scholarly journals Universal functors on symmetric quotient stacks of Abelian varieties

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Andreas Krug ◽  
Ciaran Meachan
Keyword(s):  
Elliptic Curve ◽  
Abelian Varieties ◽  
Abelian Surface ◽  
Hilbert Schemes ◽  
Abelian Surfaces ◽  
The One ◽  
Dimension One

AbstractWe consider certain universal functors on symmetric quotient stacks of Abelian varieties. In dimension two, we discover a family of $${{\mathbb {P}}}$$ P -functors which induce new derived autoequivalences of Hilbert schemes of points on Abelian surfaces; a set of braid relations on a holomorphic symplectic sixfold; and a pair of spherical functors on the Hilbert square of an Abelian surface, whose twists are related to the well-known Horja twist. In dimension one, our universal functors are fully faithful, giving rise to a semiorthogonal decomposition for the symmetric quotient stack of an elliptic curve (which we compare to the one discovered by Polishchuk–Van den Bergh), and they lift to spherical functors on the canonical cover, inducing twists which descend to give new derived autoequivalences here as well.

The Chowla–Selberg Formula and The Colmez Conjecture

2010 ◽  
Vol 62 (2) ◽  
pp. 456-472 ◽  
Author(s):  
Tonghai Yang
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Abelian Varieties ◽  
Siegel Modular Forms ◽  
Abelian Surface ◽  
Intersection Numbers ◽  
Abelian Surfaces ◽  
Elliptic Modular Form ◽  
Faltings Height

AbstractIn this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.

scholarly journals Pseudoeffective and nef classes on abelian varieties

2011 ◽  
Vol 147 (6) ◽  
pp. 1793-1818 ◽  
Author(s):  
Olivier Debarre ◽  
Lawrence Ein ◽  
Robert Lazarsfeld ◽  
Claire Voisin
Keyword(s):  
Elliptic Curve ◽  
Abelian Varieties ◽  
Complex Multiplication ◽  
The Self ◽  
Abelian Surface

AbstractWe study the cones of pseudoeffective and nef cycles of higher codimension on the self product of an elliptic curve with complex multiplication, and on the product of a very general abelian surface with itself. In both cases, we find for instance the existence of nef classes that are not pseudoeffective, answering in the negative a question raised by Grothendieck in correspondence with Mumford. We also discuss several problems and questions for further investigation.

scholarly journals A Study of Bitcoin and Blockchain

2021 ◽  
Author(s):  
Jingyi Cai
Keyword(s):  
Graph Theory ◽  
Elliptic Curve ◽  
Distributed Database ◽  
Blockchain Technology ◽  
The One ◽  
Mathematical Foundations

Blockchain technology is a distributed database and a public ledger. It records every transaction that has been made from its inception. Once entered, these records cannot be modified or erased. The technology utilizes various algorithms of a cryptographic nature to reach a consensus. These cryptograhic functions also ensure the integrity and authenticity of data that has been interchanged across the network. Because of these features, blockchain technology has been implemented into various financial and non-financial fields. In this thesis, we introduce the mathematical foundations of Bitcoin, and different cryptographic functions that are used in Blockchain, especially elliptic curve multiplication. We construct two MATLAB models to study the fork events which is the one of typical consensus problems in the system. Moreover, we use graph theory and MATLAB models to represent and describe the Bitcoin protocols.

Local Bounds for Torsion Points on Abelian Varieties

2008 ◽  
Vol 60 (3) ◽  
pp. 532-555 ◽  
Author(s):  
Pete L. Clark ◽  
Xavier Xarles
Keyword(s):  
Abelian Varieties ◽  
Residue Field ◽  
Rational Numbers ◽  
Abelian Surface ◽  
Torsion Subgroup ◽  
Good Reduction ◽  
Torsion Points ◽  
Ramification Index ◽  
Numerical Invariants ◽  
Split Torus

AbstractWe say that an abelian variety over a p-adic field K has anisotropic reduction (AR) if the special fiber of its Néronminimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the K-rational torsion subgroup of a g-dimensional AR variety depending only on g and the numerical invariants of K (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of g, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an AR abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.

scholarly journals Renormalization of a model for spin-1 matter fields

2020 ◽  
Vol 80 (7) ◽  
Author(s):  
Ailier Rivero-Acosta ◽  
Carlos A. Vaquera-Araujo
Keyword(s):  
Lorentz Group ◽  
Homogeneous Lorentz Group ◽  
Mass Dimension ◽  
The One ◽  
Dimension One ◽  
Matter Fields

