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We present a necessary and sufficient condition for a maximal curve, defined over the algebraic closure of a finite field, to be realised as an HKG-cover. We use an approach via pole numbers in a rational point of the curve. For this class of curves, we compute their Weierstrass semigroup as well as the jumps of their higher ramification filtrations at this point, the unique ramification point of the cover.

Let $X$ be a smooth projective geometrically irreducible curve over a perfect field $k$ of positive characteristic $p$. Suppose $G$ is a finite group acting faithfully on $X$ such that $G$ has non-trivial cyclic Sylow $p$-subgroups. We show that the decomposition of the space of holomorphic differentials of $X$ into a direct sum of indecomposable $k[G]$-modules is uniquely determined by the lower ramification groups and the fundamental characters of closed points of $X$ that are ramified in the cover $X\to X/G$. We apply our method to determine the $\mathrm{PSL}(2,\mathbb{F}_\ell)$-module structure of the space of holomorphic differentials of the reduction of the modular curve $\mathcal{X}(\ell)$ modulo $p$ when $p$ and $\ell$ are distinct odd primes and the action of $\mathrm{PSL}(2,\mathbb{F}_\ell)$ on this reduction is not tamely ramified. This provides some non-trivial congruences modulo appropriate maximal ideals containing $p$ between modular forms arising from isotypic components with respect to the action of $\mathrm{PSL}(2,\mathbb{F}_\ell)$ on $\mathcal{X}(\ell)$., 51 pages. In this version, we corrected some typos

We use tools from combinatorial group theory in order to study actions of three types on groups acting on a curve, namely the automorphism group of a compact Riemann surface, the mapping class group acting on a surface (which now is allowed to have some points removed) and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ in the case of cyclic covers of the projective line., Comment: 23 pages

We study p p -group Galois covers X → P 1 X \rightarrow \mathbb {P}^1 with only one fully ramified point in characteristic p > 0 p>0 . These covers are important because of the Harbater–Katz–Gabber compactification theorem of Galois actions on complete local rings. The sequence of ramification jumps is related to the Weierstrass semigroup of the global cover at the stabilized point. We determine explicitly the jumps of the ramification filtrations in terms of pole numbers. We give applications for curves with zero p p -rank: we focus on curves that admit a big action. Moreover, we initiate the study of the Galois module structure of polydifferentials.

We consider a Fermat curve $F_n:x^n+y^n+z^n=1$ over an algebraically closed field $k$ of characteristic $p\geq0$ and study the action of the automorphism group $G=\left(\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}\right)\rtimes S_3$ on the canonical ring $R=\bigoplus H^0(F_n,\Omega_{F_n}^{\otimes m})$ when $p>3$, $p\nmid n$ and $n-1$ is not a power of $p$. In particular, we explicitly determine the classes $[H^0(F_n,\Omega_{F_n}^{\otimes m})]$ in the Grothendieck group $K_0(G,k)$ of finitely generated $k[G]$-modules, describe the respective equivariant Hilbert series $H_{R,G}(t)$ as a rational function, and use our results to write a program in Sage that computes $H_{R,G}(t)$ for an arbitrary Fermat curve.

The canonical ideal for Harbater Katz Gabber covers satisfying the conditions of Petri's theorem is studied and an explicit non-singular model of the above curves is given., Comment: 17 pages

The Heisenberg curve is defined to be the curve corresponding to an extension of the projective line by the Heisenberg group modulo $n$, ramified above three points. This curve is related to the Fermat curve and its group of automorphisms is studied. Also we give an explicit equation for the curve $C_3$., 13 pages

Let K be an algebraically closed field of characteristic p ≥ 0 . A generalized Fermat curve of type ( k , n ) , where k , n ≥ 2 are integers (for p ≠ 0 we also assume that k is relatively prime to p), is a non-singular irreducible projective algebraic curve F k , n defined over K admitting a group of automorphisms H ≅ Z k n so that F k , n / H is the projective line with exactly ( n + 1 ) cone points, each one of order k. Such a group H is called a generalized Fermat group of type ( k , n ) . If ( n − 1 ) ( k − 1 ) > 2 , then F k , n has genus g n , k > 1 and it is known to be non-hyperelliptic. In this paper, we prove that every generalized Fermat curve of type ( k , n ) has a unique generalized Fermat group of type ( k , n ) if ( k − 1 ) ( n − 1 ) > 2 (for p > 0 we also assume that k − 1 is not a power of p). Generalized Fermat curves of type ( k , n ) can be described as a suitable fiber product of ( n − 1 ) classical Fermat curves of degree k. We prove that, for ( k − 1 ) ( n − 1 ) > 2 (for p > 0 we also assume that k − 1 is not a power of p), each automorphism of such a fiber product curve can be extended to an automorphism of the ambient projective space. In the case that p > 0 and k − 1 is a power of p, we use tools from the theory of complete projective intersections in order to prove that, for k and n + 1 relatively prime, every automorphism of the fiber product curve can also be extended to an automorphism of the ambient projective space. In this article we also prove that the set of fixed points of the non-trivial elements of the generalized Fermat group coincide with the hyper-osculating points of the fiber product model under the assumption that the characteristic p is either zero or p > k n − 1 .

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 framization of the Temperley-Lieb algebra is defined as a quotient of the Yokonuma-Hecke 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. Using this we construct 1-variable invariants for classical knots and links, which, as we show, are not topologically equivalent to the Jones polynomial.

The automorphism group of a curve is studied from the viewpoint of the canonical embedding and Petri's theorem. A criterion for identifying the automorphism group as an algebraic subgroup the general linear group is given. Furthermore the action of the automorphism group is extended to an action of the minimal free resolution of the canonical ring of the curve $X$., 16 pages

The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups $H$. We provide a unified study of the action of both cover Galois group $H$ and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated., 35 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1804.07021

The theory of R. Crowell on derived modules is approached within the theory of non-commutative differential modules. We also seek analogies to the theory of cotangent complex from differentials in the commutative ring setting. Finally we give examples motivated from the theory of Galois coverings of curves., Comment: 9 pages

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., Comment: 15 pages

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., Comment: 25 pages

The main theorem of [2], claiming to give an upper bound for the number of automorphisms of a Mumford curve in characteristic

This is a survey article concerning the groups of automorphisms of curves defined over algebraically closed fields of positive characteristic, their representations and applications to their deformation theory.

We study integral representations of holomorphic differentials on the Oort-Sekiguci-Suwa component of deformations of curves with cyclic group actions., Comment: 12 pages 1 figure

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.

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 pℓn. We also study the relation to the Weierstrass semigroup for the case of Galois Weierstrass points.

In this paper we compute the minimal polynomials of Ramanujan values [Formula: see text] for discriminants D ≡ 5 ( mod 24). Our method is based on Shimura Reciprocity Law as which was made computationally explicit by Gee and Stevenhagen in [Generating class fields using Shimura reciprocity, in Algorithmic Number Theory, Lecture Notes in Computer Science, Vol. 1423 (Springer, Berlin, 1998), pp. 441–453; MR MR1726092 (2000m:11112)]. However, since these Ramanujan values are not class invariants, we present a modification of the method used in [Generating class fields using Shimura reciprocity, in Algorithmic Number Theory, Lecture Notes in Computer Science, Vol. 1423 (Springer, Berlin, 1998), pp. 441–453; MR MR1726092 (2000m:11112)] which can be applied on modular functions that do not necessarily yield class invariants.

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., 30 pages, to appear in Ann. Inst. Fourier (Grenoble)

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

The Complex Multiplication (CM) method is a method frequently used for the generation of elliptic curves (ECs) over a prime field Fp. 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. However, there are several polynomials, called class polynomials, which can be used in the CM method, having much smaller coefficients, and fulfilling the prerequisite that their roots can be easily transformed to the roots of the corresponding Hilbert polynomials.In this paper, we propose the use of a new class of polynomials which are derived from Ramanujan’s class invariants tn. We explicitly describe the algorithm for the construction of the new polynomials and give the necessary transformation of their roots to the roots of the corresponding Hilbert polynomials. We provide a theoretical asymptotic bound for the bit precision requirements of all class polynomials and, also using extensive experimental assessments, we compare the efficiency of using the new polynomials against the use of the other class polynomials. Our comparison shows that the new class of polynomials clearly surpass all of the previously used polynomials when they are used in the generation of prime order elliptic curves.

