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El. knyga: Cubic Forms and the Circle Method

  • Formatas: EPUB+DRM
  • Serija: Progress in Mathematics 343
  • Išleidimo metai: 19-Nov-2021
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030868727
Kitos knygos pagal šią temą:
Cubic Forms and the Circle MethodDidesnis paveikslėlis
  • Formatas: EPUB+DRM
  • Serija: Progress in Mathematics 343
  • Išleidimo metai: 19-Nov-2021
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030868727
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The Hardy–Littlewood circle method was invented over a century ago to study integer solutions to special Diophantine equations, but it has since proven to be one of the most successful all-purpose tools available to number theorists. Not only is it capable of handling remarkably general systems of polynomial equations defined over arbitrary global fields, but it can also shed light on the space of rational curves that lie on algebraic varieties.  This book, in which the arithmetic of cubic polynomials takes centre stage, is aimed at bringing beginning graduate students into contact with some of the many facets of the circle method, both classical and modern. This monograph is the winner of the 2021 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.

Recenzijos

This marvelous and very clearly written book is a very valuable addition to the literature ... . I think it is a great read for every circle method practitioner, especially those coming from the more classical side but eager to learn more about the function field or geometric side, as well as for graduate students in analytic number theory, to be studied along with the more classical texts by Davenport and Vaughan. (Rainer Dietmann, Mathematical Reviews, March, 2024)





It is recommended for readers with a solid background in abstract algebra, local fields, algebraic varieties and some analytic and algebraic number theory. (Franz Lemmermeyer, zbMATH 1493.11003, 2022)

1 Cubic Forms Over Local Fields
1(24)
1.1 The Hasse Principle
2(4)
1.2 Systems of Equations Over Finite Fields
6(7)
1.3 Solubility Over Local Fields
13(6)
1.4 Local Densities
19(6)
2 Waring's Problem for Cubes
25(32)
2.1 Weyl Differencing
29(11)
2.2 The Asymptotic Formula
40(8)
2.3 Analysis of the Singular Series
48(9)
3 Cubic Forms via Weyl Differencing
57(32)
3.1 Cubic Forms in Many Variables
58(8)
3.2 The Minor Arcs
66(13)
3.3 The Major Arcs
79(10)
4 Norm Forms Over Number Fields
89(24)
4.1 Background on Algebraic Number Theory
92(4)
4.2 The Circle Method Over Number Fields
96(7)
4.3 Singular Integral
103(4)
4.4 Singular Series
107(6)
5 Diagonal Cubic Forms Over Function Fields
113(28)
5.1 Background on Function Fields
115(9)
5.2 The Circle Method
124(7)
5.3 The Major Arcs and the Asymptotic Formula
131(10)
6 Lines on Cubic Hypersurfaces
141(18)
6.1 Transition to Counting Functions
145(3)
6.2 Dimension and Irreducibility via the Circle Method
148(3)
6.3 Singular Locus via the Circle Method
151(8)
References 159(6)
Index 165
Tim Browning is a professor of number theory with a focus on analytic number theory and Diophantine geometry. 
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