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Journal articles on the topic 'Galois cover'

Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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1

Baik, Hyungryul, Farbod Shokrieh, and Chenxi Wu. "Limits of canonical forms on towers of Riemann surfaces." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 764 (July 1, 2020): 287–304. http://dx.doi.org/10.1515/crelle-2019-0007.

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AbstractWe prove a generalized version of Kazhdan’s theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence {\{S_{n}\rightarrow S\}} of finite Galois covers of a hyperbolic Riemann surface S, converging to the universal cover. The theorem states that the sequence of forms on S inherited from the canonical forms on {S_{n}}’s converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover. Along the way, we also prove a Gauss–Bonnet-type theorem in the context of arbitrary infinite Galois covers.

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2

Kontogeorgis, Aristides, and Panagiotis Paramantzoglou. "Galois action on homology of generalized Fermat Curves." Quarterly Journal of Mathematics 71, no. 4 (November 28, 2020): 1377–417. http://dx.doi.org/10.1093/qmath/haaa038.

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Abstract 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 $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.

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3

König, Joachim. "On the mod-p distribution of discriminants of G-extensions." International Journal of Number Theory 16, no. 04 (November 11, 2019): 767–85. http://dx.doi.org/10.1142/s1793042120500396.

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This paper was motivated by a recent paper by Krumm and Pollack ([Twists of hyperelliptic curves by integers in progressions modulo [Formula: see text], preprint (2018); https://arXiv.org/abs/1807.00972] ) investigating modulo-[Formula: see text] behavior of quadratic twists with rational points of a given hyperelliptic curve, conditional on the abc-conjecture. We extend those results to twisted Galois covers with arbitrary Galois groups. The main point of this generalization is to interpret those results as statements about the sets of specializations of a given Galois cover under restrictions on the discriminant. In particular, we make a connection with existing heuristics about the distribution of discriminants of Galois extensions such as the Malle conjecture: our results show in a precise sense the non-existence of “local obstructions” to such heuristics, in many cases essentially only under the assumption that [Formula: see text] occurs as the Galois group of a Galois cover defined over [Formula: see text]. This complements and generalizes a similar result in the direction of the Malle conjecture by Dèbes ([On the Malle conjecture and the self-twisted cover, Israel J. Math. 218(1) (2017) 101–131]).

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4

AMRAM, MEIRAV, MINA TEICHER, and UZI VISHNE. "THE FUNDAMENTAL GROUP OF GALOIS COVER OF THE SURFACE 𝕋 × 𝕋." International Journal of Algebra and Computation 18, no. 08 (December 2008): 1259–82. http://dx.doi.org/10.1142/s0218196708004895.

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This is the final paper in a series of four, concerning the surface 𝕋 × 𝕋 embedded in ℂℙ8, where 𝕋 is the one-dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto ℂℙ2, and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface.

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5

AMRAM, MEIRAV, MINA TEICHER, and UZI VISHNE. "THE FUNDAMENTAL GROUP OF THE GALOIS COVER OF HIRZEBRUCH SURFACE F1(2, 2)." International Journal of Algebra and Computation 17, no. 03 (May 2007): 507–25. http://dx.doi.org/10.1142/s0218196707003780.

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This paper is the second in a series of papers concerning Hirzebruch surfaces. In the first paper in this series, the fundamental group of Galois covers of Hirzebruch surfaces Fk(a, b), where a, b are relatively prime, was shown to be trivial. For the general case, the conjecture stated that the fundamental group is [Formula: see text] where c = gcd (a, b) and n = 2ab + kb2. In this paper, we degenerate the Hirzebruch surface F1(2, 2), compute the braid monodromy factorization of the branch curve in ℂ2, and verify that, in this case, the conjecture holds: the fundamental group of the Galois cover of F1(2, 2) with respect to a generic projection is isomorphic to [Formula: see text].

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6

Kedlaya, Kiran S. "On the Geometry ofp-Typical Covers in Characteristicp." Canadian Journal of Mathematics 60, no. 1 (February 1, 2008): 140–63. http://dx.doi.org/10.4153/cjm-2008-006-8.

