Reading seminar on the circle method, Fall 2021
For three introductory lectures on the circle method for Waring's problem see also
Kevin Hughes' online course
starting on October 11, 2021.
- Oct. 4 at 14:00 (Boaz)
Introduction to Waring's problem and the circle method
([N, §I.5.1], [D, Foreword, §§1-2], [V, §1]).
Decomposition into major and minor arcs
([N, §I.5.3], [D, §4]).
The minor arcs ([N, §I.5.4], [D, Lemma 4.1]).
([N, Theorem I.4.3], [D, Lemma 3.1], [V, Lemma 2.4]).
- Oct. 11 at 14:00 (Rosa)
Dirichlet's theorem ([N, Theorem I.4.1], [V, Lemma 2.1]).
([N, Theorem I.4.6], [D, Lemma 3.2], [V, Lemma 2.5]).
Auxiliary functions for the major arcs
([N, §I.5.5], [D, §4 till Lemma 4.3]).
- Oct. 18 at 14:00 (Miriam)
The singular integral
([N, §I.5.6], [D, Theorem 4.1]).
The singular series
([N, §I.5.7], [D, §§5-6]).
- Oct. 25 at 14:00 (Francesca)
Waring's problem vs solution of equations
Homogeneous equations and Birch's theorem
- Introduction to the problem for homogeneous forms
([D, §§11, 19 (only overview)], [B, §1]).
Introduction to the problem for cubic forms
([D, §§13, 14]).
- Cubic forms via the circle method
- Birch's theorem
([D, §19], [B, §§2-6], [V, §9], [Browning, §8 till §8.2.3])
- p-adic questions for cubic forms ([D, §18])
p-adic questions for Birch's theorem ([B, §7])
3 or 4 talks on [HB]. TBA
- Van Valckenborgh, Squareful numbers in hyperplanes,
- Browning-Yamagishi, Arithmetic of higher-dimensional orbifolds and a mixed Waring problem,
- Shute, Sums of four squareful numbers,
[B] B. J. Birch, Forms in many variables.
Proc. Roy. Soc. London Ser. A 265 (1961/62), 245–263.
[Browning] T. D. Browning, Quantitative arithmetic of projective varieties.
Progress in Mathematics, 277. Birkhäuser Verlag, Basel, 2009
[D] H. Davenport, Analytic methods for Diophantine equations and Diophantine inequalities.
Second edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2005
[HB] D. R. Heath-Brown, A new form of the circle method, and its application to quadratic forms.
J. Reine Angew. Math. 481 (1996), 149–206
[N] M. B. Nathanson, Additive number theory. The classical bases.
Graduate Texts in Mathematics, 164. Springer-Verlag, New York, 1996
[V] R. C. Vaughan, The Hardy-Littlewood circle method.
Second edition. Cambridge Tracts in Mathematics, 125. Cambridge University Press, Cambridge, 1997