Arithmetic Geometry Group

Group actions on curves

There is a variety of groups that can act on a Riemann surface/algebraic curve over \(\mathbb{C}\); the automorphism group, the mapping class group (here we might allow punctures) and if the curve is defined over \(\bar{\mathbb{Q}}\) the absolute Galois group \(\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\) is also acting on the curve. Understanding the above groups is a difficult problem and these actions provide information on both the curve and the group itself. For all the groups mentioned above, the action can often be understood in terms of linear representations, by allowing the group to act on vector spaces and modules related to the curve itself, as the (co)homology groups and section of holomorphic differentials.

Surveys on Automorphisms of Curves

J. Antoniadis, A. Kontogeorgis Automorphisms of Curves
A. Broughton, T. Shaska, and A. Wootton On automorphisms of algebraic curves

Geometric Actions

Explicit Group of Automorphisms

The group of automorphisms of the function fields of the curve x^n+y^m+1=0 and Kontogeorgis, Aristides I. J. Number Theory 1998 [Abstract]

We study the group of automorphisms of the function fields of the curves x^n+y^m-1=0. This group is bigger than μ(m)xμ(n) in case m | n. If moreover n-1 is a power of the characteristic, then the group order exceeds the Hurwitz bound
[DOI] [PDF]
We study the group of automorphisms of the function fields of the curves x^n+y^m-1=0. This group is bigger than μ(m)xμ(n) in case m | n. If moreover n-1 is a power of the characteristic, then the group order exceeds the Hurwitz bound

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The group of automorphisms of cyclic extensions of rational function fields and Kontogeorgis, Aristides J. Algebra 1999 [Abstract]

We study the automorphism groups of cyclic extensions of the rational function fields. We give conditions for the cyclic Galois group to be normal in the whole automorphism group, and then we study how the ramification type determines the structure of the whole automorphism group
[DOI] [PDF]
We study the automorphism groups of cyclic extensions of the rational function fields. We give conditions for the cyclic Galois group to be normal in the whole automorphism group, and then we study how the ramification type determines the structure of the whole automorphism group

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Automorphisms of Fermat-like varieties and Kontogeorgis, Aristides Manuscripta Math. 2002 [Abstract]

We study the automorphisms of some nice hypersurfaces and complete intersections in projective space by reducing the problem to the determination of the linear automorphisms of the ambient space that leave the algebraic set invariant
[DOI] [PDF]
We study the automorphisms of some nice hypersurfaces and complete intersections in projective space by reducing the problem to the determination of the linear automorphisms of the ambient space that leave the algebraic set invariant

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On the non-existence of exceptional automorphisms on Shimura curves Kontogeorgis, Aristides, and Rotger, Victor Bull. Lond. Math. Soc. 2008 [Abstract]

We study the group of automorphisms of Shimura curves X_0(D,N) attached to an Eichler order of square-free level N in an indefinite rational quaternion algebra of discriminant D>1. We prove that, when the genus g of the curve is greater than or equal to 2, \mathrmAut(X_0(D,N)) is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions W_0(D,N) as a subgroup of index 1 or 2. It is conjectured that \mathrmAut(X_0(D,N))=W_0(D,N) except for finitely many values of (D,N) and we provide criteria that allow us to show that this is indeed often the case. Our methods are based on the theory of complex multiplication and integral models of Shimura curves
[DOI] [PDF]
We study the group of automorphisms of Shimura curves X_0(D,N) attached to an Eichler order of square-free level N in an indefinite rational quaternion algebra of discriminant D>1. We prove that, when the genus g of the curve is greater than or equal to 2, \mathrmAut(X_0(D,N)) is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions W_0(D,N) as a subgroup of index 1 or 2. It is conjectured that \mathrmAut(X_0(D,N))=W_0(D,N) except for finitely many values of (D,N) and we provide criteria that allow us to show that this is indeed often the case. Our methods are based on the theory of complex multiplication and integral models of Shimura curves

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Bielliptic and Hyperelliptic Modular curves X(N) and the group Aut(X(N)). Bars, Francesc, Kontogeorgis, Aristides, and Xarles, Xavier Acta Arithmetica 2013 [Abstract]

