Machine learning predicts arithmetic curve invariants with high accuracy.
problem Classifying arithmetic curves based on their invariants.
method Training machine learning algorithms on datasets of elliptic and genus 2 curves.
result High accuracy in classifying curves, including rank, torsion, and integral points.
Minimal crossing number found in arithmetic curve systems.
problem Finding the minimal crossing number in arithmetic curve systems.
method Analyzing systoles of hyperbolic surfaces associated with congruence lattices in SL2(Z).
result Minimal crossing number is asymptotically achieved.
In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsific…
Study on curves on moduli spaces of local systems, proving structure and integral point determination.
problem Arithmetic of algebraic curves on moduli spaces of local systems.
method Proved structure theorem for morphisms and effectively determined integral points on curves.
result Effective determination of integral points on nondegenerate algebraic curves on moduli space.
Study on curves on specific arithmetic quotients of hyperbolic 2-ball.
problem No complex curves of certain genus on these arithmetic quotients.
method Volume estimates and understanding special subvarieties.
result For large discriminants, no complex curves of fixed genus.
The paper proves the monodromy of a specific curve family is arithmetic.
problem Proving the arithmeticity of monodromy for a specific curve family.
method Analyzing the Wiman-Edge pencil and its monodromy, proving commensurability with a Hilbert modular group.
result The monodromy of the Wiman-Edge pencil is arithmetic.
New non-arithmetic ball quotients from elliptic curves on Abelian surfaces.
problem Constructing non-arithmetic ball quotients from Abelian surfaces.
method Branched covers of Abelian surface quotients by finite groups.
result Alternative construction of lattices from elliptic curves.
Study shows Betti numbers of curves and orbifolds relate to volume and genus.
problem Understanding Betti numbers of Shimura curves and 3-orbifolds.
method Analyzes asymptotic behavior of Betti numbers in relation to volume and genus.
result Gauss-Bonnet equality for Shimura curves and vanishing Betti numbers for 3-orbifolds.
Characterizes arithmetic and commensurable links in curved surfaces.
problem Classifying arithmetic and commensurable links in curved surfaces.
method Combines symmetry arguments, combinatorial geometry, and number-theoretic data.
result Characterizes arithmetic and commensurable right-angled tiling links.
The paper characterizes arithmetic metrics in coarsely geometric settings.
problem Characterizing arithmetic metrics in coarsely geometric settings.
method Using coarse-geometric commensurators and under the Hilbert-Smith conjecture.
result Positive answer in general and unconditional for specific cases.
In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose co…
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…
Study finds bounds for systole length on arithmetic punctured spheres.
problem Finding the shortest essential curve on arithmetic punctured spheres.
method Correspondence between surfaces and planar triangulations to bound systole length.
result Arithmetic surfaces do not achieve maximal systole length for n=7,10,11. Finite totally geodesic hypersurfaces in curved manifolds proven.
problem Characterizing totally geodesic hypersurfaces in curved manifolds.
method Analytic Riemannian manifold analysis with negative sectional curvature.
result Closed manifolds with negative curvature have only finitely many totally geodesic hypersurfaces.
Extends shape analysis to framed space curves using quaternionic arithmetic.
problem Matching and classifying shapes of framed space curves.
method Extends square root transform to framed curves using quaternionic arithmetic and Hopf fibration properties. Describes geodesics in framed curve space explicitly.
result Explicit descriptions of geodesics in framed curve space and averages of collections of curves.
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…
We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residua…
We prove that every algebraic curve X defined over the algebraic closure of the rationals is birational over the complex numbers to a Teichmuller curve.
Study confirms nullity of biharmonic maps family, linking spectral and arithmetic geometry.
problem Prove nullity of biharmonic maps from flat 2-torus to round 2-sphere.
method Use spectral geometry, construct polynomial isomorphism with elliptic curve, determine rational points on spectral curve.
result Nullity of every map in the family is 5, confirming conjecture.
Study non-arithmetic orbifold ball quotients using Jacobian of Bolza curve.
problem Identify and study non-arithmetic orbifold ball quotients.
method Use Jacobian of Bolza curve and birational transformations.
result Obtain orbifold ball quotient surfaces with interesting configurations.
Study transcendence of abelian differential periods from bi-algebraic perspective.
problem Arithmetic and functional transcendence of periods of abelian differentials.
method Bi-algebraic structure on strata of abelian differentials.
result Characterization of arithmetic points and proof of linear bi-algebraic curves.
New arithmetic phenomenon 'murmurations' detected using AI.
problem Detecting new arithmetic patterns in large datasets.
method Machine learning interpretability tools (PCA, saliency, convolutional filters).
result Murmurations encode Frobenius traces and connect to number theory.
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…
Paper proves non-arithmetic Teichmüller length spectra for subgroup of mapping class groups.
problem Proving non-arithmetic Teichmüller length spectra for subgroups of mapping class groups.
method Introducing cross-ratios on Teichmüller and projectable mapping classes, studying their geometric and dynamical properties.
result Every non-elementary subgroup of the mapping class group has non-arithmetic Teichmüller length spectrum.
We solve a long-standing question about the monodromy of certain complex surfaces.
problem When is the monodromy group of an algebraic family of complex varieties arithmetic?
method Topological analysis of the 'geometric' monodromy, valued in the mapping class group of the fiber.
result We resolve the question affirmatively for Atiyah-Kodaira manifolds.
