Mod $ell$ representations of arithmetic fundamental groups II: A conjecture of A. J. de Jong
Mod $ell$ representations of arithmetic fundamental groups II: A conjecture of A. J. de Jong
复制标题
算术基本群的 Mod $ell$ 表示 II：A. J. de Jong 的猜想
作者：
Gebhard Boeckle;C. Khare
我们研究了n维表示$overline{
定义在特征为e 11的有限域上的算术基本群pi 1（X）的一条光滑曲线，其中X是特征为p（x）的有限域k上的一条几何不可约光滑曲线
Eq ell$）。当$overline{
ho}$具有大的象，我们能够证明所得到的环是有限平坦的。证明主要使用伽罗瓦理论的提升结果的作者在第一部分，这两部分的工作，一个提升结果的尖点模$ell$形式的Ogilvie，泰勒-怀尔斯系统和结果的Lafforgue。这意味着de Jong的一个猜想，对于特征为$ell$的有限域上的幂级数环中的系数的$pi_1（X）$表示，具有此mod $ell$表示$overline{
h 0}$作为其减少。一个证明的所有情况下的猜想为$ell>2$如下结果宣布Gaitsgory。方法不同。
We study deformation rings of an n-dimensional representation $overline{
ho}$, defined over a finite field of characteristic $ell$, of the arithmetic fundamental group $pi_1(X)$, where X is a geometrically irreducible, smooth curve over a finite field k of characteristic p ($
eq ell$). When $overline{
ho}$ has large image, we are able to show that the resulting rings are finite flat over $mathbf{Z}_ell$. The proof principally uses a Galois-theoretic lifting result of the authors in Part I of this two-part work, a lifting result for cuspidal mod $ell$ forms of Ogilvie, Taylor–Wiles systems and the result of Lafforgue. This implies a conjecture of de Jong for representations of $pi_1(X)$ with coefficients in power series rings over finite fields of characteristic $ell$, that have this mod $ell$ representation $overline{
ho}$ as their reduction. A proof of all cases of the conjecture for $ell>2$ follows from a result announced by Gaitsgory. The methods are different.