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Journal articles on the topic 'Image de Galois'

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Published: 4 June 2021

Last updated: 4 February 2022

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1

TOMAŠIĆ, IVAN. "TWISTED GALOIS STRATIFICATION." Nagoya Mathematical Journal 222, no. 1 (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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2

Greenberg, Ralph. "Galois representations with open image." Annales mathématiques du Québec 40, no. 1 (2016): 83–119. http://dx.doi.org/10.1007/s40316-015-0050-6.

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3

Boston, Nigel, and Rafe Jones. "The Image of an Arboreal Galois Representation." Pure and Applied Mathematics Quarterly 5, no. 1 (2009): 213–25. http://dx.doi.org/10.4310/pamq.2009.v5.n1.a6.

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4

Gekeler, Ernst-Ulrich. "The Galois image of twisted Carlitz modules." Journal of Number Theory 163 (June 2016): 316–30. http://dx.doi.org/10.1016/j.jnt.2015.11.021.

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5

Moon, Hyunsuk, and Yuichiro Taguchi. "Mod $p$ Galois representations of solvable image." Proceedings of the American Mathematical Society 129, no. 9 (2001): 2529–34. http://dx.doi.org/10.1090/s0002-9939-01-05894-4.

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6

Ramakrishna, Ravi. "Constructing Galois Representations with Very Large Image." Canadian Journal of Mathematics 60, no. 1 (2008): 208–21. http://dx.doi.org/10.4153/cjm-2008-009-7.

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AbstractStarting with a 2-dimensional mod p Galois representation, we construct a deformation to a power series ring in infinitely many variables over the p-adics. The image of this representation is full in the sense that it contains SL2 of this power series ring. Furthermore, all Zp specializations of this deformation are potentially semistable at p.

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7

Sharma, Devika. "Locally indecomposable Galois representations with full residual image." International Journal of Number Theory 13, no. 05 (2017): 1191–211. http://dx.doi.org/10.1142/s1793042117500646.

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We consider certain [Formula: see text]-ordinary non-CM Hida families with full residual Galois representation and give mild conditions under which every arithmetic point in these families is locally indecomposable when [Formula: see text]. The proof uses methods from deformation theory and mostly works for any odd prime [Formula: see text], but ultimately relies on the existence of a weight [Formula: see text] form in an auxiliary family which is available only for [Formula: see text]. We end by giving several non-trivial examples of [Formula: see text]-ordinary non-CM locally indecomposable modular forms of small level with full residual Galois representation.

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8

Boston, Nigel, and David T. Ose. "Characteristic p Galois Representations That Arise from Drinfeld Modules." Canadian Mathematical Bulletin 43, no. 3 (2000): 282–93. http://dx.doi.org/10.4153/cmb-2000-035-5.

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AbstractWe examine which representations of the absolute Galois group of a field of finite characteristic with image over a finite field of the same characteristic may be constructed by the Galois group’s action on the division points of an appropriate Drinfeld module.

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9

Koch, Alan. "Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures." Proceedings of the American Mathematical Society, Series B 8, no. 16 (2021): 189–203. http://dx.doi.org/10.1090/bproc/87.

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Let G G be a finite nonabelian group, and let ψ : G → G \psi :G\to G be a homomorphism with abelian image. We show how ψ \psi gives rise to two Hopf-Galois structures on a Galois extension L / K L/K with Galois group (isomorphic to) G G ; one of these structures generalizes the construction given by a “fixed point free abelian endomorphism” introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equation as well as a pair of Hopf-Galois structures on a (potentially) different finite Galois extension.

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10

Conti, Andrea. "Galois level and congruence ideal for -adic families of finite slope Siegel modular forms." Compositio Mathematica 155, no. 4 (2019): 776–831. http://dx.doi.org/10.1112/s0010437x19007048.

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We consider families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call the Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zero locus defined by the Galois level of the family admits an automorphic description: it is the locus of points that arise from overconvergent eigenforms for $\operatorname{GL}_{2}$, via a $p$-adic Langlands lift attached to the symmetric cube representation. Our proof goes via the comparison of the Galois level with a ‘fortuitous’ congruence ideal. Some of the $p$-adic lifts are interpolated by a morphism of rigid analytic spaces from an eigencurve for $\operatorname{GL}_{2}$ to an eigenvariety for $\operatorname{GSp}_{4}$, while the remainder appear as isolated points on the eigenvariety.

