Relevant bibliographies by topics / Number theory – Arithmetic algebraic geometry (Diophantine geometry) – Varieties over global fields

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  1. Journal articles
  2. Dissertations / Theses
  3. Books

Academic literature on the topic 'Number theory – Arithmetic algebraic geometry (Diophantine geometry) – Varieties over global fields'

Author: Grafiati
Published: 4 June 2021
Last updated: 15 February 2022

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Journal articles on the topic "Number theory – Arithmetic algebraic geometry (Diophantine geometry) – Varieties over global fields"

1

Huang, Ming-Deh, and Wayne Raskind. "Global Duality, Signature Calculus and the Discrete Logarithm Problem." LMS Journal of Computation and Mathematics 12 (2009): 228–63. http://dx.doi.org/10.1112/s1461157000001509.

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AbstractWe develop a formalism for studying the discrete logarithm problem for the multiplicative group and for elliptic curves over finite fields by lifting the respective group to an algebraic number field and using global duality. One of our main tools is the signature of a Dirichlet character (in the multiplicative group case) or principal homogeneous space (in the elliptic curve case), which is a measure of its ramification at certain places. We then develop signature calculus, which generalizes and refines the index calculus method. Finally, using some heuristics, we show the random polynomial time equivalence for these two cases between the problem of computing signatures and the discrete logarithm problem. This relates the discrete logarithm problem to some very well-known problems in algebraic number theory and arithmetic geometry.
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Ito, Kazuhiro, Tetsushi Ito, and Teruhisa Koshikawa. "CM liftings of surfaces over finite fields and their applications to the Tate conjecture." Forum of Mathematics, Sigma 9 (2021). http://dx.doi.org/10.1017/fms.2021.24.

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Abstract We give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of $K3$ surfaces over finite fields. We prove that every $K3$ surface of finite height over a finite field admits a characteristic $0$ lifting whose generic fibre is a $K3$ surface with complex multiplication. Combined with the results of Mukai and Buskin, we prove the Tate conjecture for the square of a $K3$ surface over a finite field. To obtain these results, we construct an analogue of Kisin’s algebraic group for a $K3$ surface of finite height and construct characteristic $0$ liftings of the $K3$ surface preserving the action of tori in the algebraic group. We obtain these results for $K3$ surfaces over finite fields of any characteristics, including those of characteristic $2$ or $3$ .
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Dissertations / Theses on the topic "Number theory – Arithmetic algebraic geometry (Diophantine geometry) – Varieties over global fields"

1

Tyler, Michael Peter. "On the birational section conjecture over function fields." Thesis, University of Exeter, 2017. http://hdl.handle.net/10871/31600.

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The birational variant of Grothendieck's section conjecture proposes a characterisation of the rational points of a curve over a finitely generated field over Q in terms of the sections of the absolute Galois group of its function field. While the p-adic version of the birational section conjecture has been proven by Jochen Koenigsmann, and improved upon by Florian Pop, the conjecture in its original form remains very much open. One hopes to deduce the birational section conjecture over number fields from the p-adic version by invoking a local-global principle, but if this is achieved the problem remains to deduce from this that the conjecture holds over all finitely generated fields over Q. This is the problem that we address in this thesis, using an approach which is inspired by a similar result by Mohamed Saïdi concerning the section conjecture for étale fundamental groups. We prove a conditional result which says that, under the condition of finiteness of certain Shafarevich-Tate groups, the birational section conjecture holds over finitely generated fields over Q if it holds over number fields.
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Derenthal, Ulrich. "Geometry of universal torsors." Doctoral thesis, [S.l.] : [s.n.], 2006. http://webdoc.sub.gwdg.de/diss/2006/derenthal.

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Books on the topic "Number theory – Arithmetic algebraic geometry (Diophantine geometry) – Varieties over global fields"

1

Shparlinski, Igor E., and David R. Kohel. Frobenius distributions: Lang-Trotter and Sato-Tate conjectures : Winter School on Frobenius Distributions on Curves, February 17-21, 2014 [and] Workshop on Frobenius Distributions on Curves, February 24-28, 2014, Centre International de Rencontres Mathematiques, Marseille, France. Providence, Rhode Island: American Mathematical Society, 2016.

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Arithmetic of L-functions. Providence, R.I: American Mathematical Society, 2011.

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Brauer groups, Tamagawa measures, and rational points on algebraic varieties. Providence, Rhode Island: American Mathematical Society, 2014.

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1965-, Aubry Yves, Ritzenthaler Christophe 1976-, Zykin Alexey 1984-, and Geocrypt Conference (2011 : Bastia, France), eds. Arithmetic, geometry, cryptography and coding theory: 13th Conference on Arithmetic, Geometry, Cryptography and Coding Theory, March 14-18, 2011, CIRM, Marseille, France : Geocrypt 2011, June 19-24, 2011, Bastia, France. Providence, R.I: American Mathematical Society, 2012.

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Capacity theory with local rationality: The strong Fekete-Szegő theorem on curves. Providence, Rhode Island: American Mathematical Society, 2013.

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Elliptic curves, modular forms, and their L-functions. Providence, R.I: American Mathematical Society, 2011.

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Fermat's last theorem. Providence, Rhode Island: American Mathematical Society, 2013.

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Alta.) WIN (Conference) (2nd 2011 Banff. Women in Numbers 2: Research directions in number theory : BIRS Workshop, WIN2 - Women in Numbers 2, November 6-11, 2011, Banff International Research Station, Banff, Alberta, Canada. Edited by David Chantal 1964-, Lalín Matilde 1977-, and Manes Michelle 1970-. Providence, Rhode Island: American Mathematical Society, 2013.

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1978-, Ghioca Dragos, and Tucker Thomas J. 1969-, eds. The dynamical Mordell-Lang conjecture. Providence, Rhode Island: American Mathematical Society, 2016.

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1953-, Campillo Antonio, ed. Zeta functions in algebra and geometry: Second International Workshop on Zeta Functions in Algebra and Geometry, May 3-7, 2010, Universitat de Les Illes Balears, Palma de Mallorca, Spain. Providence, R.I: American Mathematical Society, 2012.

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