Cubic Forms and the Circle Method

1. Front Matter

   Pages i-xiv

   PDF

2. Cubic Forms Over Local Fields

   • Tim Browning

   Pages 1-24

3. Waring’s Problem for Cubes

   • Tim Browning

   Pages 25-56

4. Cubic Forms via Weyl Differencing

   • Tim Browning

   Pages 57-87

5. Norm Forms Over Number Fields

   • Tim Browning

   Pages 89-112

6. Diagonal Cubic Forms Over Function Fields

   • Tim Browning

   Pages 113-139

7. Lines on Cubic Hypersurfaces

   • Tim Browning

   Pages 141-157

8. Back Matter

   Pages 159-166

   PDF

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.

• Circle method
• Cubic forms
• Diophantine equations
• Fourier analysis
• Global fields
• Hasse principle
• Local fields
• Moduli spaces
• Norm forms
• Waring's problem

“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)

• Institute of Science and Technology Austria, Klosterneuburg, Austria

  Tim Browning

Tim Browning is a professor of number theory with a focus on analytic number theory and Diophantine geometry. 

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