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AbstractForpa prime, ap-typical cover of a connected scheme on whichp= 0 is a finite étale cover whose monodromy group (i.e.,the Galois group of its normal closure) is ap-group. The geometry of such covers exhibits some unexpectedly pleasant behaviors; building on work of Katz, we demonstrate some of these. These include a criterion for when a morphism induces an isomorphism of thep-typical quotients of the étale fundamental groups, and a decomposition theorem forp-typical covers of polynomial rings over an algebraically closed field.

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7

Shirane, Taketo. "Connected Numbers and the Embedded Topology of Plane Curves." Canadian Mathematical Bulletin 61, no. 3 (September 1, 2018): 650–58. http://dx.doi.org/10.4153/cmb-2017-066-5.

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AbstractThe splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree b ≥ 4, where an Artal arrangement of degree b is a plane curve consisting of one smooth curve of degree b and three of its total inflectional tangents.

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8

PARDINI, RITA, and FRANCESCA TOVENA. "ON THE FUNDAMENTAL GROUP OF AN ABELIAN COVER." International Journal of Mathematics 06, no. 05 (October 1995): 767–89. http://dx.doi.org/10.1142/s0129167x9500033x.

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Let X, Y be smooth complex projective varieties of dimension n≥2 and let f: Y→X be a totally ramified abelian cover. Assume that the components of the branch divisor of f are ample. Then the map f*: π1(Y)→π1(X) is surjective and gives rise to a central extension: [Formula: see text] where K is a finite group. Here we show how the kernel K and the cohomology class c(f) ∈ H2(π1(X), K) of (1) can be computed in terms of the Chern classes of the components of the branch divisor of f and of the eigensheaves of [Formula: see text] under the action of the Galois group. Using this result, for any integer m>0, we construct m varieties X1,…, Xm no two of which are homeomorphic, even though they have the same numerical invariants and they are realized as covers of the same projective variety X with the same Galois group, branch locus and inertia subgroups.

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9

Ksir, Amy E. "Dimensions of Prym varieties." International Journal of Mathematics and Mathematical Sciences 26, no. 2 (2001): 107–16. http://dx.doi.org/10.1155/s016117120101153x.

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Given a tame Galois branched cover of curvesπ:X→Ywith any finite Galois groupGwhose representations are rational, we compute the dimension of the (generalized) Prym varietyPrymρ(X)corresponding to any irreducible representationρofG. This formula can be applied to the study of algebraic integrable systems using Lax pairs, in particular systems associated with Seiberg-Witten theory. However, the formula is much more general and its computation and proof are entirely algebraic.

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10

OTSUKI, Hideaki, and Tomio HIRATA. "The Biclique Cover Problem and the Modified Galois Lattice." IEICE Transactions on Information and Systems E98.D, no. 3 (2015): 497–502. http://dx.doi.org/10.1587/transinf.2014fcp0019.

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11

Bhatt, Bhargav, Javier Carvajal-Rojas, Patrick Graf, Karl Schwede, and Kevin Tucker. "Étale Fundamental Groups of Strongly $\boldsymbol{F}$-Regular Schemes." International Mathematics Research Notices 2019, no. 14 (October 30, 2017): 4325–39. http://dx.doi.org/10.1093/imrn/rnx253.

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Abstract We prove that a strongly $F$-regular scheme $X$ admits a finite, generically Galois, and étale-in-codimension-one cover $\tilde X \to X$ such that the étale fundamental groups of $\tilde X$ and $\tilde X_{{\mathrm{reg}}}$ agree. Equivalently, every finite étale cover of $\tilde X_{{\mathrm{reg}}}$ extends to a finite étale cover of $\tilde X$. This is analogous to a result for complex klt varieties by Greb, Kebekus, and Peternell.

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12

Amram, Meirav, David Goldberg, Mina Teicher, and Uzi Vishne. "The fundamental group of a Galois cover of ℂℙ1×T." Algebraic & Geometric Topology 2, no. 1 (May 25, 2002): 403–32. http://dx.doi.org/10.2140/agt.2002.2.403.

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13

Anderson, David E. "A cusp singularity with no Galois cover by a complete intersection." Proceedings of the American Mathematical Society 132, no. 9 (April 8, 2004): 2517–27. http://dx.doi.org/10.1090/s0002-9939-04-07302-2.

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14

Hwangk, Yoon Sung, and Bill Jacob. "Brauer Group Analogues of Results Relating the Witt Ring to Valuations and Galois Theory." Canadian Journal of Mathematics 47, no. 3 (June 1, 1995): 527–43. http://dx.doi.org/10.4153/cjm-1995-029-4.

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AbstractLet F be a field of characteristic different from p containing a primitive p-th root of unity. This paper studies the cup product pairing Hl(F, p) x Hl(F, p) → H2(F, p) and its relationship to valuation theory and Galois theory. Sufficient conditions on the pairing which guarantee the existence of a valuation on the field are described. In the non p-adic case these results provide a converse to the well-known structure theory in this situation. In the p-adic case, the pairing is described using the notion of "relative rigidity". These results are analogues of results in quadratic form theory developed in the past decade, which cover the special case p = 2. Applications to the maximal pro-p Galois group of F are also described.

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15

Abughazalah, Nabilah, and Majid Khan. "An Efficient Information Hiding Mechanism Based on Confusion Component over Local Ring and Moore-Penrose Pseudo Inverse." WSEAS TRANSACTIONS ON MATHEMATICS 20 (March 2, 2021): 24–36. http://dx.doi.org/10.37394/23206.2021.20.3.

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The basic requirement by adding confusion is to ensure the confidentiality of the secret information. In the present article, we have suggested new methodology for the construction of nonlinear confusion component. This confusion component is used for enciphering the secret information and hiding it in a cover medium by proposed scheme. The proposed scheme is based on ring structure instead of Galois field mechanism. To provide multi-layer security, secret information is first encrypted by using confusion component and then utilized three different substitution boxes (S-boxes) to hide into the cover medium

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16

Corvaja, Pietro, Julian Lawrence Demeio, Ariyan Javanpeykar, Davide Lombardo, and Umberto Zannier. "On the distribution of rational points on ramified covers of abelian varieties." Compositio Mathematica 158, no. 11 (November 2022): 2109–55. http://dx.doi.org/10.1112/s0010437x22007746.

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We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $\pi : X \to A$ , where $A$ is an abelian variety over $k$ with a dense set of $k$ -rational points, we prove that there is a finite-index coset $C \subset A(k)$ such that $\pi (X(k))$ is disjoint from $C$ . Our results do not seem to be in the range of other methods available at present; they confirm predictions coming from Lang's conjectures on rational points, and also go in the direction of an issue raised by Serre regarding possible applications to the inverse Galois problem. Finally, the conclusions of our work may be seen as a sharp version of Hilbert's irreducibility theorem for abelian varieties.

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17

Ford, Timothy J. "The group of units on an affine variety." Journal of Algebra and Its Applications 13, no. 08 (June 24, 2014): 1450065. http://dx.doi.org/10.1142/s0219498814500650.

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The object of study is the group of units 𝒪*(X) in the coordinate ring of a normal affine variety X over an algebraically closed field k. Methods of Galois cohomology are applied to those varieties that can be presented as a finite cyclic cover of a rational variety. On a cyclic cover X → 𝔸m of affine m-space over k such that the ramification divisor is irreducible and the degree is prime, it is shown that 𝒪*(X) is equal to k*, the non-zero scalars. The same conclusion holds, if X is a sufficiently general affine hyperelliptic curve. If X has a projective completion such that the divisor at infinity has r components, then sufficient conditions are given for 𝒪*(X)/k* to be isomorphic to ℤ(r-1).

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18

Amram, Meirav, Mina Teicher, and Uzi Vishne. "The Coxeter Quotient of the Fundamental Group of a Galois Cover of 𝕋 × 𝕋." Communications in Algebra 34, no. 1 (January 2006): 89–106. http://dx.doi.org/10.1080/00927870500346024.

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19

Amram, Meirav, Sheng Li Tan, Wan Yuan Xu, and Michael Yoshpe. "Calculating the Fundamental Group of Galois Cover of the (2,3)-embedding of ℂℙ1 × T." Acta Mathematica Sinica, English Series 36, no. 3 (February 15, 2020): 273–91. http://dx.doi.org/10.1007/s10114-020-9220-9.

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20

Castryck, Wouter, Floris Vermeulen, and Yongqiang Zhao. "Scrollar invariants, syzygies and representations of the symmetric group." Journal für die reine und angewandte Mathematik (Crelles Journal) 2023, no. 796 (February 28, 2023): 117–59. http://dx.doi.org/10.1515/crelle-2022-0088.

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Abstract We give an explicit minimal graded free resolution, in terms of representations of the symmetric group S d {S_{d}} , of a Galois-theoretic configuration of d points in 𝐏 d - 2 {\mathbf{P}^{d-2}} that was studied by Bhargava in the context of ring parametrizations. When applied to the geometric generic fiber of a simply branched degree d cover of 𝐏 1 {\mathbf{P}^{1}} by a relatively canonically embedded curve C, our construction gives a new interpretation for the splitting types of the syzygy bundles appearing in its relative minimal resolution. Concretely, our work implies that all these splitting types consist of scrollar invariants of resolvent covers. This vastly generalizes a prior observation due to Casnati, namely that the first syzygy bundle of a degree 4 cover splits according to the scrollar invariants of its cubic resolvent. Our work also shows that the splitting types of the syzygy bundles, together with the multi-set of scrollar invariants, belong to a much larger class of multi-sets of invariants that can be attached to C → 𝐏 1 {C\to\mathbf{P}^{1}} : one for each irreducible representation of S d {S_{d}} , i.e., one for each partition of d.

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21

Rito, Carlos. "Cuspidal quintics and surfaces with , and 5-torsion." LMS Journal of Computation and Mathematics 19, no. 1 (2016): 42–53. http://dx.doi.org/10.1112/s1461157015000315.

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If $S$ is a quintic surface in $\mathbb{P}^{3}$ with singular set 15 3-divisible ordinary cusps, then there is a Galois triple cover ${\it\phi}:X\rightarrow S$ branched only at the cusps such that $p_{g}(X)=4$, $q(X)=0$, $K_{X}^{2}=15$ and ${\it\phi}$ is the canonical map of $X$. We use computer algebra to search for such quintics having a free action of $\mathbb{Z}_{5}$, so that $X/\mathbb{Z}_{5}$ is a smooth minimal surface of general type with $p_{g}=0$ and $K^{2}=3$. We find two different quintics, one of which is the van der Geer–Zagier quintic; the other is new.We also construct a quintic threefold passing through the 15 singular lines of the Igusa quartic, with 15 cuspidal lines there. By taking tangent hyperplane sections, we compute quintic surfaces with singular sets $17\mathsf{A}_{2}$, $16\mathsf{A}_{2}$, $15\mathsf{A}_{2}+\mathsf{A}_{3}$ and $15\mathsf{A}_{2}+\mathsf{D}_{4}$.

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22

Siddiqui, Nasir, Fahim Yousaf, Fiza Murtaza, Muhammad Ehatisham-ul-Haq, M. Usman Ashraf, Ahmed M. Alghamdi, and Ahmed S. Alfakeeh. "A highly nonlinear substitution-box (S-box) design using action of modular group on a projective line over a finite field." PLOS ONE 15, no. 11 (November 12, 2020): e0241890. http://dx.doi.org/10.1371/journal.pone.0241890.

