An official website of the United States government
Abstract Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy $$\text {PGL}_2(\mathbb {F}_q)$$ PGL 2 ( F q ) or $$\text {PSL}_2(\mathbb {F}_q)$$ PSL 2 ( F q ) . We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus.
more »
« less
Borel and Volume Classes for Dense Representations of Discrete Groups
Abstract We show that the bounded Borel class of any dense representation $$\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $$G$$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $$\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$$ is uniformly separated in semi-norm from any other representation $$\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$$ for which there is a subgroup $$H\le G$$ on which $$\rho $$ is still dense but $$\rho ^{\prime}$$ is discrete or indiscrete but stabilizes a point, line, or plane in $${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of $${\operatorname{PSL}}_2{\mathbb{R}}$$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties.
more »
« less
- Award ID(s):
- 1902896
- PAR ID:
- 10466055
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2022
- Issue:
- 15
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 11891 to 11956
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We generalize work by Bourgain and Kontorovich [ On the local-global conjecture for integral Apollonian gaskets , Invent. Math. 196 (2014), 589–650] and Zhang [ On the local-global principle for integral Apollonian 3-circle packings , J. Reine Angew. Math. 737 , (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group $${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$$ satisfying certain conditions, where $$K$$ is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that $${\mathcal{A}}$$ possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in $$\operatorname{PSL}_{2}({\mathcal{O}}_{K})$$ containing a Zariski dense subgroup of $$\operatorname{PSL}_{2}(\mathbb{Z})$$ .more » « less
-
Let ρ ¯ : G Q → GSp 4 ( F 3 ) \overline {\rho }: G_{\mathbf {Q}} \rightarrow \operatorname {GSp}_4(\mathbf {F}_3) be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider ρ ¯ \overline {\rho } to be the Galois representation associated to the 3 3 -torsion of a principally polarized abelian surface A / Q A/\mathbf {Q} . We prove that the moduli space A 2 ( ρ ¯ ) \mathcal {A}_2(\overline {\rho }) of principally polarized abelian surfaces B / Q B/\mathbf {Q} admitting a symplectic isomorphism B [ 3 ] ≃ ρ ¯ B[3] \simeq \overline {\rho } of Galois representations is never rational over Q \mathbf {Q} when ρ ¯ \overline {\rho } is surjective, even though it is both rational over C \mathbf {C} and unirational over Q \mathbf {Q} via a map of degree 6 6 .more » « less
-
Abstract We construct a moduli space $$\textsf {LP}_{G}$$ of $$\operatorname {SL}_{2}$$-parameters over $${\mathbb {Q}}$$, and show that it has good geometric properties (e.g., explicitly parametrized geometric connected components and smoothness). We construct a Jacobson–Morozov morphism$$\textsf {JM}\colon \textsf {LP}_{G}\to \textsf {WDP}_{G}$$ (where $$\textsf {WDP}_{G}$$ is the moduli space of Weil–Deligne parameters considered by several other authors). We show that $$\textsf {JM}$$ is an isomorphism over a dense open of $$\textsf {WDP}_{G}$$, that it induces an isomorphism between the discrete loci $$\textsf {LP}^{\textrm {disc}}_{G}\to \textsf {WDP}_{G}^{\textrm {disc}}$$, and that for any $${\mathbb {Q}}$$-algebra $$A$$ it induces a bijection between Frobenius semi-simple equivalence classes in $$\textsf {LP}_{G}(A)$$ and Frobenius semi-simple equivalence classes in $$\textsf {WDP}_{G}(A)$$ with constant (up to conjugacy) monodromy operator.more » « less
-
We introduce the notions of symmetric and symmetrizable representations of . The linear representations of arising from modular tensor categories are symmetric and have congruence kernel. Conversely, one may also reconstruct modular data from finite-dimensional symmetric, congruence representations of . By investigating a -symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of that are subrepresentations of a symmetric one.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnab078
