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Relevant bibliographies by topics / Bloch-Kato conjecture

Academic literature on the topic 'Bloch-Kato conjecture'

Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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Journal articles on the topic "Bloch-Kato conjecture"

1

Huber, Annette, and Guido Kings. "A cohomological Tamagawa number formula." Nagoya Mathematical Journal 202 (June 2011): 45–75. http://dx.doi.org/10.1215/00277630-1260441.

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AbstractFor smooth linear group schemes over ℤ, we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori, the cohomological and the motivic Tamagawa measures coincide, which proves again the Bloch-Kato conjecture for motives associated to tori.

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Huber, Annette, and Guido Kings. "A cohomological Tamagawa number formula." Nagoya Mathematical Journal 202 (June 2011): 45–75. http://dx.doi.org/10.1017/s0027763000010242.

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AbstractFor smooth linear group schemes over ℤ, we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori, the cohomological and the motivic Tamagawa measures coincide, which proves again the Bloch-Kato conjecture for motives associated to tori.

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Swinnerton-Dyer, Sir Peter. "Diagonal hypersurfaces and the Bloch-Kato conjecture, I." Journal of the London Mathematical Society 90, no. 3 (October 20, 2014): 845–60. http://dx.doi.org/10.1112/jlms/jdu055.

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Guo, Li. "On the Bloch–Kato Conjecture for HeckeL-Functions." Journal of Number Theory 57, no. 2 (April 1996): 340–65. http://dx.doi.org/10.1006/jnth.1996.0053.

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Asok, Aravind. "Rationality problems and conjectures of Milnor and Bloch–Kato." Compositio Mathematica 149, no. 8 (June 3, 2013): 1312–26. http://dx.doi.org/10.1112/s0010437x13007021.

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AbstractWe show how the techniques of Voevodsky’s proof of the Milnor conjecture and the Voevodsky–Rost proof of its generalization the Bloch–Kato conjecture can be used to study counterexamples to the classical Lüroth problem. By generalizing a method due to Peyre, we produce for any prime number $\ell $ and any integer $n\geq 2$, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree $n$ unramified étale cohomology class with $\ell $-torsion coefficients. When $\ell = 2$, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified étale cohomology class of lower degree.

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Tamiozzo, Matteo. "On the Bloch–Kato conjecture for Hilbert modular forms." Mathematische Zeitschrift 299, no. 1-2 (January 30, 2021): 427–58. http://dx.doi.org/10.1007/s00209-020-02689-0.

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AbstractThe aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

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Voevodsky, Vladimir. "Motives over simplicial schemes." Journal of K-Theory 5, no. 1 (February 2010): 1–38. http://dx.doi.org/10.1017/is010001030jkt107.

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AbstractThis paper was written as a part of [8] and is intended primarily to provide the definitions and results concerning motives over simplicial schemes, which are used in the proof of the Bloch-Kato conjecture.

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Kings, Guido, and Annette Huber. "Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters." Duke Mathematical Journal 119, no. 3 (September 2003): 393–464. http://dx.doi.org/10.1215/s0012-7094-03-11931-6.

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DUMMIGAN, NEIL. "RATIONAL TORSION ON OPTIMAL CURVES." International Journal of Number Theory 01, no. 04 (December 2005): 513–31. http://dx.doi.org/10.1142/s1793042105000340.

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Vatsal has proved recently a result which has consequences for the existence of rational points of odd prime order ℓ on optimal elliptic curves over ℚ. When the conductor N is squarefree, ℓ ∤ N and the local root number wp= -1 for at least one prime p | N, we offer a somewhat different proof, starting from an explicit cuspidal divisor on X0(N). We also prove some results linking the vanishing of L(E,1) with the divisibility by ℓ of the modular parametrization degree, fitting well with the Bloch–Kato conjecture for L( Sym2E,2), and with an earlier construction of elements in Shafarevich–Tate groups. Finally (following Faltings and Jordan) we prove an analogue of the result on ℓ-torsion for cuspidal Hecke eigenforms of level one (and higher weight), thereby strengthening some existing evidence for another case of the Bloch–Kato conjecture.

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DUMMIGAN, NEIL. "SYMMETRIC SQUARE L-FUNCTIONS AND SHAFAREVICH–TATE GROUPS, II." International Journal of Number Theory 05, no. 07 (November 2009): 1321–45. http://dx.doi.org/10.1142/s1793042109002699.

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We re-examine some critical values of symmetric square L-functions for cusp forms of level one. We construct some more of the elements of large prime order in Shafarevich–Tate groups, demanded by the Bloch–Kato conjecture. For this, we use the Galois interpretation of Kurokawa-style congruences between vector-valued Siegel modular forms of genus two (cusp forms and Klingen–Eisenstein series), making further use of a construction due to Urban. We must assume that certain 4-dimensional Galois representations are symplectic. Our calculations with Fourier expansions use the Eholzer–Ibukiyama generalization of the Rankin–Cohen brackets. We also construct some elements of global torsion which should, according to the Bloch–Kato conjecture, contribute a factor to the denominator of the rightmost critical value of the standard L-function of the Siegel cusp form. Then we prove, under certain conditions, that the factor does occur.

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Dissertations / Theses on the topic "Bloch-Kato conjecture"

1

Harrison, Michael Corin. "On the conjecture of Bloch-Kato for Grossencharacters over Q(i)." Thesis, University of Cambridge, 1992. https://www.repository.cam.ac.uk/handle/1810/251690.

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Tamiozzo, Matteo [Verfasser], and Massimo [Akademischer Betreuer] Bertolini. "On the Bloch-Kato conjecture for Hilbert modular forms / Matteo Tamiozzo ; Betreuer: Massimo Bertolini." Duisburg, 2019. http://d-nb.info/1191690938/34.

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Lin, Qiang Flach Matthias. "Bloch-Kato conjecture for the adjoint of H1(X0(N)) with integral Hecke algebra /." Diss., Pasadena, Calif. : California Institute of Technology, 2004. http://resolver.caltech.edu/CaltechETD:etd-11182003-084742.

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QUADRELLI, CLAUDIO. "Cohomology of Absolute Galois Groups." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2014. http://hdl.handle.net/10281/56993.

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The main problem this thesis deals with is the characterization of profinite groups which are realizable as absolute Galois groups of fields: this is currently one of the major problems in Galois theory. Usually one reduces the problem to the pro-p case, i.e., one would like to know which pro-p groups occur as maximal pro-p Galois groups, i.e., maximal pro-p quotients of absolute Galois groups. Indeed, pro-p groups are easier to deal with than general profinite groups, yet they carry a lot of information on the whole absolute Galois group. We define a new class of pro-p groups, called Bloch-Kato pro-p group, whose Galois cohomology satisfies the consequences of the Bloch-Kato conjecture. Also we introduce the notion of cyclotomic orientation for a pro-p group. With this approach, we are able to recover new substantial information about the structure of maximal pro-p Galois groups, and in particular on theta-abelian pro-p groups, which represent the "upper bound" of such groups. Also, we study the restricted Lie algebra and the universal envelope induced by the Zassenhaus filtration of a maximal pro-p Galois group, and their relations with Galois cohomology via Koszul duality. Altogether, this thesis provides a rather new approach to maximal pro-p Galois groups, besides new substantial results.

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Chenevier, Gaëtan. "Familles p-adiques de formes automorphes et applications aux conjectures de Bloch-Kato." Paris 7, 2003. http://www.theses.fr/2003PA077027.

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Mundy, Samuel Raymond. "Eisenstein series for G₂ and the symmetric cube Bloch--Kato conjecture." Thesis, 2021. https://doi.org/10.7916/d8-k3ys-vh32.

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The purpose of this thesis is to construct nontrivial elements in the Bloch--Kato Selmer group of the symmetric cube of the Galois representation attached to a cuspidal holomorphic eigenform 𝐹 of level 1. The existence of such elements is predicted by the Bloch--Kato conjecture. This construction is carried out under certain standard conjectures related to Langlands functoriality. The broad method used to construct these elements is the one pioneered by Skinner and Urban in [SU06a] and [SU06b]. The construction has three steps, corresponding to the three chapters of this thesis. The first step is to use parabolic induction to construct a functorial lift of 𝐹 to an automorphic representation π of the exceptional group G₂ and then locate every instance of this functorial lift in the cohomology of G₂. In Eisenstein cohomology, this is done using the decomposition of Franke--Schwermer [FS98]. In cuspidal cohomology, this is done assuming Arthur's conjectures in order to classify certain CAP representations of G₂ which are nearly equivalent to π, and also using the work of Adams--Johnson [AJ87] to describe the Archimedean components of these CAP representations. This step works for 𝐹 of any level, even weight 𝑘 ≥ 4, and trivial nebentypus, as long as the symmetric cube 𝐿-function of 𝐹 vanishes at its central value. This last hypothesis is necessary because only then will the Bloch--Kato conjecture predict the existence of nontrivial elements in the symmetric cube Bloch--Kato Selmer group. Here this hypothesis is used in the case of Eisenstein cohomology to show the holomorphicity of certain Eisenstein series via the Langlands--Shahidi method, and in the case of cuspidal cohomology it is used to ensure that relevant discrete representations classified by Arthur's conjecture are cuspidal and not residual. The second step is to use the knowledge obtained in the first step to 𝓅-adically deform a certain critical 𝓅-stabilization 𝜎π of π in a generically cuspidal family of automorphic representations of G₂. This is done using the machinery of Urban's eigenvariety [Urb11]. This machinery operates on the multiplicities of automorphic representations in certain cohomology groups; in particular, it can relate the location of π in cohomology to the location of 𝜎π in an overconvergent analogue of cohomology and, under favorable circumstances, use this information to 𝓅-adically deform 𝜎π in a generically cuspidal family. We show that these circumstances are indeed favorable when the sign of the symmetric functional equation for 𝐹 is -1 either under certain conditions on the slope of 𝜎π, or in general when 𝐹 has level 1. The third and final step is to, under the assumption of a global Langlands correspondence for cohomological automorphic representations of G₂, carry over to the Galois side the generically cuspidal family of automorphic representations obtained in the second step to obtain a family of Galois representations which factors through G₂ and which specializes to the Galois representation attached to π. We then show this family is generically irreducible and make a Ribet-style construction of a particular lattice in this family. Specializing this lattice at the point corresponding to π gives a three step reducible Galois representation into GL₇, which we show must factor through, not only G₂, but a certain parabolic subgroup of G₂. Using this, we are able to construct the desired element of the symmetric cube Bloch--Kato Selmer group as an extension appearing in this reducible representation. The fact that this representation factors through the aforementioned parabolic subgroup of G₂ puts restrictions on the extension we obtain and guarantees that it lands in the symmetric cube Selmer group and not the Selmer group of 𝐹 itself. This step uses that 𝐹 is level 1 to control ramification at places different from 𝓅, and to ensure that 𝐹 is not CM so as to guarantee that the Galois representation attached to π has three irreducible pieces instead of four.