This note has two main facets, first we determine all modular curves X(N) (with N≥7) which are hyperelliptic or bielliptic. As a consequence, the set \Gamma_2(X(N),M):={P∈X(N)(L)|[L:M]\leq2} is not finite for some number field M if and only if N=7 or N=8. Moreover, we obtain that \Gamma_2(X(N),\mathbbQ(\zeta_N)) is always finite where \zeta_N is an N-th primitive root of unity, hence, in particular, the number of quadratic points is finite for any model X_N over \mathbbQ of X(N). Secondly we make available a proof that the automorphism group of X(N) coincides with the normalizer of Γ(N) in \mathrmPSL_2(\mathbbR) modulo Γ(N), in particular no exceptional automorphisms appear for X(N)
[DOI] [PDF]
This note has two main facets, first we determine all modular curves X(N) (with N≥7) which are hyperelliptic or bielliptic. As a consequence, the set \Gamma_2(X(N),M):={P∈X(N)(L)|[L:M]\leq2} is not finite for some number field M if and only if N=7 or N=8. Moreover, we obtain that \Gamma_2(X(N),\mathbbQ(\zeta_N)) is always finite where \zeta_N is an N-th primitive root of unity, hence, in particular, the number of quadratic points is finite for any model X_N over \mathbbQ of X(N). Secondly we make available a proof that the automorphism group of X(N) coincides with the normalizer of Γ(N) in \mathrmPSL_2(\mathbbR) modulo Γ(N), in particular no exceptional automorphisms appear for X(N)

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Automorphisms of generalized Fermat curves Hidalgo, Rubén A., Kontogeorgis, Aristides, Leyton-Álvarez, Maximiliano, and Paramantzoglou, Panagiotis J. Pure Appl. Algebra 2017 [Abstract] [DOI] [PDF]
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Bounding Automorphism Order

Discontinuous groups in positive characteristic and automorphisms of Mumford curves Cornelissen, Gunther, Kato, Fumiharu, and Kontogeorgis, Aristides Math. Ann. 2001 [Abstract]

In this paper, we prove that the order of the automorphism group of a Mumford curve in positive characteristic is bounded by a cubic polynomial of square-root of its genus, and moreover, this bound is sharp. We also show that the automorphisms of Drinfeld modular curves (with the full level) are, under some conditions on the characteristic, all modular in the sense that they come from the normalizer of the corresponding modular group
[DOI] [PDF]
In this paper, we prove that the order of the automorphism group of a Mumford curve in positive characteristic is bounded by a cubic polynomial of square-root of its genus, and moreover, this bound is sharp. We also show that the automorphisms of Drinfeld modular curves (with the full level) are, under some conditions on the characteristic, all modular in the sense that they come from the normalizer of the corresponding modular group

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On abelian automorphism groups of Mumford curves Kontogeorgis, Aristides, and Rotger, Victor Bull. Lond. Math. Soc. 2008 [Abstract]

We use rigid analytic uniformization by Schottky groups to give a bound for the order of the abelian subgroups of the automorphism group of a Mumford curve in terms of its genus
[DOI] [PDF]
We use rigid analytic uniformization by Schottky groups to give a bound for the order of the abelian subgroups of the automorphism group of a Mumford curve in terms of its genus

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Galois module structure of polydifferentials

Quadratic Differentials and Equivariant Deformation Theory of Curves Koeck, Benrhard, and Kontogeorgis, Aristides Annales de L’institut Fourier 2012 [Abstract]

Given a finite p-group G acting on a smooth projective curve X over an algebraically closed field k of characteristic p, the dimension of the tangent space of the associated equivariant deformation functor is equal to the dimension of the space of coinvariants of G acting on the space V of global holomorphic quadratic differentials on X. We apply known results about the Galois module structure of Riemann-Roch spaces to compute this dimension when G is cyclic or when the action of G on X is weakly ramified. Moreover we determine certain subrepresentations of V, called p-rank representations.
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Given a finite p-group G acting on a smooth projective curve X over an algebraically closed field k of characteristic p, the dimension of the tangent space of the associated equivariant deformation functor is equal to the dimension of the space of coinvariants of G acting on the space V of global holomorphic quadratic differentials on X. We apply known results about the Galois module structure of Riemann-Roch spaces to compute this dimension when G is cyclic or when the action of G on X is weakly ramified. Moreover we determine certain subrepresentations of V, called p-rank representations.