Abstract In this work, the one-loop renormalization of a theory for fields transforming in the $$(1,0)\oplus (0,1)$$(1,0)⊕(0,1) representation of the Homogeneous Lorentz Group is studied. The model includes an arbitrary gyromagnetic factor and self-interactions of the spin 1 field, which has mass dimension one. The model is shown to be renormalizable for any value of the gyromagnetic factor.

scholarly journals Multivariate Markov Families of Copulas

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Ludger Overbeck ◽  
Wolfgang M. Schmidt
Keyword(s):  
Markov Property ◽  
Dimensional Case ◽  
Dependence Structure ◽  
Multivariate Process ◽  
One Dimensional ◽  
Multivariate Case ◽  
Finite Dimensional ◽  
Different Dimensions ◽  
The One ◽  
Dimension One

AbstractFor the Markov property of a multivariate process, a necessary and suficient condition on the multidimensional copula of the finite-dimensional distributions is given. This establishes that the Markov property is solely a property of the copula, i.e., of the dependence structure. This extends results by Darsow et al. [11] from dimension one to the multivariate case. In addition to the one-dimensional case also the spatial copula between the different dimensions has to be taken into account. Examples are also given.

The wall-crossing behavior for Bridgeland’s stability conditions on abelian and K3 surfaces

2018 ◽  
Vol 2018 (735) ◽  
pp. 1-107 ◽  
Author(s):  
Hiroki Minamide ◽  
Shintarou Yanagida ◽  
Kōta Yoshioka
Keyword(s):  
Stability Condition ◽  
Derived Category ◽  
The Other ◽  
Abelian Surface ◽  
Stability Conditions ◽  
Abelian Surfaces ◽  
Coherent Sheaves ◽  
Fundamental Properties ◽  
Wall Crossing ◽  
Wall Crossing Formula

AbstractThe wall-crossing behavior for Bridgeland’s stability conditions on the derived category of coherent sheaves on K3 or abelian surface is studied. We introduce two types of walls. One is called the wall for categories, where thet-structure encoded by stability condition is changed. The other is the wall for stabilities, where stable objects with prescribed Mukai vector may get destabilized. Some fundamental properties of walls and chambers are studied, including the behavior under Fourier–Mukai transforms. A wall-crossing formula of the counting of stable objects will also be derived. As an application, we will explain previous results on the birational maps induced by Fourier–Mukai transforms on abelian surfaces. These transformations turns out to coincide with crossing walls of certain property.

scholarly journals Abelian varieties isogenous to a power of an elliptic curve

2018 ◽  
Vol 154 (5) ◽  
pp. 934-959 ◽  
Author(s):  
Bruce W. Jordan ◽  
Allan G. Keeton ◽  
Bjorn Poonen ◽  
Eric M. Rains ◽  
Nicholas Shepherd-Barron ◽  
...  
Keyword(s):  
Elliptic Curve ◽  
Abelian Variety ◽  
Opposite Direction ◽  
Abelian Varieties ◽  
Sufficient Conditions ◽  
Higher Dimensional ◽  
Free Left ◽  
Necessary And Sufficient ◽  
Finitely Presented ◽  
Torsion Free

Let $E$ be an elliptic curve over a field $k$. Let $R:=\operatorname{End}E$. There is a functor $\mathscr{H}\!\mathit{om}_{R}(-,E)$ from the category of finitely presented torsion-free left $R$-modules to the category of abelian varieties isogenous to a power of $E$, and a functor $\operatorname{Hom}(-,E)$ in the opposite direction. We prove necessary and sufficient conditions on $E$ for these functors to be equivalences of categories. We also prove a partial generalization in which $E$ is replaced by a suitable higher-dimensional abelian variety over $\mathbb{F}_{p}$.

scholarly journals Kummer surfaces associated to (1, 2)-polarized abelian surfaces

2011 ◽  
Vol 202 ◽  
pp. 127-143
Author(s):  
Afsaneh Mehran
Keyword(s):  
Double Cover ◽  
Pezzo Surface ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Abelian Surfaces ◽  
Del Pezzo Surface ◽  
Singular Fibers ◽  
Kummer Surfaces ◽  
Del Pezzo

AbstractThe aim of this paper is to describe the geometry of the generic Kummer surface associated to a (1, 2)-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an interesting elliptic fibration with twelve singular fibers of typeI2.

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