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Lin, Qiang. "Bloch-Kato Conjecture for the Adjoint of H¹(X₀(N)) with Integral Hecke Algebra." Thesis, 2004. https://thesis.library.caltech.edu/4595/1/BurnsFlachConjectureForIntegralHeckeAlgebra.pdf.

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Let M be a motive that is defined over a number field and admits an action of a finite dimensional semisimple Q-algebra T. David Burns and Matthias Flach formulated a conjecture, which depends on a choice of Z-order T in T, for the leading coefficient of the Taylor expansion at 0 of the T-equivariant L-function of M. For primes l outside a finite set we prove the l-primary part of this conjecture for the specific case where M is the trace zero part of the adjoint of H¹(X₀(N)) for prime N and where T is the (commutative) integral Hecke algebra for cusp forms of weight 2 and the congruence group Γ₀(N), thus providing one of the first nontrivial supporting examples for the conjecture in a geometric situation where T is not the maximal order of T.

We also compare two Selmer groups, one of which appears in Bloch-Kato conjecture and the other a slight variant of what is defined by A. Wiles. A result on the Fontaine-Laffaille modules with coefficients in a local ring finite free over Z_ℓ is obtained.

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Books on the topic "Bloch-Kato conjecture"

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Coates, John, A. Raghuram, Anupam Saikia, and R. Sujatha, eds. The Bloch–Kato Conjecture for the Riemann Zeta Function. Cambridge: Cambridge University Press, 2015. http://dx.doi.org/10.1017/cbo9781316163757.

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John, Coates, R. Sujatha, A. Raghuram, and Anupam Saikia. Bloch-Kato Conjecture for the Riemann Zeta Function. Cambridge University Press, 2015.

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John, Coates, A. Raghuram, and Anupam Saikia. Bloch Kato Conjecture for the Riemann Zeta Function. Cambridge University Press, 2015.

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John, Coates, R. Sujatha, A. Raghuram, and Anupam Saikia. Bloch-Kato Conjecture for the Riemann Zeta Function. Cambridge University Press, 2015.

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Bloch-Kato Conjecture for the Riemann Zeta Function. Cambridge University Press, 2015.

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Haesemeyer, Christian, and Charles A. Weibel. The Norm Residue Theorem in Motivic Cohomology. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691191041.001.0001.

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This book presents the complete proof of the Bloch–Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups. Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The book draws on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduces the key figures behind its development. It proceeds to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. It then addresses symmetric powers of motives and motivic cohomology operations. The book unites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.

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Book chapters on the topic "Bloch-Kato conjecture"

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Hulsbergen, Wilfred W. J. "The Bloch-Kato conjecture." In Conjectures in Arithmetic Algebraic Geometry, 207–27. Wiesbaden: Vieweg+Teubner Verlag, 1994. http://dx.doi.org/10.1007/978-3-663-09505-7_12.

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Suslin, Andrei, and Vladimir Voevodsky. "Bloch-Kato Conjecture and Motivic Cohomology with Finite Coefficients." In The Arithmetic and Geometry of Algebraic Cycles, 117–89. Dordrecht: Springer Netherlands, 2000. http://dx.doi.org/10.1007/978-94-011-4098-0_5.

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Conference papers on the topic "Bloch-Kato conjecture"

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Chebolu, Sunil, and Ján Mináč. "Absolute Galois groups viewed from small quotients and the Bloch–Kato conjecture." In New topological contexts for Galois theory and algebraic geometry. Mathematical Sciences Publishers, 2009. http://dx.doi.org/10.2140/gtm.2009.16.31.

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