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Representation of cyclic groups in positive characteristic and Weierstrass semigroups Karanikolopoulos, Sotiris, and Kontogeorgis, Aristides Journal of Number Theory 2013 [Abstract]

We study the k[G]-module structure of the space of holomorphic differentials of a curve defined over an algebraically closed field of positive characteristic, for a cyclic group G of order p∤n. We also study the relation to the Weierstrass semigroup for the case of Galois Weierstrass points
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We study the k[G]-module structure of the space of holomorphic differentials of a curve defined over an algebraically closed field of positive characteristic, for a cyclic group G of order p∤n. We also study the relation to the Weierstrass semigroup for the case of Galois Weierstrass points

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Integral representations of cyclic groups acting on relative holomorphic differentials of deformations of curves with automorphisms Karanikolopoulos, Sotiris, and Kontogeorgis, Aristides Proc. Amer. Math. Soc. 2014 [Abstract]

We study integral representations of holomorphic differentials on the Oort-Sekiguci-Suwa component of deformations of curves with cyclic group actions
[DOI] [PDF]
We study integral representations of holomorphic differentials on the Oort-Sekiguci-Suwa component of deformations of curves with cyclic group actions

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Automorphisms of Curves and Weierstrass semigroups Karanikolopoulos, Sotiris, and Kontogeorgis, Aristides 2010 [Abstract]

The relation of the Weierstrass semigroup with several invariants of a curve is studied. For Galois covers of curves with group G we introduce a new filtration of the group decomposition subgroup of G. The relation to the ramification filtration is investigated in the case of cyclic covers. We relate our results to invariants defined by Boseck and we study the one point ramification case. We also give applications to Hasse-Witt invariant and symmetric semigroups.
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The relation of the Weierstrass semigroup with several invariants of a curve is studied. For Galois covers of curves with group G we introduce a new filtration of the group decomposition subgroup of G. The relation to the ramification filtration is investigated in the case of cyclic covers. We relate our results to invariants defined by Boseck and we study the one point ramification case. We also give applications to Hasse-Witt invariant and symmetric semigroups.

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On the Galois-module structure of polydifferentials of Artin-Schreier-Mumford curves, modular and integral representation theory Kato, Fumiharu, and Kontogeorgis, Aristides 2016 [Abstract]

We study the Galois-module structure of polydifferentials for Mumford curves, defined over a field of positive charactersitic, using the theory of harmonic cocycles. For the case of Artin-Schreier-Mumford curves the structure of holomorphic polydifferentials is explicitly computed.
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We study the Galois-module structure of polydifferentials for Mumford curves, defined over a field of positive charactersitic, using the theory of harmonic cocycles. For the case of Artin-Schreier-Mumford curves the structure of holomorphic polydifferentials is explicitly computed.

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Galois structure of the holomorphic differential of curves Bleher, Frauke, Chinburg, Ted, and Kontogeorgis, Aristides 2017 [Abstract] [PDF]
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Arithmetic actions on cyclotomic function fields Kontogeorgis, Aristides, and Ward, Kenneth 2018 [Abstract]

We derive the group structure for cyclotomic function fields obtained by applying the Carlitz action for extensions of an initial constant field. The tame and wild structures are isolated to describe the Galois action on differentials. We show that the associated invariant rings are not polynomial.
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We derive the group structure for cyclotomic function fields obtained by applying the Carlitz action for extensions of an initial constant field. The tame and wild structures are isolated to describe the Galois action on differentials. We show that the associated invariant rings are not polynomial.

