Study ranks of elliptic curves via prime averages.
problem Classifying elliptic curves by rank.
method Average Frobenius trace over primes, data science experiments.
result Oscillating pattern in average trace values, correlates with rank.
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48 results for “Elliptic curves”
New proof for curved 3-cohom manifold rational ellipticity.
problem Rational ellipticity of curved manifolds with specific cohomogeneity.
method Proved rationally elliptic for cohomogeneity-three manifolds with positive curvature and no boundary quotient.
result Closed, simply connected, positively curved, cohomogeneity-three manifolds without boundary are rationally elliptic.
We show that one can define a spectral curve for the Cauchy-Riemann operator on a punctured elliptic curve if one imposes appropriate boundary conditions. Algebraic curves of the type thus obtained appear as irreducible components of spectral curves of minimal tori with planar ends in R^3. It appears that these curves …
No projective structure found on foliations of elliptic curves.
problem Existence of transversely projective structures on foliations.
method Analyzing foliations on the product of two elliptic curves.
result Generic nonsingular turbulent foliations do not have transversely projective structures.
In this short note we prove that in the case of elliptic curves, the isomorphism of generalized complex structure between -dual manifolds described by Cavalcanti-Gualtieri coincides with the mirror map for elliptic curves described by Polishchuk and Zaslow.
Machine learning predicts Shafarevich-Tate group orders of elliptic curves.
problem Predicting the order of the Shafarevich-Tate group of elliptic curves.
method Train feed-forward neural network and regression models on elliptic curve invariants.
result Models achieve high accuracy () and predict orders not seen during training.
Mathematical formulas for elliptic curve integrals solve anomaly equations.
problem Mathematical formulation of contact term singularities on elliptic curves.
method Residue formulas and holomorphic anomaly equations.
result Regularized integrals on elliptic curves satisfy holomorphic anomaly equations.
Study Kähler geometry on vector bundles over elliptic curves.
problem Characterize Kähler metrics on vector bundle total spaces.
method Analyzing function theory and Kähler geometry on vector bundles of degree zero.
result Biholomorphic total spaces correspond to isomorphic vector bundles.
The moduli space of stable rank 2 parabolic bundles over an elliptic curve with 3 marked pointsmath.AG
We describe the moduli space of stable rank 2 parabolic bundles over an elliptic curve with 3 marked points.
problem Characterize the moduli space of stable rank 2 parabolic bundles over an elliptic curve with marked points.
method Explicitly describe the moduli space as a blow-up of an embedded elliptic curve in and interpret it as the character variety of the 3-punctured torus.
result The moduli space can be described as a blow-up of an embedded elliptic curve in and interpreted as the character variety of the 3-punctured torus.
Convolutional neural networks predict the analytic rank of elliptic curves accurately.
problem Predicting the analytic rank of elliptic curves over Q.
method Applied one-dimensional convolutional neural networks to Frobenius traces.
result High accuracy predictions for analytic rank across various conductors.
These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and stacks) important in the study of moduli spaces of curves and abelian varieties thro…
We determine obstructedness or unobstructedness of (holomorphic) Poisson deformations of ruled surfaces over an elliptic curve.
Elliptic curve governs Hopf linking in symmetric tensegrity.
problem Understanding Hopf linking in A4-symmetric tensegrity.
method Parametrized by points on the elliptic curve 30a2.
result Only one rational point corresponds to a stable configuration.
We consider an almost complex structure J on CP2, or more generally an elliptic structure E which is tamed by the standard symplectic structure. An E-curve is a surface tangent to E (this generalizes the notion of J(holomorphic)-curve), and an E-line is an E-curve of degree 1. We prove that the space of E-lines is agai…
Constructs the moduli stack of elliptic curves as an orbifold.
problem Moduli stack construction for elliptic curves.
method Analytic orbifold construction, non-effective actions.
result Provides a self-contained account for young researchers.
Classifies charge-3 monopoles with symmetry, identifying new spectral curves.
problem Classifying charge-3 monopoles with symmetry.
method Constructing Nahm data and identifying spectral curves with elliptic quotients.
result New monopole spectral curves with and symmetry identified.