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Cryptography is commonly used to secure communication and data transmission over insecure networks through the use of cryptosystems. A cryptosystem is a set of cryptographic algorithms offering security facilities for maintaining more cover-ups. A substitution-box (S-box) is the lone component in a cryptosystem that gives rise to a nonlinear mapping between inputs and outputs, thus providing confusion in data. An S-box that possesses high nonlinearity and low linear and differential probability is considered cryptographically secure. In this study, a new technique is presented to construct cryptographically strong 8×8 S-boxes by applying an adjacency matrix on the Galois field GF(28). The adjacency matrix is obtained corresponding to the coset diagram for the action of modular group PSL(2,Z) on a projective line PL(F7) over a finite field F7. The strength of the proposed S-boxes is examined by common S-box tests, which validate their cryptographic strength. Moreover, we use the majority logic criterion to establish an image encryption application for the proposed S-boxes. The encryption results reveal the robustness and effectiveness of the proposed S-box design in image encryption applications.

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23

Iovita, Adrian, Jackson S. Morrow, and Alexandru Zaharescu. "On p-adic uniformization of abelian varieties with good reduction." Compositio Mathematica 158, no. 7 (July 2022): 1449–76. http://dx.doi.org/10.1112/s0010437x22007643.

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Let $p$ be a rational prime, let $F$ denote a finite, unramified extension of ${{\mathbb {Q}}}_p$ , let $K$ be the maximal unramified extension of ${{\mathbb {Q}}}_p$ , ${{\overline {K}}}$ some fixed algebraic closure of $K$ , and ${{\mathbb {C}}}_p$ be the completion of ${{\overline {K}}}$ . Let $G_F$ be the absolute Galois group of $F$ . Let $A$ be an abelian variety defined over $F$ , with good reduction. Classically, the Fontaine integral was seen as a Hodge–Tate comparison morphism, i.e. as a map $\varphi _{A} \otimes 1_{{{\mathbb {C}}}_p}\colon T_p(A)\otimes _{{{\mathbb {Z}}}_p}{{\mathbb {C}}}_p\to \operatorname {Lie}(A)(F)\otimes _F{{\mathbb {C}}}_p(1)$ , and as such it is surjective and has a large kernel. This paper starts with the observation that if we do not tensor $T_p(A)$ with ${{\mathbb {C}}}_p$ , then the Fontaine integral is often injective. In particular, it is proved that if $T_p(A)^{G_K} = 0$ , then $\varphi _A$ is injective. As an application, we extend the Fontaine integral to a perfectoid like universal cover of $A$ and show that if $T_p(A)^{G_K} = 0$ , then $A(\overline {K})$ has a type of $p$ -adic uniformization, which resembles the classical complex uniformization.

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24

BISWAS, INDRANIL. "PRINCIPAL BUNDLES ON RATIONALLY CONNECTED FIBRATIONS OVER ABELIAN VARIETIES." International Journal of Mathematics 20, no. 02 (February 2009): 167–88. http://dx.doi.org/10.1142/s0129167x0900525x.

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Let f : M → A be a smooth surjective algebraic morphism, where M is a connected complex projective manifold and A a complex abelian variety, such that all the fibers of f are rationally connected. We show that an algebraic principal G-bundle EG over M admits a flat holomorphic connection if EG admits a holomorphic connection; here G is any connected reductive linear algebraic group defined over ℂ. We also show that EG admits a holomorphic connection if and only if any of the following three statements holds. (1) The principal G-bundle EG is semistable, c2( ad (EG)) = 0, and all the line bundles associated to EG for the characters of G have vanishing rational first Chern class. (2) There is an algebraic principal G-bundle E'G on A such that f*E'G = EG, and all the translations of E'G by elements of A are isomorphic to E'G itself. (3) There is a finite étale Galois cover [Formula: see text] and a reduction of structure group [Formula: see text] to a Borel subgroup B ⊂ G such that all the line bundles associated to ÊB for the characters of B have vanishing rational first Chern class. In particular, the above three statements are equivalent.

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25

van Bommel, Raymond. "Inverse Galois problem for ordinary curves." International Journal of Number Theory 14, no. 05 (May 28, 2018): 1305–15. http://dx.doi.org/10.1142/s1793042118500811.

