arXiv research

Study calculates Gothic Teichmüller curves' Euler characteristics, differing from usual.

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmueller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and deve…

We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus two square-tiled surfaces with these invariants. The main technical tool is the co…

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…

We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmu…

We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms…

Unified method for computing modular curvature on toric noncommutative manifolds using hypergeometric functions.

We study how the systole of principal congruence coverings of a Hilbert modular variety grows when the degree of the covering goes to infinity. We prove that given a Hilbert modular variety $M$ of real dimension $2n$, the sequence of principal congruence coverings $M_{I}$ eventually satisfies $$sysπ_{1}(M_{I})\geq \fra…

We give formulas for the Whitehead groups and the rational $K$-theory groups of the (integer group ring of the) Hilbert modular group in terms of its maximal finite subgroups.

In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of $C^*$-algebras, after tensoring with $\mathbb{Q}$, by computing the source of the assembly maps in the Farrell-Jones and the Baum-Con…

New fundamental domain for Hilbert-Blumenthal cusp shapes.

Study periodic points on genus two surfaces, solving dynamics and geometry problems.

The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact lo…

The paper proves the monodromy of a specific curve family is arithmetic.

Research classifies quadratic forms over various fields.

Study modular forms over Γ^0(2) and anomaly cancellation formulas.

Discoveries new symmetries in 3d topological and physical systems.

New modular forms for anomaly cancellation formulas on any dimensional manifolds.

A Teichmüller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichmüller curves of positive genus. Our methods are based on the study of certain Hilbert modular forms and the use of Ahlfors's variational formu…

The paper introduces elliptic quasi-modular forms via moduli spaces.

Constructs modular forms and proves divisibility results for odd-dimensional manifolds.

New formulas derived for anomaly cancellation using modular forms and E8 bundles.

New formulas derived from modular forms for manifold indices.

Study shows non-positivity of Einstein-Hilbert action for certain metrics.

A new method integrates forms on Riemann surfaces, leading to modular forms.

We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomor…

The paper defines new modular forms from almost complex manifolds and derives anomaly cancellation formulas.

New anomaly formulas derived from $E_8$ bundles.

The paper connects Hodge theory and modular forms to prove inequalities.

Abstract: Generalizes modular forms to family case and finds new anomaly cancellation formulas.

SL(2,Z) forms lead to new anomaly formulas.

New anomaly cancellation formulas for E8*E8*E8 gauge group.

Holomorphic quantum modular forms linked to knot volumes.

Formula conjectured for refined SU(3) Vafa-Witten invariants of surfaces.

The abstract discusses fiber sum formulas for 4-manifolds using topological modular forms.

Study linking numbers in hyperbolic 3-folds, linking to Siegel modular forms.

Connects 4-manifold topology to topological modular forms.

We give some new methods, based on Lipschitz extension theorems, for bounding filling invariants of subsets of nonpositively curved spaces. We apply our methods to find sharp bounds on higher-order Dehn functions of Sol_{2n+1}, horospheres in euclidean buildings, Hilbert modular groups, and certain S-arithmetic groups.

Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.

New quantum modular forms for specific 3-manifolds discovered.

In this paper, we study modularity of several functions which naturally arose in a recent paper of Lau and Zhou on open Gromov-Witten potentials of elliptic orbifolds. They derived a number of examples of indefinite theta functions, and we provide modular completions for several such functions which involve more compli…

New elliptic genera defined for spin manifolds.

Note on new cancellation formulas for manifolds.

14 homogeneous convex foliations of degree 5 found on complex projective plane.

By studying modular invariance properties of some characteristic forms, we prove some new anomaly cancellation formulas which generalize the Han-Zhang and Han-Liu-Zhang anomaly cancellation formulas

Study Alexander polynomials of modular knots, revealing finite and infinite coefficient properties.

Proves modular functors for SO(3) have integral Hodge structures.

We show that the Atiyah-Patodi-Singer reduced $η$-invariant of the twisted Dirac operator on a closed $4m-1$ dimensional spin manifold, with the twisted bundle being the Witten bundle appearing in the theory of elliptic genus, is a meromorphic modular form of weight $2m$ up to an integral $q$-series. We prove this resu…