# Tannakian group of Galois representations coming from geometry

Let $$K$$ be a number field. Let $$G_K$$ be its absolute Galois group.
Let $$p$$ be a rational prime.

Let $$mathcal{R}_{K,p}^g$$ be the category of finite-dimensional continuous $$p$$-adic representations of $$G_K$$ that come from geometry. Thus, an object $$V$$ of $$RG_{K}^p$$ is isomorphic to a sub-quotient of $$H^i_{ét}(X, mathbb{Q}_p)(n)$$ for some smooth projective variety $$X$$ and integers $$i$$ and $$n$$.

$$mathcal{R}_{K,p}^g$$ should be a Tannakian category, and therefore, have a Tannakian fundamental group scheme, $$mathcal{G}_{K,p}^g$$, associated to it.

How is $$mathcal{G}_{K,p}^g$$ related to $$G_K$$?

MathOverflow Asked by smn on December 27, 2020

## Related Questions

### Closure of the product of subfunctors

1  Asked on December 8, 2021 by nate-gallup

### Small-$r$ asymptotics of an integral of $1/log ^alpha r$

1  Asked on December 8, 2021 by yongpan-huang

### Does $mathcal{A}otimesmathbb{C}(t)congmathcal{D}otimesmathbb{C}(t)$ imply an isomorphism of Lie algebras?

1  Asked on December 8, 2021 by solver6

### When does $det begin{pmatrix} A & X \ X^T & A end{pmatrix} = (det A)^2 + (det X)^2$?

1  Asked on December 8, 2021

### When does cohomology of a pro-algebraic group commute with filtered colimits of coefficients?

1  Asked on December 8, 2021 by patrick-elliott

### Bounded weak and weak-$star$ topologies and metrics

0  Asked on December 8, 2021 by jorge-e-cardona

### Is semi-simplicity of Galois representations local?

0  Asked on December 8, 2021

### Maximum eigenvalue of a covariance matrix of Brownian motion

2  Asked on December 6, 2021 by weiqiang-yang

### Identity theorem in $p$-adic geometry/analysis

1  Asked on December 6, 2021

### Updates to Stanley’s 1999 survey of positivity problems in algebraic combinatorics?

1  Asked on December 5, 2021 by sam-hopkins

### Construction of $K(Gal(bar{k}/k), 1)$

0  Asked on December 5, 2021 by jog

### Forcing as a tool to prove theorems

9  Asked on December 3, 2021

### The determinant of the sum of normal matrices

1  Asked on December 3, 2021 by denis-serre

### Block matrices and their determinants

1  Asked on December 3, 2021 by t-amdeberhan

### Extending submodular functions from a sublattice

1  Asked on December 3, 2021 by darij-grinberg

### Regularity properties of Minakshisundaram–Pleijel zeta function

0  Asked on December 3, 2021 by alephbeth

### On proving the absence of limit cycles in a dynamical system

1  Asked on December 3, 2021 by norregaard

### Computing the infinite dimensional Lebesgue measure of “cubes”

0  Asked on December 3, 2021

### Why do Todd classes appear in Grothendieck-Riemann-Roch formula?

4  Asked on December 1, 2021 by ilya-nikokoshev