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11

Reverter, Amadeu, and Núria Vila. "Images of mod p Galois Representations Associated to Elliptic Curves." Canadian Mathematical Bulletin 44, no. 3 (2001): 313–22. http://dx.doi.org/10.4153/cmb-2001-031-1.

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AbstractWe give an explicit recipe for the determination of the images associated to the Galois action on p-torsion points of elliptic curves. We present a table listing the image for all the elliptic curves defined over without complex multiplication with conductor less than 200 and for each prime number p.

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12

Katz, Nicholas. "A note on Galois representations with big image." L’Enseignement Mathématique 65, no. 3 (2020): 271–301. http://dx.doi.org/10.4171/lem/65-3/4-1.

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13

Bisatt, Matthew. "Frobenius elements in Galois representations with SL image." Journal of Number Theory 188 (July 2018): 165–71. http://dx.doi.org/10.1016/j.jnt.2018.01.010.

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14

Fischman, Ami. "On the image of $\Lambda$-adic Galois representations." Annales de l’institut Fourier 52, no. 2 (2002): 351–78. http://dx.doi.org/10.5802/aif.1890.

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15

Hida, Haruzo. "Image of Λ-adic Galois representations modulo p". Inventiones mathematicae 194, № 1 (2012): 1–40. http://dx.doi.org/10.1007/s00222-012-0439-7.

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16

LOEFFLER, DAVID. "IMAGES OF ADELIC GALOIS REPRESENTATIONS FOR MODULAR FORMS." Glasgow Mathematical Journal 59, no. 1 (2016): 11–25. http://dx.doi.org/10.1017/s0017089516000367.

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AbstractWe show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic subgroup of GL2 (defined by F. Momose). We also show a similar result for the adelic Galois representation attached to a finite set of modular forms.

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17

PANDE, AFTAB. "DEFORMATIONS OF GALOIS REPRESENTATIONS AND THE THEOREMS OF SATO–TATE AND LANG–TROTTER." International Journal of Number Theory 07, no. 08 (2011): 2065–79. http://dx.doi.org/10.1142/s1793042111004939.

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We construct infinitely ramified Galois representations ρ such that the al(ρ)'s have distributions in contrast to the statements of Sato–Tate, Lang–Trotter and others. Using similar methods we deform a residual Galois representation for number fields and obtain an infinitely ramified representation with very large image, generalizing a result of Ramakrishna.

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18

Wang, Nan, Guixin Di, Xiaolan Lv, et al. "Galois Field-Based Image Encryption for Remote Transmission of Tumor Ultrasound Images." IEEE Access 7 (2019): 49945–50. http://dx.doi.org/10.1109/access.2019.2910563.

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19

Coppola, Nirvana. "Wild Galois representations: Elliptic curves over a 2-adic field with non-abelian inertia action." International Journal of Number Theory 16, no. 06 (2020): 1199–208. http://dx.doi.org/10.1142/s179304212050061x.

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In this paper, we present a description of the [Formula: see text]-adic Galois representation attached to an elliptic curve defined over a [Formula: see text]-adic field [Formula: see text], in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely [Formula: see text] and [Formula: see text], and in each case, we need to distinguish whether the inertia degree of [Formula: see text] over [Formula: see text] is even or odd. The results presented here are being implemented in an algorithm to compute explicitly the Galois representation in these four cases.

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20

Arias-de-Reyna, Sara, Luis Dieulefait, and Gabor Wiese. "Compatible systems of symplectic Galois representations and the inverse Galois problem II: Transvections and huge image." Pacific Journal of Mathematics 281, no. 1 (2016): 1–16. http://dx.doi.org/10.2140/pjm.2016.281.1.

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21

Hellmann, Eugen. "On arithmetic families of filtered -modules and crystalline representations." Journal of the Institute of Mathematics of Jussieu 12, no. 4 (2012): 677–726. http://dx.doi.org/10.1017/s1474748012000813.

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AbstractWe consider stacks of filtered$\varphi $-modules over rigid analytic spaces and adic spaces. We show that these modules parameterize$p$-adic Galois representations of the absolute Galois group of a$p$-adic field with varying coefficients over an open substack containing all classical points. Further, we study a period morphism (defined by Pappas and Rapoport) from a stack parameterizing integral data, and determine the image of this morphism.