We compute the minimal polynomials of the Ramanujan values tn, where n ≡ 11 mod 24, using the Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field and have much smaller coefficients than the Hilbert polynomials.

We consider the generation of prime-order elliptic curves (ECs) over a prime field $\mathbb{F}_{p}$ 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?3mod8), 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 $\mathbb{F}_{p}$ . 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 $\mathbb{F}_{p^{3}}$ and present a set of transformations for mapping roots of Weber polynomials in $\mathbb{F}_{p^{3}}$ to roots of their corresponding Hilbert polynomials in $\mathbb{F}_{p}$ . We also show how an alternative class of polynomials, with degree equal to their corresponding Hilbert counterparts (and hence having roots in $\mathbb{F}_{p}$ ), 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.

Nous donnons un critere, dependant du groupe des automorphismes, pour que certains revetements cycliques de la droite projective soient definis sur leur corps de modules. Nous donnons aussi un exemple de revetement cyclique de la droite projective complexe de corps de module R qui ne peut pas etre defini sur R.

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.

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 and 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.

In this article we give a detailed overview of the Complex Multiplication (CM) method for constructing elliptic curves with a given number of points. In the core of this method, there is a special polynomial called Hilbert class polynomial which is constructed with input a fundamental discriminantd < 0. The construction of this polynomial is the most demanding and time-consuming part of the method and thus the use of several alternative polynomials has been proposed in previous work. All these polynomials are calledclass polynomials and they are generated by proper values of modular functions calledclass invariants. Besides an analysis on these polynomials, in this paper we will describe our results about finding new class invariants using the Shimura reciprocity law. Finally, we will see how the choice of the discriminant can affect the degree of the class polynomial and consequently the efficiency of the whole CM method.

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.

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.

We determine all modular curves X(N) (with $N\geq 7$) which are hyperelliptic or bielliptic. We make available a proof that the automorphism group of X(N) coincides with the normalizer of $\Gamma(N)$ in $\operatorname{PSL}_2(\mathbb{R})$., Comment: Corrected typos. Reorganized the whole paper. Main results unchanged

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 $N$., 12 pages

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.

We study the automorphism groups of the reduction$X_0(N) \times \bar {\mathbb {F}}_p$of a modular curveX0(N) over primespN.

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, $\Aut (X_0(D, N))$ 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 $\Aut (X_0(D, N)) = W_0(D, N)$ 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 of Shimura curves and the Cerednik-Drinfeld theory on their rigid analytic uniformization at primes $p\mid D$., Title correction. 15 pages. This is an expanded version of the second half of version 1 of math/0604099

In this paper we investigate the connection between relations among various invariants of number fields L H corresponding to subgroups H acting on L and of linear relations among norm idempotents.

We consider the generation of prime order elliptic curves (ECs) over a prime field $\mathbb{F}_p$ 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, uniquely determined by the CM discriminant D. 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 ≡ 3 (mod 8), 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 $\mathbb{F}_p$. In this paper we show how to overcome the above difficulties and provide efficient methods for generating ECs of prime order supported by a thorough experimental study. In particular, we show that such Weber polynomials have roots in $\mathbb{F}_{p^3}$ and present a set of transformations for mapping roots of Weber polynomials in $\mathbb{F}_{p^3}$ to roots of their corresponding Hilbert polynomials in $\mathbb{F}_{p}$. We also show how a new class of polynomials, with degree equal to their corresponding Hilbert counterparts (and hence having roots in $\mathbb{F}_{p}$), can be used in the CM method to generate prime order ECs. Finally, we compare experimentally the efficiency of using this new class against the use of the aforementioned Weber polynomials.

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., 33 pages, A gap in the proof of lemma 1.5 is filled

We will study the group of automorphisms of the function fields of the curves x n + y m +1=0, for n ≠ m . This groups is bigger than μ ( n )× μ ( m ) in case m | n . If moreover n −1 is a power of the characteristic, then the group order exceeds the Hurwitz bound.