Study finds negatively curved spheres in elliptic surfaces and their modifications.
problem Finding negatively curved spheres in elliptic surfaces.
method Using elliptic fibrations with specific singular fibers.
result Identifies spheres with very negative self-intersections in elliptic surfaces and their modifications.
Elliptic Chern characters and Atiyah-Witten formula generalized to double loop spaces.
problem Generalizing classical Atiyah-Witten formula to double loop spaces.
method Constructing elliptic Chern and Bismut-Chern characters, defining elliptic holonomy, and using equivariant twisted parallel transport.
result Established elliptic Atiyah-Witten formula on double loop space.
Study shows unbounded Pontryagin numbers on curved manifolds.
problem Understanding unbounded Pontryagin numbers on curved manifolds.
method Analyzing rational linear combinations of Pontryagin numbers and their relation to the universal elliptic genus.
result Proves existence of unbounded Pontryagin numbers on nonnegatively curved spin manifolds.
The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.
problem Characterizing the locus of residueless meromorphic differentials on elliptic curves.
method Multi-scale compactification of strata, formulas for genus and degree of maps, distinguishing components.
result Complete classification of connected components of residueless loci in exceptional strata.
Using the Weil-Brezin-Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely-generated projective modules over the algebra of the noncommutative torus. We show that such -modules have a natural interpretatio…
Study invariants of elliptic curves in LCS manifolds, leading to new phenomena in Riemann-Finsler geometry.
problem Defining and studying invariants of elliptic curves in locally conformally symplectic manifolds.
method Using -holomorphic curves and Gromov-Witten theory to define and study invariants.
result Found new phenomena in Riemann-Finsler geometry and an analogue of the Weinstein conjecture.
We explain the correspondences between twisted monopoles with Dirac type singularity and polystable twisted mini-holomorphic bundles with Dirac type singularity on a 3-dimensional torus. We also explain that they are equivalent to polystable parabolic twisted difference modules on elliptic curves.
It is an interesting question whether a given equation of motion has a periodic solution or not, and in the positive case to describe them. We investigate periodic magnetic curves in elliptic Sasakian space forms and we obtain a quantization principle for periodic magnetic flowlines on Berger spheres. We give a criteri…
We study the conformally invariant variational problem for time-like curves in the -dimensional Einstein universe defined by the conformal strain functional. We prove that the stationary curves are trapped into an Einsetin universe of dimension , or . We study the linearly-full stationary curves in a four-…
Elliptic curves and braid groups linked through configuration spaces.
problem Understanding the relationship between elliptic curves and braid groups via configuration spaces.
method Constructing isomorphisms between configuration spaces and triples of elliptic curves, points, and holomorphic differentials.
result Unified exceptional sequences involving braid groups and automorphisms of free groups.
Holomorphic maps between configuration spaces are classified, resolving quartic and elliptic curve problems.
problem Classifying holomorphic maps between configuration spaces.
method Using braid groups, elliptic curves, and complex analysis, the authors classify maps and families of elliptic curves.
result The only non-trivial, non-identity holomorphic maps are the resolving quartic map and a map from elliptic curves.
The Frey--Mazur conjecture states that an elliptic curve over is determined up to isogeny by its -torsion Galois representation for . We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multip…
Study trisections on rational elliptic surfaces to find new Zariski pairs.
problem Constructing trisections and related plane curves on rational elliptic surfaces.
method Utilized Mumford representations of semi-reduced divisors to construct trisections and plane curves.
result Existence of a family of Zariski pairs degenerating to the same conic-line arrangement.
This paper establishes an interesting connection between the family of CMC surfaces of revolution in and some specific families of elliptic curves. As a consequence of this connection, we show in the class of spacelike CMC surfaces of revolution in the , only spacelike cylinders and stand…
Takamura established a theory on splitting families of degenerations of complex curves. He introduced a powerful method for constructing a splitting family, called a barking family, in which there appear not only a singular fiber over the origin but also singular fibers over other points, called subordinate fibers. In …
Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed that the B-model of mirror symmetry should be described by a quantum field theory on a Calabi-Yau variety, which they called the Kodaira-Spenser theory (we call it the BCOV theory). This is the first of three papers in which we construct and analyze the quantum BCOV theory…
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.
Every fibration of a projective hyper-Kähler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified by Markushevich. We study those cases where the Abelian surface is a product of two elliptic curves, under some mild genericity hypothes…
We consider the stable ruled surface over an elliptic curve. There is a unique foliation on transverse to the fibration. The minimal self-intersection sections also define a 2-web. We prove that the 4-web defined by the fibration, the foliation and the 2-web is locally parallelizable.
The paper conjectures and proves fixed points for certain group actions on nonpositively curved spaces.
problem Actions by automorphisms of finitely generated groups on nonpositively curved complexes without fixed points.
method Use of Helly graphs and geodesic clique paths to prove ellipticity results.
result Finitely generated torsion groups cannot act without fixed points on nonpositively curved spaces.
We prove several results about the vanishing of the elliptic genus on positively curved Spin manifolds with logarithmic symmetry rank. The proofs are based on the rigidity of the elliptic genus and Kennard's improvement of the Connectedness Lemma for transversely intersecting, totally geodesic submanifolds.
In this paper we consider the isoptic curves on the 2-dimensional geometries of constant curvature $\bE^2,~\bH^2,~\cE^2$. The topic is widely investigated in the Euclidean plane $\bE^2$ see for example \cite{CMM91} and \cite{Wi} and the references given there, but in the hyperbolic and elliptic plane there are few resu…
Elliptic bouquets defined for spin manifolds with circular actions.
problem Defining integration in elliptic cohomology.
method Introducing elliptic bouquets of germs of holomorphic equivariant cohomology classes, integrating them as in K-theory.
result Witten's rigidity theorem follows from integration of elliptic bouquets.
Veech fibrationsmath.GT
Study complex surfaces fibered over Teichmüller curves with Veech fibers.
problem Understanding the geometric and topological properties of complex surfaces.
method Compute invariants via monodromy action on fiber's mod- homology.
result Exact values of invariants for all known algebraically primitive Teichmüller curves.
Modern treatment of Winger's pencil reveals deep connections to modular curves and monodromy.
problem Understanding the structure and deformations of genus ten curves with icosahedral symmetry.
method Analyzing the Jacobian of the Winger pencil and its monodromy properties.
result The Jacobian of the Winger pencil contains an elliptic curve with a distinguished point of order 3 and a monodromy group isomorphic to Γ1(3).
Characterizes totally elliptic surface group representations into Lie groups.
problem Understanding totally elliptic surface group representations into Lie groups.
method Characterization of representations into and by their mapping properties.
result They are either into a compact subgroup or Deroin--Tholozan representations.
We define a Fourier-Mukai transform for a triple consisting of two holomorphic vector bundles over an elliptic curve and a homomorphism between them. We prove that in some cases the transform preserves the natural stability condition for a triple. We also define a Nahm transform for solutions to natural gauge-theoretic…
The paper studies self-Bäcklund curves in centroaffine geometry using elliptic functions.
problem Understanding self-Bäcklund curves in centroaffine geometry.
method Description of general properties and detailed analysis using elliptic functions.
result Provides a detailed description of self-Bäcklund centroaffine curves in terms of elliptic functions.
We study fixed points of smooth torus actions on closed manifolds using fixed point formulas and equivariant elliptic genera. We also give applications to positively curved Riemannian manifolds with symmetry.
The study counts closed elliptic curves and Reeb orbits on Vaisman and Sasakian manifolds.
problem Counting closed orbits and elliptic curves on Vaisman and Sasakian manifolds.
method Analyzes the structure of Vaisman and Sasakian manifolds, uses quasi-regular and -quotients, and counts closed orbits and curves.
result The number of closed elliptic curves and Reeb orbits is either infinite or equal to the sum of all Betti numbers of a Kähler orbifold.
Over a closed manifold, we consider the sectorial projection of an elliptic pseudo-differential operator A of positive order with two rays of minimal growth. We show that it depends continuously on A when the space of pseudo-differential operators is equipped with a certain topology which we explicitly describe. Our ma…
Study homeomorphisms on fine curve graph of surfaces, revealing new types of dynamics.
problem Understanding dynamics of homeomorphisms on fine curve graphs of surfaces.
method Analyzing the action of homeomorphisms on the fine curve graph and relating to classical curve graphs.
result Homeomorphisms induce parabolic isometries, and all positive reals are realized as asymptotic translation lengths.