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We consider the inverse Galois problem over function fields of positive characteristic [Formula: see text], for example, over the projective line. We describe a method to construct certain Galois covers of the projective line and other curves, which are ordinary in the sense that their Jacobian has maximal [Formula: see text]-torsion. We do this by constructing Galois covers of ordinary semi-stable curves, and then deforming them into smooth Galois covers.

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26

TOMAŠIĆ, IVAN. "TWISTED GALOIS STRATIFICATION." Nagoya Mathematical Journal 222, no. 1 (May 13, 2016): 1–60. http://dx.doi.org/10.1017/nmj.2016.9.

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We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic–geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulas associated with finite Galois covers of difference schemes.

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27

Amram, Meirav, Cheng Gong, Sheng-Li Tan, Mina Teicher, and Wan-Yuan Xu. "The fundamental groups of Galois covers of planar Zappatic deformations of type Ek." International Journal of Algebra and Computation 29, no. 06 (August 26, 2019): 905–25. http://dx.doi.org/10.1142/s0218196719500358.

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In this paper, we investigate the fundamental groups of Galois covers of planar Zappatic deformations of type [Formula: see text]. Using Moishezon–Teicher’s algorithm, we prove that the Galois covers of the generic fiber of planar Zappatic deformations of type [Formula: see text] [Formula: see text] are simply-connected; we also compute their Chern numbers.

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Malle, Gunter, and David P. Roberts. "Number Fields with Discriminant ±2a3b and Galois Group An or Sn." LMS Journal of Computation and Mathematics 8 (2005): 80–101. http://dx.doi.org/10.1112/s1461157000000905.

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AbstractThe authors present three-point and four-point covers having bad reduction at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 12, 18, 28, and 33. By specializing these covers, they obtain number fields ramified at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 11, 12, 17, 18, 25, 28, 30, and 33.

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29

Xiao, Gang. "Galois covers between $K3$ surfaces." Annales de l’institut Fourier 46, no. 1 (1996): 73–88. http://dx.doi.org/10.5802/aif.1507.

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30

Harbater, David. "Galois Covers of an Arithmetic Surface." American Journal of Mathematics 110, no. 5 (October 1988): 849. http://dx.doi.org/10.2307/2374696.

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31

TONINI, FABIO. "RAMIFIED GALOIS COVERS VIA MONOIDAL FUNCTORS." Transformation Groups 22, no. 3 (June 23, 2016): 845–68. http://dx.doi.org/10.1007/s00031-016-9395-4.

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32

Moishezon, B., A. Robb, and M. Teicher. "On Galois covers of Hirzebruch surfaces." Mathematische Annalen 305, no. 1 (May 1996): 493–539. http://dx.doi.org/10.1007/bf01444235.

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33

Crespo, Teresa, and Zbigniew Hajto. "Differential Galois realization of double covers." Annales de l’institut Fourier 52, no. 4 (2002): 1017–25. http://dx.doi.org/10.5802/aif.1908.

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34

Pappas, Georgios. "Galois module structure of unramified covers." Mathematische Annalen 341, no. 1 (November 20, 2007): 71–97. http://dx.doi.org/10.1007/s00208-007-0183-2.

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35

AMRAM, MEIRAV, RUTH LAWRENCE, and UZI VISHNE. "ARTIN COVERS OF THE BRAID GROUPS." Journal of Knot Theory and Its Ramifications 21, no. 07 (April 7, 2012): 1250061. http://dx.doi.org/10.1142/s0218216512500617.

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Computation of fundamental groups of Galois covers recently led to the construction and analysis of Coxeter covers of the symmetric groups [L. H. Rowen, M. Teicher and U. Vishne, Coxeter covers of the symmetric groups, J. Group Theory8 (2005) 139–169]. In this paper we consider analog covers of Artin's braid groups, and completely describe the induced geometric extensions of the braid group.

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36

Poonen, Bjorn. "Unramified covers of Galois covers of low genus curves." Mathematical Research Letters 12, no. 4 (2005): 475–81. http://dx.doi.org/10.4310/mrl.2005.v12.n4.a3.