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Deformation theory of curves

Polydifferentials and the deformation functor of curves with automorphisms and Kontogeorgis, A. J. Pure Appl. Algebra 2007 [Abstract]

We give a relation between the dimension of the tangent space of the deformation functor of curves with automorphisms and the Galois module structure of the space of 2-holomorphic differentials. We prove a homological version of the local-global principle similar to the one of J.Bertin, A. Mezard. Let G be a cyclic subgroup of the group of automorphisms of a curve X, so that the order of G is equal to the characteristic. By using the results of S. Nakajima on the Galois module structure of the space of 2-holomorphic differentials, we compute the dimension of the tangent space of the deformation functor
[DOI] [PDF]
We give a relation between the dimension of the tangent space of the deformation functor of curves with automorphisms and the Galois module structure of the space of 2-holomorphic differentials. We prove a homological version of the local-global principle similar to the one of J.Bertin, A. Mezard. Let G be a cyclic subgroup of the group of automorphisms of a curve X, so that the order of G is equal to the characteristic. By using the results of S. Nakajima on the Galois module structure of the space of 2-holomorphic differentials, we compute the dimension of the tangent space of the deformation functor

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On the tangent space of the deformation functor of curves with automorphisms and Kontogeorgis, Aristides Algebra Number Theory 2007 [Abstract]

We provide a method to compute the dimension of the tangent space to the global infinitesimal deformation functor of a curve together with a subgroup of the group of automorphisms. The computational techniques we developed are applied to several examples including Fermat curves, p-cyclic covers of the affine line and to Lehr-Matignon curves
[DOI] [PDF]
We provide a method to compute the dimension of the tangent space to the global infinitesimal deformation functor of a curve together with a subgroup of the group of automorphisms. The computational techniques we developed are applied to several examples including Fermat curves, p-cyclic covers of the affine line and to Lehr-Matignon curves

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The ramification sequence for a fixed point of an automorphism of a curve and the Weierstrass gap sequence and Kontogeorgis, A. Math. Z. 2008 [Abstract]

For nonsingular projective curves defined over algebraically closed fields of positive characteristic the dependence of the ramification filtration of decomposition groups of the automorphism group with Weierstrass semigroups attached at wild ramification points is studied. A faithful representation of the p-part of the decomposition group at each wild ramified point to a Riemann-Roch space is defined
[DOI] [PDF]
For nonsingular projective curves defined over algebraically closed fields of positive characteristic the dependence of the ramification filtration of decomposition groups of the automorphism group with Weierstrass semigroups attached at wild ramification points is studied. A faithful representation of the p-part of the decomposition group at each wild ramified point to a Riemann-Roch space is defined

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Arithmetic equivalence for function fields, the Goss zeta function and a generalisation Cornelissen, Gunther, Kontogeorgis, Aristides, and Zalm, Lotte J. Number Theory 2010 [Abstract]

We consider three examples of families of curves over a non-archimedean valued field which admit a non-trivial group action. These equivariant deformation spaces can be described by algebraic parameters (in the equation of the curve), or by rigid-analytic parameters (in the Schottky group of the curve). We study the relation between these parameters as rigid-analytic self-maps of the disk
[DOI] [PDF]
We consider three examples of families of curves over a non-archimedean valued field which admit a non-trivial group action. These equivariant deformation spaces can be described by algebraic parameters (in the equation of the curve), or by rigid-analytic parameters (in the Schottky group of the curve). We study the relation between these parameters as rigid-analytic self-maps of the disk

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The relation between rigid-analytic and algebraic deformation parameters for Artin-Schreier-Mumford curves Cornelissen, Gunther, Kato, Fumiharu, and Kontogeorgis, Aristides Israel J. Math. 2010 [Abstract]

We consider three examples of families of curves over a non-archimedean valued field which admit a non-trivial group action. These equivariant deformation spaces can be described by algebraic parameters (in the equation of the curve), or by rigid-analytic parameters (in the Schottky group of the curve). We study the relation between these parameters as rigid-analytic self-maps of the disk
[DOI] [PDF]
We consider three examples of families of curves over a non-archimedean valued field which admit a non-trivial group action. These equivariant deformation spaces can be described by algebraic parameters (in the equation of the curve), or by rigid-analytic parameters (in the Schottky group of the curve). We study the relation between these parameters as rigid-analytic self-maps of the disk

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Field of moduli-Definition

On cyclic covers of the projective line Antoniadis, Jannis A., and Kontogeorgis, Aristides Manuscripta Math. 2006 [Abstract]