The paper studies representations of braid groups via curves and finds conditions for their Zariski closure and arithmeticity.
problem Representations of braid groups via specific families of Riemann surfaces.
method Consider families of Riemann surfaces defined by plane curves and study their monodromy representations into symplectic groups.
result Criterions for the Zariski closure of the image of the representation to be maximal and for the image to be an arithmetic lattice.
We consider the interplay of point counts, singular cohomology, étale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree d rational curves in Pn. We deduce as special cases algebro-geome…
Study intersections of curves on translation surfaces, focusing on regular polygons and their Teichmüller disks.
problem Maximizing algebraic intersection between curves of given lengths.
method Investigate the quantity KVol defined for any closed orientable surface, focusing on regular n-gons for even n ≥ 8.
result Maximize algebraic intersection between curves of given lengths.
We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…
In this paper a new intrinsic geometric characterization of the symmetric square of a curve and of the ordinary product of two curves is given. More precisely it is shown that the existence on a surface of general type S of irregularity q of an effective divisor D having self-intersection D^2>0 and arithmetic genus q i…
We consider the analogue of Hurwitz curves, smooth projective curves C of genus g≥2 that realize equality in the Hurwitz bound ∣Aut(C)∣≤84(g−1), to smooth compact quotients S of the unit ball in C2. When S is arithmetic, we show that ∣Aut(S)∣≤288e(S), where $e(S…
We determine the spectral curve of charge 3 BPS su(2) monopoles with C_3 cyclic symmetry. The symmetry means that the genus 4 spectral curve covers a (Toda) spectral curve of genus 2. A well adapted homology basis is presented enabling the theta functions and monopole data of the genus 4 curve to be given in terms of g…
Study on knot 74 surgeries reveals infinite residue characteristics and infinite order points.
problem Arithmetic properties of Dehn surgery points on knot 74. method Analyzing the canonical component of the SL2(C)-character variety. result Infinite set of ramified places and infinite order points in the Mordell-Weil group.
Survey of Weber's class number problem and related topics.
problem Weber's class number problem and its variants.
method Arithmetic topology, units, generalized Pell's equation, p-adic limits of class numbers. result Numerical investigation of class numbers in p-adic towers for knots and elliptic curves. Paper finds new ball quotients from curve products.
problem Finding rational deformations of surfaces.
method Using cocompact lattices in PU(2,1).
result Shows existence of new ball quotients for surfaces.
Develops Weil bundles over \( p \)-adic manifolds for arithmetic geometry.
problem Connecting differential calculus and arithmetic geometry over \( p \)-adic fields.
method Systematic theory of Weil bundles, developing analytic structures.
result Establishes canonical analytic structures on Weil bundles and their cohomological comparison.
Paper extends algebraic geometry results to hyperbolic link complements.
problem Understanding algebraic and number-theoretic properties of canonical curves.
method Generalizes Chinburg-Reid-Stover's results to hyperbolic link complements.
result Azumaya algebra does not extend to canonical surfaces.
The study quantifies distances between certain hyperbolic surfaces and bounds their number.
problem Quantifying distances and bounding numbers of hyperbolic surfaces.
method Parametrization of Teichmüller space by length functions.
result Two coverings of arithmetic surfaces are $P( rac{1}{g})$ apart, with P depending only on the surface. We prove that if the Lyapunov spectrum of the Kontsevich-Zorich cocycle over an affine SL(2,R)-invariant submanifold is completely degenerate, i.e. λ2=⋯=λg=0, then the submanifold must be an arithmetic Teichmueller curve in the moduli space of Abelian differentials over surfaces of genus three…
Computes canonical heights for arithmetic log surfaces using Hurwitz zeta function.
problem Computing canonical heights for arithmetic log surfaces.
method Introduces a canonical height and uses limits of periods to compute it.
result Explicit formulas for canonical heights of arithmetic log surfaces, including (P_1,D).
This paper proves an upper limit on rational points on curves.
problem Finding rational points on curves of genus at least two.
method Arithmetic and analytic estimates.
result Explicit upper bounds on rational points.
Study on moduli spaces of sextic curves with simple singularities and their compactifications.
problem Understanding moduli spaces of sextic curves with simple singularities.
method Using period maps of K3 surfaces with ADE singularities, algebraic open embeddings into arithmetic quotients of type IV domains, and GIT and Looijenga compactifications.
result Identifications of GIT and Looijenga compactifications for all cases.
In this paper we will describe an approach to mirror symmetry for appropriate 1-dimensional DM stacks of arithmetic genus g≤1, called tcnc curves, which was developed by the author with Treumann and Zaslow in arXiv:1103.2462 . This involves introducing a conjectural sheaf-theoretic model for the Fukaya category …
New hyperbolic manifolds with diverse features created.
problem Creating new types of hyperbolic manifolds.
method Using cubulations and hyperbolization procedures.
result First examples of hyperbolic manifolds with non-trivial Stiefel-Whitney and Pontryagin classes.
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.
We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard manifolds are established for singular spaces; new arithmeticity and rigidity state…
In this note, we study the integral of the 1-form logxydy−logyxdx over certain plane curves defined by A-polynomials of knots. It is quite surprising that a Chern-Simons type invariant of 3-manifolds, which can be geometrically computed, may be used to get the exact values of those integrals. Th…
Method calculates systolic length of modular curves.
problem Computing upper bounds on systolic length of Riemann surfaces.
method Using congruence subgroups of hyperbolic triangle groups and traces of generators.
result Systolic length grows logarithmically with genus.