## Geometry of Numbers VIGRE Research Group (2011-2012)

Instructor: Associate Professor Pete L. Clark, pete (at) math (at) uga (dot) edu

Course webpage: http://math.uga.edu/~pete/GeometryofNumbers.html (i.e., right here)

Office Hours: By appointment.

Course text: none required. The following texts are recommended as (broadly) useful:

Cassels, An Introduction to the Geometry of Numbers, Springer Classics in Mathematics, 1997.
Siegel, Lectures on the Geometry of Numbers, Springer 1989.
Gruber, Convex and Discrete Geometry, Springer Grundlehren der mathematischen Wissenschaften, 2007.
Beck and Robins, Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra, Springer Undergraduate Texts in Mathematics, 2007.

Research Goal: The goal of our VRG is to find and explore open questions in both geometry of numbers -- e.g. Lattice Point Enumerators, the Ehrhart (Quasi)-Polynomial, Minkowski's Convex Body Theorems, Lattice Constants for Ellipsoids, Minkowski-Hlawka Theorem,... -- and its applications to number theory, especially to solutions of Diophantine equations (and especially, to integers represented by quadratic forms).

Lecture Notes: a work in progress! (pdf) (currently 107 pages)

Geometry of Numbers Explained (pdf) (currently 16 pages)
This document (very much a work in progress) abstracts the features of the "classical" GoN arguments for studying binary and quaternary quadratic forms. In particular it gives a "master theorem" enabling the pursuit of representation theorems for (certain) quadratic forms over (certain) normed rings.

If you were to read even half of the papers below, you will be far ahead of me, which is sort of sad because it takes some time and effort to post these papers, whether I read them or not. But probably each of you should try reading at least one or two of the papers below: please tell me which you are interested in.

Early Papers on the Geometry of Numbers
H.F. Blichfeldt, A new principle in the geometry of numbers, with some applications. Trans. Amer. Math. Soc. 15 (1914), 227235. (pdf)
L.E. Dickson, Applications of the geometry of numbers to algebraic numbers. Bull. Amer. Math. Soc. 25 (1919), 453-455. (pdf)
O. Frink, Jordan measure and Riemann integration. Ann. of Math. (2) 46 (1933), 518-526. (pdf)
L.J. Mordell, On some arithmetical results in the geometry of numbers. Compositio Math. 1 (1935), 248-253. (pdf)
H. Davenport, The geometry of numbers. Math. Gaz. 31, (1947), 206-210. (pdf)

On universal quaternary quadratic forms and the 15, 290 Theorems
Ramanujan, On the expression of a number in the form ax2 + by2 + cz2 + du2, Proceedings of the Cambridge Philosophical Society 19(1917), 1121. Available online (here)
L.E. Dickson, Integers represented by positive ternary quadratic forms. Bull. Amer.Math. Soc. 33 (1927), 63-70. (pdf)
P.R. Halmos, Note on almost-universal forms. Bull. Amer. Math. Soc. 44 (1938), 141144. (pdf)
J.H. Conway, Universal Quadratic Forms and the Fifteen Theorem, Quadratic forms and their applications (Dublin, 1999), 23-26, Contemp. Math., 272, 2000. (pdf)
M. Bhargava, On the Conway-Schneeberger fifteen theorem. Quadratic forms and their applications (Dublin, 1999), 2737, Contemp. Math., 272, 2000. (pdf)
M. Bhargava and J.P. Hanke, Universal Quadratic Forms and the 290-Theorem, 2005 preprint.

The Minkowski-Hlawka Theorem
C.L. Siegel, A mean value theorem in geometry of numbers. Ann. of Math. (2) 46 (1945), 340-347. (pdf)
C.A. Rogers, Existence theorems in the geometry of numbers. Ann. of Math. (2) 48 (1947), 9941002. (pdf)
C.A. Rogers, The number of lattice points in a star body. J. London Math. Soc. 26 (1951), 307310. (pdf)

Lattice Points in Regions with Fractal Boundary
L. Colzani, Approximation of Lebesgue integrals by Riemann sums and lattice points in domains with fractal boundary. Monatsh. Math. 123 (1997), no. 4, 299308. (pdf)

