{
"id": "2001.04956",
"version": "v1",
"published": "2020-01-14T18:28:31.000Z",
"updated": "2020-01-14T18:28:31.000Z",
"title": "Galois deformation spaces with a sparsity of automorphic points",
"authors": [
"Kevin Childers"
],
"comment": "39 pages",
"categories": [
"math.NT"
],
"abstract": "Let $k/\\mathbb F_p$ denote a finite field. For any split connected reductive group $G/W(k)$ and certain CM number fields $F$, we deform certain Galois representations $\\overline\\rho:Gal(\\overline F/F) \\to G(k)$ to continuous families $X_{\\overline\\rho}$ of Galois representations $Gal(\\overline F/F) \\to G(\\overline{\\mathbb Q_p})$ lifting $\\overline\\rho$ such that the space of points of $X_{\\overline\\rho}$ which are geometric (in the sense of the Fontaine-Mazur conjecture) with parallel Hodge-Tate weights has positive codimension in $X_{\\overline\\rho}$. Thus the set of points in $X_{\\overline\\rho}$ which could (conjecturally) be associated to automorphic forms is sparse. This generalizes a result of Calegari and Mazur for $F/\\mathbb Q$ quadratic imaginary and $G = GL_2$. The sparsity of automorphic points for $F$ a CM field contrasts with the situation when $F$ is a totally real field, where automorphic points are often provably dense.",
"revisions": [
{
"version": "v1",
"updated": "2020-01-14T18:28:31.000Z"
}
],
"analyses": {
"keywords": [
"galois deformation spaces",
"automorphic points",
"galois representations",
"cm field contrasts",
"cm number fields"
],
"note": {
"typesetting": "TeX",
"pages": 39,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}