## Wieferich Primes and a mod \$p\$ Leopoldt Conjecture. (arXiv:1805.00131v2 [math.NT] UPDATED)

We consider questions in Galois cohomology which arise by considering mod \$p\$ Galois representations arising from automorphic forms. We consider a Galois cohomological analog for the standard heuristics about the distribution of Wieferich primes, i.e. prime \$p\$ such that \$2^{p-1}\$ is 1 mod \$p^2\$. Our analog relates to asking if in a compatible system of Galois representations, for almost all primes \$p\$, the residual mod \$p\$ representation arising from it has unobstructed deformation theory. This analog leads in particular to formulating a mod \$p\$ analog for almost all primes \$p\$ of the classical Leopoldt conjecture, which has been considered previously by G. Gras. Leopoldt conjectured that for a number field \$F\$, and a prime \$p\$, the \$p\$-adic regulator \$R_{F,p}\$ is non-zero. The mod \$p\$ analog is that for a fixed number field \$F\$, for almost all primes \$p\$, the \$p\$-adic regulator \$R_{F,p}\$ is a unit at \$p\$.查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 We consider questions in Galois cohomology which arise by considering mod \$p\$ Galois representations arising from automorphic forms. We consider a Galois cohomological analog for the standard heuristics about the distribution of Wieferich primes, i.e. prime \$p\$ such that \$2^{p-1}\$ is 1 mod \$p^2\$. Our analog relates to asking if in a compatible system of Galois representations, for almost all primes \$p\$, the residual mod \$p\$ representation arising from it has unobstructed deformation theory. This analog leads in particular to formulating a mod \$p\$ analog for almost all primes \$p\$ of the classical Leopoldt conjecture, which has been considered previously by G. Gras. Leopoldt conjectured that for a number field \$F\$, and a prime \$p\$, the \$p\$-adic regulator \$R_{F,p}\$ is non-zero. The mod \$p\$ analog is that for a fixed number field \$F\$, for almost all primes \$p\$, the \$p\$-adic regulator \$R_{F,p}\$ is a unit at \$p\$.