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Abstract Let K/\mathbf{Q}_{p}be unramified.Inside the Emerton–Gee stack \mathcal{X}_{2}, one can consider the locus of two-dimensional mod 𝑝 representations of \mathrm{Gal}(\overline{K}/K)having a crystalline lift with specified Hodge–Tate weights.We study the case where the Hodge–Tate weights are irregular, which is an analogue for Galois representations of the partial weight one condition for Hilbert modular forms.We prove that if the gap between each pair of weights is bounded by 𝑝 (the irregular analogue of a Serre weight), then this locus is irreducible.We also establish various inclusion relations between these loci.
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Hodge Representations
Abstract Green–Griffiths–Kerr introduced Hodge representations to classify the Hodge groups of polarized Hodge structures, and the corresponding Mumford–Tate subdomains. We summarize how, given a fixed period domain $$ \mathcal{D} $$ , to enumerate the Hodge representations and corresponding Mumford–Tate subdomains $$ D \subset \mathcal{D} $$ . The procedure is illustrated in two examples: (i) weight two Hodge structures with $$ {p}_g={h}^{2,0}=2 $$ ; and (ii) weight three CY-type Hodge structures.
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- Award ID(s):
- 1906352
- PAR ID:
- 10249678
- Editor(s):
- Clingher, Adrian
- Date Published:
- Journal Name:
- Experimental Results
- Volume:
- 1
- ISSN:
- 2516-712X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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The term degenerate is used to describe abelian varieties whose Hodge rings contain exceptional cycles -- Hodge cycles that are not generated by divisor classes. We can see the effect of the exceptional cycles on the structure of an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate group. In this article we examine degeneracy through these different but related lenses. We specialize to a family of abelian varieties of Fermat type, namely Jacobians of hyperelliptic curves of the form $y^2=x^m-1$. We prove that the Jacobian of the curve is degenerate whenever $$m$$ is an odd, composite integer. We explore the various forms of degeneracy for several examples, each illustrating different phenomena that can occur.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/exp.2020.55