We construct configuration spaces for cyclic covers of the projective line that admit extra automorphisms and we describe the locus of curves with given automorphism group. As an application we provide examples of arbitrary high genus that are defined over their field of moduli and are not hyperelliptic
[DOI] [PDF]
We construct configuration spaces for cyclic covers of the projective line that admit extra automorphisms and we describe the locus of curves with given automorphism group. As an application we provide examples of arbitrary high genus that are defined over their field of moduli and are not hyperelliptic

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Field of moduli versus field of definition for cyclic covers of the projective line and Kontogeorgis, Aristides J. Théor. Nombres Bordeaux 2009 [Abstract]

We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli \mathbbR that can not be defined over ℝ is given
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We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli \mathbbR that can not be defined over ℝ is given

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HKG-covers Nottingham-group

A cohomological treatise of HKG-covers with applications to the Nottingham group Kontogeorgis, Aristides, and Tsouknidas, Ioannis 2019 [Abstract]

We characterize Harbater-Katz-Gabber curves in terms of a family of cohomology classes satisfying a compatibility condition. Our construction is applied to the description of finite subgroups of the Nottingham Group.
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We characterize Harbater-Katz-Gabber curves in terms of a family of cohomology classes satisfying a compatibility condition. Our construction is applied to the description of finite subgroups of the Nottingham Group.

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Arithmetic Actions - Arithmetic Topology

On the principal ideal theorem in arithmetic topology. Goundaroulis, Dimocles, and Kontogeorgis, Aristides 2008 [Abstract]

In this paper we state and prove the analogous of the principal ideal theorem of algebraic number theory for the case of 3-manifolds from the point of view of arithmetic topology.
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In this paper we state and prove the analogous of the principal ideal theorem of algebraic number theory for the case of 3-manifolds from the point of view of arithmetic topology.

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Actions on Symmetric curves from the Arithmetic Topology viewpoint Kontogeorgis, Aristides, and Paramantzoglou, Panagiotis 2018 [Abstract]

We give an explanation of the MKR dictionary in Arithmetic topology using Ihara’s theory of profinite braid groups. Motivated by the analogy we perform explicit computations for representations of both braid groups and the absolute Galois group \mathrm{Gal}(\bar{\Q}/\Q) for cyclic covers of the projective line and generalized Fermat curves.
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We give an explanation of the MKR dictionary in Arithmetic topology using Ihara’s theory of profinite braid groups. Motivated by the analogy we perform explicit computations for representations of both braid groups and the absolute Galois group \mathrm{Gal}(\bar{\Q}/\Q) for cyclic covers of the projective line and generalized Fermat curves.

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Group Actions on cyclic covers of the projective line Kontogeorgis, Aristides, and Paramantzoglou, Panagiotis 2018 [Abstract] [PDF]
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A non-commutative differential module approach to Alexander modules Kontogeorgis, Aristides, and Paramantzoglou, Panagiotis 2019 [Abstract] [PDF]
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Galois action on homology of generalized Fermat Curves Kontogeorgis, Aristides, and Paramantzoglou, Panagiotis 2019 [Abstract] [PDF]
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Explicit class field theory-Elliptic curves with complex multiplication

Generating prime order elliptic curves: difficulties and efficiency considerations Konstantinou, Elisavet, Kontogeorgis, Aristides, Stamatiou, Yannis C., and Zaroliagis, Christos 2005 [Abstract] [DOI] [PDF]
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Computing polynomials of the Ramanujan t_n class invariants Konstantinou, Elisavet, and Kontogeorgis, Aristides Canad. Math. Bull. 2009 [Abstract]

We compute the minimal polynomials of the Ramanujan values tn , where n = 11 mod 24, using Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field \mathbbQ(\sqrt-n, and have much smaller coeffcients than the Hilbert polynomials
[DOI] [PDF]
We compute the minimal polynomials of the Ramanujan values tn , where n = 11 mod 24, using Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field \mathbbQ(\sqrt-n, and have much smaller coeffcients than the Hilbert polynomials

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Ramanujan’s class invariants and their use in elliptic curve cryptography Konstantinou, Elisavet, and Kontogeorgis, Aristides Comput. Math. Appl. 2010 [Abstract]