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22

Loisant, Erwan, José Martinez, Hiroshi Ishikawa, and Kaoru Katayama. "Galois' Lattices as a Classification Technique for Image Retrieval." IPSJ Digital Courier 2 (2006): 1–13. http://dx.doi.org/10.2197/ipsjdc.2.1.

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23

Caraiani, Ana, and Bao V. Le Hung. "On the image of complex conjugation in certain Galois representations." Compositio Mathematica 152, no. 7 (2016): 1476–88. http://dx.doi.org/10.1112/s0010437x16007016.

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We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $\text{GL}_{n}$ over a totally real field $F$.

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24

DELBOURGO, DANIEL, and PAUL SMITH. "Kummer theory for big Galois representations." Mathematical Proceedings of the Cambridge Philosophical Society 142, no. 2 (2007): 205–17. http://dx.doi.org/10.1017/s0305004106009868.

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AbstractIn their 1990 paper, Bloch and Kato described the image of the Kummer map on an abelian variety over a local field, as the group of 1-cocycles which trivialise after tensoring by Fontaine's mysterious ring BdR. We prove the analogue of this statement for the universal nearly-ordinary Galois representation. The proof uses a generalisation of the Tate local pairing to representations over affinoid K-algebras.

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25

Cadoret, Anna, та Akio Tamagawa. "Genus of abstract modular curves with level-ℓ structures". Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, № 752 (2019): 25–61. http://dx.doi.org/10.1515/crelle-2016-0057.

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Abstract We prove – in arbitrary characteristic – that the genus of abstract modular curves associated to bounded families of continuous geometrically perfect {\mathbb{F}_{\ell}} -linear representations of étale fundamental groups of curves goes to infinity with {\ell} . This applies to the variation of the Galois image on étale cohomology groups with coefficients in {\mathbb{F}_{\ell}} in 1-dimensional families of smooth proper schemes or, under certain assumptions, to specialization of first Galois cohomology groups.

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26

Loeffler, David, and Sarah Livia Zerbes. "Iwasawa theory and p-adic L-functions over ${\mathbb Z}_{p}^{2}$-extensions." International Journal of Number Theory 10, no. 08 (2014): 2045–95. http://dx.doi.org/10.1142/s1793042114500699.

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We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localization at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.

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27

HUANG, CHIN-PAN, and CHING-CHUNG LI. "SECURE AND PROGRESSIVE IMAGE TRANSMISSION THROUGH SHADOWS GENERATED BY MULTIWAVELET TRANSFORM." International Journal of Wavelets, Multiresolution and Information Processing 06, no. 06 (2008): 907–31. http://dx.doi.org/10.1142/s0219691308002677.

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In this paper, multiwavelet transform is used in the context of image sharing employing the SPIHT-generated bit stream with the modified Shamir's (r, m) threshold scheme over the Galois field. This provides multiwavelet-based embedded shadow images for progressive transmission and reconstruction. Both integer multiwavelet transform and balanced multiwavelet transform are considered in the method. It is secure and fault-tolerant, and gives small shadow size and excellent rate-distortion performance. Experimental results and sample applications are given to demonstrate the characteristics of this method.

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28

Dieulefait, Luis V. "A non-solvable extension of ℚ unramified outside 7". Compositio Mathematica 148, № 3 (2012): 669–74. http://dx.doi.org/10.1112/s0010437x11007408.

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AbstractWe consider a mod 7 Galois representation attached to a genus 2 Siegel cusp form of level 1 and weight 28 and using some of its Fourier coefficients and eigenvalues computed by N. Skoruppa and the classification of maximal subgroups of PGSp(4,p) we show that its image is as large as possible. This gives a realization of PGSp(4,7) as a Galois group over ℚ and the corresponding number field provides a non-solvable extension of ℚ which ramifies only at 7.

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29

MANOHARMAYUM, JAYANTA. "LIFTINGN-DIMENSIONAL GALOIS REPRESENTATIONS TO CHARACTERISTIC ZERO." Glasgow Mathematical Journal 61, no. 1 (2018): 115–50. http://dx.doi.org/10.1017/s0017089518000149.

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AbstractLetFbe a number field, letN≥ 3 be an integer, and letkbe a finite field of characteristic ℓ. We show that ifρ:GF→GLN(k) is a continuous representation with image ofρcontainingSLN(k) then, under moderate conditions at primes dividing ℓ∞, there is a continuous representation ρ:GF→GLN(W(k)) unramified outside finitely many primes withρ~ρ mod ℓ. Stronger results are presented forρ:Gℚ→GL3(k).

