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

5. Nov. 1 at 14:00 (Marta)
   Introduction to the problem for homogeneous forms ([D, §§11, 19 (only overview)], [B, §1]).
   Introduction to the problem for cubic forms ([D, §§13, 14]).
6. Nov. 8 at 14:00 (Judith)
   Cubic forms via the circle method ([D, §§15-17]).
7. Nov. 15 at 14:00 (Miriam)
   p-adic questions for cubic forms ([D, §18])
   p-adic questions for Birch's theorem ([B, §7])
8. Nov. 22 at 14:00 (Julia)
   Birch's theorem ([D, §19], [B, §§2-6], [V, §9], [Browning, §8 till §8.2.3])

Heath-Brown's delta method

9. Nov. 29 at 14:00 (Rosa)
   Introduction, weight functions, and theorems 1 and 2 ([HB, §§1-3]).
10. Dec. 6 at 14:00 (Judith)
    Description of the singular integral: Theorem 3 ([HB, §§4-6]).
11. Dec. 13 at 14:00 (Shuntaro)
    Singular integral estimates and the singular series ([HB, §§7-9]).
12. Dec. 20 at 14:00 (Francesca)
    Applications to quadratic forms ([HB, §§10-14]).

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