Complex Multiplication (CM) method is a frequently used method for the generation of prime order elliptic curves (ECs) over a prime field \mathbbF_p. The most demanding and complex step of this method is the computation of the roots of a special type of class polynomials, called Hilbert polynomials. These polynonials are uniquely determined by the CM discriminant \(D\). The disadvantage of these polynomials is that they have huge coefficients and thus they need high precision arithmetic for their construction. Alternatively, Weber polynomials can be used in the CM method. These polynomials have much smaller coefficients and their roots can be easily transformed to the roots of the corresponding Hilbert polynomials. However, in the case of prime order elliptic curves, the degree of Weber polynomials is three times larger than the degree of the corresponding Hilbert polynomials and for this reason the calculation of their roots involves computations in the extension field \(\mathbbF_p^3\). Recently, two other classes of polynomials, denoted by \(M_D,l(x)\)and \(M_D,p_1,p_2(x)\)respectively, were introduced which can also be used in the generation of prime order elliptic curves. The advantage of these polynomials is that their degree is equal to the degree of the Hilbert polynomials and thus computations over the extension field can be avoided. In this paper, we propose the use of a new class of polynomials. We will call them Ramanujan polynomials named after Srinivasa Ramanujan who was the first to compute them for few values of \(D\). We explicitly describe the algorithm for the construction of the new polynomials, show that their degree is equal to the degree of the corresponding Hilbert polynomials and give the necessary transformation of their roots (to the roots of the corresponding Hilbert polynomials). Moreover, we compare (theoretically and experimentally) the efficiency of using this new class against the use of the aforementioned Weber, \(M_D,l(x)\)and \(M_D,p_1,p_2(x)\)polynomials and show that they clearly outweigh all of them in the generation of prime order elliptic curves
[DOI] [PDF]
Complex Multiplication (CM) method is a frequently used method for the generation of prime order elliptic curves (ECs) over a prime field \mathbbF_p. The most demanding and complex step of this method is the computation of the roots of a special type of class polynomials, called Hilbert polynomials. These polynonials are uniquely determined by the CM discriminant \(D\). The disadvantage of these polynomials is that they have huge coefficients and thus they need high precision arithmetic for their construction. Alternatively, Weber polynomials can be used in the CM method. These polynomials have much smaller coefficients and their roots can be easily transformed to the roots of the corresponding Hilbert polynomials. However, in the case of prime order elliptic curves, the degree of Weber polynomials is three times larger than the degree of the corresponding Hilbert polynomials and for this reason the calculation of their roots involves computations in the extension field \(\mathbbF_p^3\). Recently, two other classes of polynomials, denoted by \(M_D,l(x)\)and \(M_D,p_1,p_2(x)\)respectively, were introduced which can also be used in the generation of prime order elliptic curves. The advantage of these polynomials is that their degree is equal to the degree of the Hilbert polynomials and thus computations over the extension field can be avoided. In this paper, we propose the use of a new class of polynomials. We will call them Ramanujan polynomials named after Srinivasa Ramanujan who was the first to compute them for few values of \(D\). We explicitly describe the algorithm for the construction of the new polynomials, show that their degree is equal to the degree of the corresponding Hilbert polynomials and give the necessary transformation of their roots (to the roots of the corresponding Hilbert polynomials). Moreover, we compare (theoretically and experimentally) the efficiency of using this new class against the use of the aforementioned Weber, \(M_D,l(x)\)and \(M_D,p_1,p_2(x)\)polynomials and show that they clearly outweigh all of them in the generation of prime order elliptic curves

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On the efficient generation of prime-order elliptic curves Konstantinou, Elisavet, Kontogeorgis, Aristides, Stamatiou, Yannis C., and Zaroliagis, Christos J. Cryptology 2010 [Abstract]

We consider the generation of prime-order elliptic curves (ECs) over a prime field p using the Complex Multiplication (CM) method. A crucial step of this method is to compute the roots of a special type of class field polynomials with the most commonly used being the Hilbert and Weber ones. These polynomials are uniquely determined by the CM discriminant D. In this paper, we consider a variant of the CM method for constructing elliptic curves (ECs) of prime order using Weber polynomials. In attempting to construct prime-order ECs using Weber polynomials, two difficulties arise (in addition to the necessary transformations of the roots of such polynomials to those of their Hilbert counterparts). The first one is that the requirement of prime order necessitates that D\equiv3mod8), which gives Weber polynomials with degree three times larger than the degree of their corresponding Hilbert polynomials (a fact that could affect efficiency). The second difficulty is that these Weber polynomials do not have roots in p . In this work, we show how to overcome the above difficulties and provide efficient methods for generating ECs of prime order focusing on their support by a thorough experimental study. In particular, we show that such Weber polynomials have roots in the extension field p3 and present a set of transformations for mapping roots of Weber polynomials in \mathbbF_p^3 to roots of their corresponding Hilbert polynomials in p . We also show how an alternative class of polynomials, with degree equal to their corresponding Hilbert counterparts (and hence having roots in p ), can be used in the CM method to generate prime-order ECs. We conduct an extensive experimental study comparing the efficiency of using this alternative class against the use of the aforementioned Weber polynomials. Finally, we investigate the time efficiency of the CM variant under four different implementations of a crucial step of the variant and demonstrate the superiority of two of them
[DOI] [PDF]
We consider the generation of prime-order elliptic curves (ECs) over a prime field p using the Complex Multiplication (CM) method. A crucial step of this method is to compute the roots of a special type of class field polynomials with the most commonly used being the Hilbert and Weber ones. These polynomials are uniquely determined by the CM discriminant D. In this paper, we consider a variant of the CM method for constructing elliptic curves (ECs) of prime order using Weber polynomials. In attempting to construct prime-order ECs using Weber polynomials, two difficulties arise (in addition to the necessary transformations of the roots of such polynomials to those of their Hilbert counterparts). The first one is that the requirement of prime order necessitates that D\equiv3mod8), which gives Weber polynomials with degree three times larger than the degree of their corresponding Hilbert polynomials (a fact that could affect efficiency). The second difficulty is that these Weber polynomials do not have roots in p . In this work, we show how to overcome the above difficulties and provide efficient methods for generating ECs of prime order focusing on their support by a thorough experimental study. In particular, we show that such Weber polynomials have roots in the extension field p3 and present a set of transformations for mapping roots of Weber polynomials in \mathbbF_p^3 to roots of their corresponding Hilbert polynomials in p . We also show how an alternative class of polynomials, with degree equal to their corresponding Hilbert counterparts (and hence having roots in p ), can be used in the CM method to generate prime-order ECs. We conduct an extensive experimental study comparing the efficiency of using this alternative class against the use of the aforementioned Weber polynomials. Finally, we investigate the time efficiency of the CM variant under four different implementations of a crucial step of the variant and demonstrate the superiority of two of them

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Some remarks on the construction of class polynomials Konstantinou, Elisavet, and Kontogeorgis, Aristides Adv. Math. Commun. 2011 [Abstract]

Class invariants are singular values of modular functions which generate the class fields of imaginary quadratic number fields. Their minimal polynomials, called class polynomials, are uniquely determined by a discriminant −D≤0 and are used in many applications, including the generation of elliptic curves. In all these applications, it is desirable that the size of the polynomials is as small as possible. Among all class polynomials, Weber polynomials constructed with discriminants −D≡1(mod8) have the smallest height and require the least precision for their construction. In this paper, we will show that this fact does not necessarily lead to the most efficient computations, since the congruences modulo 8 of the discriminants affect the degrees of the polynomials
[DOI] [PDF]
Class invariants are singular values of modular functions which generate the class fields of imaginary quadratic number fields. Their minimal polynomials, called class polynomials, are uniquely determined by a discriminant −D≤0 and are used in many applications, including the generation of elliptic curves. In all these applications, it is desirable that the size of the polynomials is as small as possible. Among all class polynomials, Weber polynomials constructed with discriminants −D≡1(mod8) have the smallest height and require the least precision for their construction. In this paper, we will show that this fact does not necessarily lead to the most efficient computations, since the congruences modulo 8 of the discriminants affect the degrees of the polynomials

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Ramanujan invariants for discriminants congruent to 5\pmod24 Konstantinou, Elisavet, and Kontogeorgis, Aristides Int. J. Number Theory 2012 [Abstract]

