every G -extension and every integer ℓ > 1 {\ell>1} . When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

中文翻译：

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类组中ℓ扭转的点向界：基本的阿贝尔扩展

基本的阿贝尔群是形式为A =（ℤ/ p⁢）r {A =（\ mathbb {Z} / p \ mathbb {Z}）^ {r}}的形式的有限群。对于每个整数ℓ> 1 {\ ell> 1}和r> 1 {r> 1}，我们证明了每个A扩展的类组中ℓ{\ ell}-扭转上的平凡上限。我们的结果是有目的且无条件的。这建立了第一种情况，对于某些Galois组G，对于每个G-扩展和每个整数ℓ> 1 {\ ell> 1}，类别组中的ℓ{\ ell}-扭转是非平凡的。当r足够大时，我们获得的无条件逐点界线也打破了GRH下Ellenberg和Venkatesh显示的先前最著名的界线。