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37

Gekeler, Ernst-Ulrich. "Towers of GL($r$)-type of modular curves." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 754 (September 1, 2019): 87–141. http://dx.doi.org/10.1515/crelle-2017-0012.

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Abstract We construct Galois covers {X^{r,k}(N)} over {{\mathbb{P}}^{1}/{\mathbb{F}}_{q}(T)} with Galois groups close to {{\rm GL}(r,{\mathbb{F}}_{q}[T]/(N))} ( {r\geq 3} ) and rationality and ramification properties similar to those of classical modular curves {X(N)} over {{\mathbb{P}}^{1}/{\mathbb{Q}}} . As application we find plenty of good towers (with \limsup{\frac{\text{number~{}of~{}rational~{}points}}{{\rm genus}}>0} ) of curves over the field {{\mathbb{F}}_{q^{r}}} with {q^{r}} elements.

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38

Amram, Meirav, and David Goldberg. "Higher degree Galois covers of ℂℙ1×T." Algebraic & Geometric Topology 4, no. 2 (October 7, 2004): 841–59. http://dx.doi.org/10.2140/agt.2004.4.841.

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Eberhart, Ryan, and Hilaf Hasson. "Arithmetic descent of specializations of Galois covers." Functiones et Approximatio Commentarii Mathematici 56, no. 2 (June 2017): 259–70. http://dx.doi.org/10.7169/facm/1613.

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Obus, Andrew. "Good reduction of three-point Galois covers." Algebraic Geometry 4, no. 2 (March 15, 2017): 247–62. http://dx.doi.org/10.14231/ag-2017-013.

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Gallego, Francisco Javier, and Bangere P. Purnaprajna. "Classification of quadruple Galois canonical covers I." Transactions of the American Mathematical Society 360, no. 10 (May 28, 2008): 5489–507. http://dx.doi.org/10.1090/s0002-9947-08-04587-x.

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Eberhart, Ryan. "Galois branched covers with fixed ramification locus." Journal of Pure and Applied Algebra 219, no. 5 (May 2015): 1592–603. http://dx.doi.org/10.1016/j.jpaa.2014.06.017.

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Dèbes, Pierre, and Nour Ghazi. "Galois Covers and the Hilbert-Grunwald Property." Annales de l’institut Fourier 62, no. 3 (2012): 989–1013. http://dx.doi.org/10.5802/aif.2714.

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Gallego, Francisco Javier, and Bangere P. Purnaprajna. "Classification of quadruple Galois canonical covers, II." Journal of Algebra 312, no. 2 (June 2007): 798–828. http://dx.doi.org/10.1016/j.jalgebra.2006.11.011.

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Pop, Florian. "�tale Galois covers of affine smooth curves." Inventiones Mathematicae 120, no. 1 (December 1995): 555–78. http://dx.doi.org/10.1007/bf01241142.

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TAKAHASHI, Nobuyoshi. "QUANDLES ASSOCIATED TO GALOIS COVERS OF ARITHMETIC SCHEMES." Kyushu Journal of Mathematics 73, no. 1 (2019): 145–64. http://dx.doi.org/10.2206/kyushujm.73.145.

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Bartolo, Enrique Artal, José Ignacio Cogolludo, and Hiro-o. Tokunaga. "Nodal degenerations of plane curves and galois covers." Geometriae Dedicata 121, no. 1 (September 22, 2006): 129–42. http://dx.doi.org/10.1007/s10711-006-9094-8.

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Carocca, Angel, and Martha Romero Rojas. "On Galois group of factorized covers of curves." Revista Matemática Iberoamericana 34, no. 4 (December 6, 2018): 1853–66. http://dx.doi.org/10.4171/rmi/1046.

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Cadoret, Anna. "Counting real Galois covers of the projective line." Pacific Journal of Mathematics 219, no. 1 (March 1, 2005): 53–81. http://dx.doi.org/10.2140/pjm.2005.219.53.

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González-Diez, Gabino. "Galois action on universal covers of Kodaira fibrations." Duke Mathematical Journal 169, no. 7 (May 2020): 1281–303. http://dx.doi.org/10.1215/00127094-2019-0078.

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