Let G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

中文翻译：

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有限群积分表示的调节器常数

让G是一个有限群并且p成为素数。我们研究同构不变量$\mathbb{Z}_p$[G]-格子，其标量扩展为$\mathbb{Q}_p$是自对偶的，称为调节常数。这些最初是由 Dokchitser–Dokchitser 在椭圆曲线的背景下引入的。调节器常数典型地在 Brauer 关系空间之间产生配对G以及定义了调节常数的表示环的子空间。对所有人G，我们证明这种配对永远不会完全为零。出于形式上的原因，这种配对通常具有非平凡的内核。但是，如果G具有循环 Sylowp-子群，我们限制考虑置换格，然后我们证明配对是非退化模形式核。使用这个我们可以证明，对于某些组，包括 2 阶的二面体组p为了p奇怪的是，任何的同构类$\mathbb{Z}_p$[G]-格子，其标量扩展为$\mathbb{Q}_p$是自对偶的，由它的调节常数决定，它的标量扩展到$\mathbb{Q}_p$, 和 Yakovlev 的上同调不变量。