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30

Rajagopala, S., H. N. Upadhyay, J. B. Balaguru Rayappan, and R. Amirtharaj. "Galois Field Proficient Product for Secure Image Encryption on FPGA." Research Journal of Information Technology 6, no. 4 (2014): 308–24. http://dx.doi.org/10.3923/rjit.2014.308.324.

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31

Abrashkin, V. A. "The image of the Galois group for some crystalline representations." Izvestiya: Mathematics 63, no. 1 (1999): 1–36. http://dx.doi.org/10.1070/im1999v063n01abeh000226.

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32

Fernández, J., J.-C. Lario, and A. Rio. "Octahedral Galois Representations Arising From Q-Curves of Degree 2." Canadian Journal of Mathematics 54, no. 6 (2002): 1202–28. http://dx.doi.org/10.4153/cjm-2002-046-8.

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AbstractGenerically, one can attach to a Q-curve C octahedral representations ρ: Gal() → GL2() coming from the Galois action on the 3-torsion of those abelian varieties of GL2-type whose building block is C. When C is defined over a quadratic field and has an isogeny of degree 2 to its Galois conjugate, there exist such representations ρ having image into GL2(F9). Going the other way, we can ask which mod 3 octahedral representations ρ of Gal() arise from Q-curves in the above sense. We characterize those arising from quadratic Q-curves of degree 2. The approach makes use of Galois embedding techniques in GL2(F9), and the characterization can be given in terms of a quartic polynomial defining the S4-extension of Q corresponding to the projective representation .

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33

Hida, Haruzo. "Big Galois representations and -adic -functions." Compositio Mathematica 151, no. 4 (2014): 603–64. http://dx.doi.org/10.1112/s0010437x14007684.

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Let$p\geqslant 5$be a prime. If an irreducible component of the spectrum of the ‘big’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its Galois representation contains, up to finite error, a principal congruence subgroup${\rm\Gamma}(L)$of$\text{SL}_{2}(\mathbb{Z}_{p}[[T]])$for a principal ideal$(L)\neq 0$of$\mathbb{Z}_{p}[[T]]$for the canonical ‘weight’ variable$t=1+T$. If$L\notin {\rm\Lambda}^{\times }$, the power series$L$is proven to be a factor of the Kubota–Leopoldt$p$-adic$L$-function or of the square of the anticyclotomic Katz$p$-adic$L$-function or a power of$(t^{p^{m}}-1)$.

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34

Liang, Ke, and Jeremy Rouse. "Density of odd order reductions for elliptic curves with a rational point of order 2." International Journal of Number Theory 15, no. 08 (2019): 1547–63. http://dx.doi.org/10.1142/s1793042119500891.

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Suppose that [Formula: see text] is an elliptic curve with a rational point [Formula: see text] of order [Formula: see text] and [Formula: see text] is a point of infinite order. We consider the problem of determining the density of primes [Formula: see text] for which [Formula: see text] has odd order. This density is determined by the image of the arboreal Galois representation [Formula: see text]. Assuming that [Formula: see text] is primitive (that is neither [Formula: see text] nor [Formula: see text] is twice a point over [Formula: see text]) and that the image of the ordinary [Formula: see text]-adic Galois representation is as large as possible (subject to [Formula: see text] having a rational point of order [Formula: see text]), we determine that there are [Formula: see text] possibilities for the image of [Formula: see text]. As a consequence, the density of primes [Formula: see text] for which the order of [Formula: see text] is odd is between 1/14 and [Formula: see text].

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35

Huang, Lilian, Shiming Wang, Jianhong Xiang, and Yi Sun. "Chaotic Color Image Encryption Scheme Using Deoxyribonucleic Acid (DNA) Coding Calculations and Arithmetic over the Galois Field." Mathematical Problems in Engineering 2020 (March 9, 2020): 1–22. http://dx.doi.org/10.1155/2020/3965281.

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This paper proposes a chaotic color image encryption scheme based on DNA-coding calculations and arithmetic over the Galois field. Firstly, three modified one-dimensional (1D) chaotic maps with larger key space and better chaotic characteristics are presented. The experimental results show that their chaotic intervals are not only expanded to 0,15, but their average largest Lyapunov Exponent reaches 10. They are utilized as initial keys. Secondly, DNA coding and calculations are applied in order to add more permutation of the cryptosystem. Ultimately, the numeration over the Galois field ensures the effect for the diffusion of pixels. The simulation analysis shows that the encryption scheme proposed in this paper has good encryption effect, and the numerical results verify that it has higher security than some of the latest cryptosystems.

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36

Kudari, Medha, Shivashankar S., and Prakash S. Hiremath. "Illumination and Rotation Invariant Texture Representation for Face Recognition." International Journal of Computer Vision and Image Processing 10, no. 2 (2020): 58–69. http://dx.doi.org/10.4018/ijcvip.2020040105.

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This article presents a novel approach for illumination and rotation invariant texture representation for face recognition. A gradient transformation is used as illumination invariance property and a Galois Field for the rotation invariance property. The normalized cumulative histogram bin values of the Gradient Galois Field transformed image represent the illumination and rotation invariant texture features. These features are further used as face descriptors. Experimentations are performed on FERET and extended Cohn Kanade databases. The results show that the proposed method is better as compared to Rotation Invariant Local Binary Pattern, Log-polar transform and Sorted Local Gradient Pattern and is illumination and rotation invariant.

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37

Bahjat, Hala, and May A. Salih. "Speed Image Encryption Scheme using Dynamic Galois Field GF(P) Matrices." International Journal of Computer Applications 89, no. 7 (2014): 7–12. http://dx.doi.org/10.5120/15513-4218.

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38

Gardeyn, Francis. "Openness of the Galois image for τ-modules of dimension 1". Journal of Number Theory 102, № 2 (2003): 306–38. http://dx.doi.org/10.1016/s0022-314x(03)00106-9.

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39

Tezuka, M., and N. Yagita. "The image of the map from group cohomology to Galois cohomology." Transactions of the American Mathematical Society 363, no. 08 (2011): 4475. http://dx.doi.org/10.1090/s0002-9947-2011-05418-8.

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40

Arias-De-Reyna, Sara, and Christian Kappen. "Abelian varieties over number fields, tame ramification and big Galois image." Mathematical Research Letters 20, no. 1 (2013): 1–17. http://dx.doi.org/10.4310/mrl.2013.v20.n1.a1.

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41

CHEN, IMIN, and YOONJIN LEE. "EXPLICIT SURJECTIVITY RESULTS FOR DRINFELD MODULES OF RANK 2." Nagoya Mathematical Journal 234 (July 27, 2017): 17–45. http://dx.doi.org/10.1017/nmj.2017.26.

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Let $K=\mathbb{F}_{q}(T)$ and $A=\mathbb{F}_{q}[T]$. Suppose that $\unicode[STIX]{x1D719}$ is a Drinfeld $A$-module of rank $2$ over $K$ which does not have complex multiplication. We obtain an explicit upper bound (dependent on $\unicode[STIX]{x1D719}$) on the degree of primes ${\wp}$ of $K$ such that the image of the Galois representation on the ${\wp}$-torsion points of $\unicode[STIX]{x1D719}$ is not surjective, in the case of $q$ odd. Our results are a Drinfeld module analogue of Serre’s explicit large image results for the Galois representations on $p$-torsion points of elliptic curves (Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331; Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Etudes Sci. Publ. Math. 54 (1981), 323–401.) and are unconditional because the generalized Riemann hypothesis for function fields holds. An explicit isogeny theorem for Drinfeld $A$-modules of rank $2$ over $K$ is also proven.

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42

Thorne, Jack. "On the automorphy of l-adic Galois representations with small residual image With an appendix by Robert Guralnick, Florian Herzig, Richard Taylor and Jack Thorne." Journal of the Institute of Mathematics of Jussieu 11, no. 4 (2012): 855–920. http://dx.doi.org/10.1017/s1474748012000023.

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AbstractWe prove new automorphy lifting theorems for essentially conjugate self-dual Galois representations into GLn. Existing theorems require that the residual representation have ‘big’ image, in a certain technical sense. Our theorems are based on a strengthening of the Taylor–Wiles method which allows one to weaken this hypothesis.

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43

Weston, Tom. "Power Residues of Fourier Coefficients of Modular Forms." Canadian Journal of Mathematics 57, no. 5 (2005): 1102–20. http://dx.doi.org/10.4153/cjm-2005-042-5.

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AbstractLet ρ: GQ → GLn(Qℓ) be a motivic ℓ-adic Galois representation. For fixed m > 1 we initiate an investigation of the density of the set of primes p such that the trace of the image of an arithmetic Frobenius at p under ρ is an m-th power residue modulo p. Based on numerical investigations with modular forms we conjecture (with Ramakrishna) that this density equals 1/m whenever the image of ρ is open. We further conjecture that for such ρ the set of these primes p is independent of any set defined by Cebatorev-style Galois-theoretic conditions (in an appropriate sense). We then compute these densities for certain m in the complementary case of modular forms of CM-type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem(which does apply in the CM case) and reciprocity laws applied to Hecke characters. We also discuss a potential application (suggested by Ramakrishna) to computing inertial degrees at p in abelian extensions of imaginary quadratic fields unramified away from p.

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44

Ishitsuka, Yasuhiro, Tetsushi Ito, and Tatsuya Ohshita. "Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic." International Journal of Number Theory 16, no. 04 (2019): 881–905. http://dx.doi.org/10.1142/s1793042120500451.

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We use explicit methods to study the [Formula: see text]-torsion points on the Jacobian variety of the Fermat quartic. With the aid of computer algebra systems, we explicitly give a basis of the group of [Formula: see text]-torsion points. We calculate the Galois action, and show that the image of the mod [Formula: see text] Galois representation is isomorphic to the dihedral group of order [Formula: see text]. As applications, we calculate the Mordell–Weil group of the Jacobian variety of the Fermat quartic over each subfield of the [Formula: see text]th cyclotomic field. We determine all of the points on the Fermat quartic defined over quadratic extensions of the [Formula: see text]th cyclotomic field. Thus, we complete Faddeev’s work in 1960.

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., Pooja V., Ganavi M. ., Sowmya D. ., and Hiriyanna G. S. . "A COLOR IMAGE STEGANOGRAPHY USING 7TH BIT PIXEL INDICATOR AND GALOIS FIELDARITHMETIC." International Journal of Engineering Applied Sciences and Technology 4, no. 3 (2019): 353–58. http://dx.doi.org/10.33564/ijeast.2019.v04i03.058.

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Greenberg, Ralph. "The image of Galois representations attached to elliptic curves with an isogeny." American Journal of Mathematics 134, no. 5 (2012): 1167–96. http://dx.doi.org/10.1353/ajm.2012.0040.

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Tayou, Salim. "Images de représentations galoisiennes associées à certaines formes modulaires de Siegel de genre 2." International Journal of Number Theory 13, no. 05 (2017): 1129–44. http://dx.doi.org/10.1142/s1793042117500610.

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We study the image of the [Formula: see text]-adic Galois representations associated to the four vector valued Siegel modular forms appearing in the work of Chenevier and Lannes [3]. These representations are symplectic of dimension 4. Following methods used by Dieulefait in [4], we determine the primes [Formula: see text] for which these representations are absolutely irreducible. In addition, we show that their image is “full” for all primes [Formula: see text] such that the associated residual representation is absolutely irreducible, except in two cases.

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Cadoret, Anna, and Ben Moonen. "INTEGRAL AND ADELIC ASPECTS OF THE MUMFORD–TATE CONJECTURE." Journal of the Institute of Mathematics of Jussieu 19, no. 3 (2018): 869–90. http://dx.doi.org/10.1017/s1474748018000233.

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Let $Y$ be an abelian variety over a subfield $k\subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford–Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre are true for $Y$. In particular, if a certain Hodge-maximality condition is satisfied, we obtain an adelic open image theorem for the Galois representation on the (full) Tate module of $Y$. We also obtain an (unconditional) adelic open image theorem for K3 surfaces. These results are special cases of a more general statement for the image of a natural adelic representation of the fundamental group of a Shimura variety.

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Pink, Richard, and Egon Rütsche. "Image of the group ring of the Galois representation associated to Drinfeld modules." Journal of Number Theory 129, no. 4 (2009): 866–81. http://dx.doi.org/10.1016/j.jnt.2008.12.003.

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Stalder, Nicolas. "The semisimplicity conjecture for A-motives." Compositio Mathematica 146, no. 3 (2010): 561–98. http://dx.doi.org/10.1112/s0010437x09004448.

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AbstractWe prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.

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