## 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.

### Program

Waring's problem

1. 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]).
Weyl's inequality ([N, Theorem I.4.3], [D, Lemma 3.1], [V, Lemma 2.4]).
2. Oct. 11 at 14:00 (Rosa)
Dirichlet's theorem ([N, Theorem I.4.1], [V, Lemma 2.1]).
Hua's Lemma ([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]).
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]).
4. Oct. 25 at 14:00 (Francesca)
Waring's problem vs solution of equations ([D, §7-10]).

Homogeneous equations and Birch's theorem

1. Introduction to the problem for homogeneous forms ([D, §§11, 19 (only overview)], [B, §1]).
Introduction to the problem for cubic forms ([D, §§13, 14]).
2. Cubic forms via the circle method ([D, §§15-17]).
3. Birch's theorem ([D, §19], [B, §§2-6], [V, §9], [Browning, §8 till §8.2.3])
4. p-adic questions for cubic forms ([D, §18])
p-adic questions for Birch's theorem ([B, §7])

Heath-Brown's refinement

3 or 4 talks on [HB]. TBA

Possible expansions

• Van Valckenborgh, Squareful numbers in hyperplanes, arXiv link
• Browning-Yamagishi, Arithmetic of higher-dimensional orbifolds and a mixed Waring problem, arXiv link
• Shute, Sums of four squareful numbers, arXiv link

### References

[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