In this paper we compute the minimal polynomials of Ramanujan values 27t−12n for discriminants D≡5mod24. Our method is based on Shimura Reciprocity Law as which was made computationally explicit by A.Gee and P. Stevenhagen. However, since these Ramanujan values are not class invariants, we present a modification of the above method which can be applied on modular functions that do not necessarily yield class invariants
[DOI] [PDF]
In this paper we compute the minimal polynomials of Ramanujan values 27t−12n for discriminants D≡5mod24. Our method is based on Shimura Reciprocity Law as which was made computationally explicit by A.Gee and P. Stevenhagen. However, since these Ramanujan values are not class invariants, we present a modification of the above method which can be applied on modular functions that do not necessarily yield class invariants

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What is your “birthday elliptic curves”?" Chan, Heng Huat, Konstantinou, Elisavet, Kontogeorgis, Aristides, and Tan, Chik How Finite Fields and Their Applications 2012 [Abstract]

In this article, Ramanujan-Weber class invariants and its analogue are used to derive birthday elliptic curves.
[DOI] [PDF]
In this article, Ramanujan-Weber class invariants and its analogue are used to derive birthday elliptic curves.

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Constructing class invariants and Kontogeorgis, Aristides Math. Comp. 2014 [Abstract]

Shimura reciprocity law allows us to verify that a modular function is a class invariant. Here we present a new method based on Shimura reciprocity that allows us not only to verify but to find new class invariants from a modular function of level N.
[DOI] [PDF]
Shimura reciprocity law allows us to verify that a modular function is a class invariant. Here we present a new method based on Shimura reciprocity that allows us not only to verify but to find new class invariants from a modular function of level N.

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Revisiting the Complex Multiplication Method for the Construction of Elliptic Curves Konstantinou, Elisavet, and Kontogeorgis, Aristides 2015 [Abstract] [PDF]
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Knot theory

Framization of the Temperley-Lieb Algebra Goundaroulis, Dimos, Juyumaya, Jesus, Kontogeorgis, Aristides, and Lambropoulou, Sofia Mathematical Research Letters 2017 [Abstract]

We propose a framization of the Temperley-Lieb algebra. The framization is a procedure that can briefly be described as the adding of framing to a known knot algebra in a way that is both algebraically consistent and topologically meaningful. Our framiza- tion of the Temperley-Lieb algebra is defined as a quotient of the Yokonuma-Hecke algebra. The main theorem provides neces- sary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra to pass through to the quotient algebra. Using this we construct 1-variable invariants for classical knots and links, which, as we show, are not topologically equivalent to the Jones polynomial.
[PDF]
We propose a framization of the Temperley-Lieb algebra. The framization is a procedure that can briefly be described as the adding of framing to a known knot algebra in a way that is both algebraically consistent and topologically meaningful. Our framiza- tion of the Temperley-Lieb algebra is defined as a quotient of the Yokonuma-Hecke algebra. The main theorem provides neces- sary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra to pass through to the quotient algebra. Using this we construct 1-variable invariants for classical knots and links, which, as we show, are not topologically equivalent to the Jones polynomial.

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The Yokonuma-Temperley-Lieb Algebra Goundaroulis, Demos, Juyumaya, Jesus, Kontogeorgis, Aristides, and Lambropoulou, Sofia 2014 [Abstract]

In this paper we introduce the Yokonuma–Temperley–Lieb algebra as a quotient of the Yokonuma–Hecke algebra over a two-sided ideal generated by an expression analogous to the one of the classical Temperley-Lieb algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma–Hecke algebra to pass through to the quotient algebra, leading to a sequence of knot invariants which coincide with the Jones polynomial.
[PDF]
In this paper we introduce the Yokonuma–Temperley–Lieb algebra as a quotient of the Yokonuma–Hecke algebra over a two-sided ideal generated by an expression analogous to the one of the classical Temperley-Lieb algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma–Hecke algebra to pass through to the quotient algebra, leading to a sequence of knot invariants which coincide with the Jones polynomial.

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A generalized skein relation for Khovanov homology and a categorification of the θ-invariant Chlouveraki, Maria, Goundaroulis, Dimos, Kontogeorgis, Aristides, and Lambropoulou, Sofia 2019 [Abstract] [PDF]
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