Applications to Linear Forms
L. Tornheim, Linear Forms in Function Fields. Bull. Amer. Math. Soc. 47 (1941), 126127. (pdf)
L.J. Mordell, On the product of two non-homogeneous linear forms. J. London Math. Soc. 16, (1941). 8688. (pdf)
D.B. Sawyer, The product of two non-homogeneous linear forms. J. London Math. Soc. 23 (1948), 250251. (pdf)
L.J. Mordell, Note on Sawyer's paper ''The product of two non-homogeneous linear forms.''J. London Math. Soc. 28 (1953), 510512. (pdf)
A. Brauer and R.L. Reynolds, On a theorem of Aubry-Thue. Canadian J. Math. 3 (1951), 367374. (pdf)
G. Alon and P.L. Clark, On the Number of Representations of an Integer by a Linear Form. Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.2. (pdf)

J.H. Grace, The Four Square Theorem. J. London Math. Soc. 2 (1927), 3-8. (pdf)
R.A. Rankin, On positive definite quadratic forms. J. London Math. Soc. 28 (1953), 309314. (pdf)
N.C. Ankeny, Sums of three squares. Proc. Amer. Math. Soc. 8 (1957), 316319. (pdf)
L.J. Mordell, On the representation of a number as a sum of three squares. Rev. Math. Pures Appl. 3 (1958), 25-27. (please contact me if you have a copy)
J. Wojcik, On sums of three squares. Colloq. Math. 24 (1971/72), 117119. (pdf)
J.I. Deutsch, Geometry of numbers proof of Gotzky's four-squares theorem. J. Number Theory 96 (2002), 417-431. (pdf)
J.I. Deutsch, An alternate proof of Cohn's four squares theorem. J. Number Theory 104 (2004), 263-278. (pdf)
J.I. Deutsch, Short proofs of the universality of certain diagonal quadratic forms. Arch. Math. (Basel) 91 (2008), 4448. (pdf)
L.M. Nunley, Geometry of Numbers Approach to Small Solutions of the Extended Legendre Equation. 2010 UGA Master's theis. (pdf)
T. Hagedorn, Primes of the form x2+ny2 and the geometry of (convenient) numbers, preprint. (pdf)

Source Material for GoN Over Number Rings / General Normed Rings
L. Tornheim, Linear forms in function fields. Bull. Amer. Math. Soc. 47, (1941). 126127. (pdf)
K. Mahler, An Analogue to Minkowski's Geometry of Numbers in a Field of Series. Annals of Math. 42 (1941), 488-522. (pdf)
F. Lemmermemyer, The Euclidean Algorithm in Algebraic Number Fields. Exposition. Math. 13 (1995), 385-416. (pdf)
R. Baeza and M.I. Icaza, On Humbert-Minkowski's constant for a number field. Proc. Amer. Math. Soc. 125 (1997), 31953202. (pdf)
M.I. Icaza, Hermite constant and extreme forms for algebraic number fields. J. London Math. Soc. 55 (1997), 1122. (pdf)
R. Baeza, R. Coulangeon, M.I. Icaza and M. O'Ryan, Hermite's constant for quadratic number fields. Experiment. Math. 10 (2001), 543551. (pdf)
S. Ohno and T. Watanabe, Estimates of Hermite constants for algebraic number fields. Comment. Math. Univ. St. Paul. 50 (2001), 5363.
R. Baeza and M.I. Icaza, On the unimodularity of minimal vectors of Humbert forms. Arch. Math. (Basel) 84 (2004), 528-535. (pdf)
W.K. Chan and J. Daniels, Definite Regular Quadratic Forms over Fq[T]. Proc. AMS 133 (2005), 3121-3131. (pdf)
R. Coulangeon, M.I. Icaza, and M. O'Ryan, Lenstra's constant and extreme forms in number fields. Experiment. Math. 16 (2007), 455462.
J. Bureau and J. Morales, Representations of Definite Binary Quadratic Forms over Fq[T]. Illinois J. Math. 53 (2009), 237-249. (pdf)

Assigned Projects:

Binary Quadratic Forms over Z: Harrison Chapman, Hans Parshall (formerly Alex Rice)

Quaternary Quadratic Forms over Z: Jacob Hicks, Kate Thompson, Nathan Walters

Ternary Quadratic Forms over Z: Allan Lacy

Quadratic Forms over Number Fields (especially, examining work of J.I. Deutsch): Jacob Hicks, Kate Thompson, Lee Troupe, Nathan Walters

Linear Forms, Linear Systems and the Frobenius Problem: Jim Stankewicz, Jun Zhang

The Minkowski-Hlawka Theorem: Lauren Huckaba, Alex Rice

Starbodies and Minkowski distance functions: Brian Bonsignore

Minkowski's Theorem on Successive Minima: Jim Stankewicz, Nathan Walters, Jun Zhang

Ehrhart (quasi-)polynomials: Nham Ngo

RESULTS: