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The modern geometry of numbers


TCC module: term 3 2024

Learn the geometry of numbers with a focus on applications in analytic number theory. Topics include Minkowski minima, reduced bases, LLL algorithm, orthogonal and dual lattices, and point-counting results. The course will cover the proof of Manin’s conjecture for $x_1y_1+\dots+x_ny_n=0$, which is a special case of a result of J L Thunder, and examine the geometry of numbers in J Maynard's work on primes with missing digits and small fractional parts of polynomials.


The assessment for this course is expected to consist of one assignment, with four equally weighted compulsory questions worth 5 marks each, having two deadlines.

First will be a check-in deadline, where students can submit partial answers and receive 25% of the mark for showing plausible strategies for at least two questions.

After the check-in, there will be a lecture to discuss the different strategies attempted by the students, led where possible by students (technology permitting) and with opportunities to discuss their questions about the assignment.

The final deadline will contribute the remaining 75% of the mark for the assignment. Students are welcome to collaborate on strategies and ideas, and should write up their answers independently.



We introduce the geometry of numbers with a focus on applications in analytic number theory, including Manin’s conjecture and recent work in prime number theory. Topics covered include:

  • Definition of lattices and their properties, including the dual lattice, Minkowski minima, reduced basis, sublattices, and projections, with an emphasis on solving problems related to point counting and lattice reduction.
  • Explanation of the Lenstra-Lenstra-Lovász algorithm and its application to lattice reduction.
  • Application of asymptotic and upper bounds to count points of bounded norm in a lattice, and to count integral lattices with a given shape.
  • Explanation of Manin's conjecture and its significance in the study of Diophantine and polynomial equations. The spirit of Thunder's proof for bilinear varieties.
  • Application of the geometry of numbers to reduce problems in exponential sums over primes to geometric problems, and use of this approach to bound expressions related to prime numbers. In particular we will discuss J Maynard's work on primes with missing digits (time permitting).

An approximate schedule is as follows.

Week 1. Definition of a lattice and its covolume. History and origins of the geometry of numbers. The problem of primes with missing digits. The shape of a lattice. Inner products, bases, quadratic forms, the Gaussian volume heuristic.
Week 2. The dual lattice. Minkowski minima. The general idea of a reduced basis. Minkowski’s theorems. Sublattices and projections.
Week 3. Lenstra–Lenstra–Lovász algorithm. Historical questions of interest in the geometry of numbers. More recent questions about lattice reduction.
Week 4. Counting points of bounded norm in a lattice: asymptotic and upper bounds. Primitive sublattices. Orthogonal sublattices. Covolume of orthogonal lattices.
Week 5. Manin’s conjecture. The equation $x_1y_1+\dots+x_ny_n=0$. Basic idea of Thunder’s proof. Assignment check-in deadline and lecture.
Week 6. Manin’s conjecture for $x_1y_1+\dots+x_ny_n=0$: completion of the proof. Review of course.
Week 7. Primes with missing digits: why geometry of numbers is relevant. Vaughan’s identity. The circle method. Statement of Maynard’s Proposition 9.3, Lemma 13.1, Proposition 13.3 and Proposition 13.4.
Week 8. Selected proofs from Maynard.
Week 9. Assignment final deadline.