Quadratic Forms over $${\mathbb{Z}}$$ from Diophantus to the 290 Theorem

Abstract

What follows 1 is a brief historical survey of the representation theory of quadratic forms over the integers. It starts with questions considered by Diophantus about 1800 years ago, dashes past classical contributions of Euler, Lagrange, and Sylvester, considers work of Ramanujan and Dickson, continues with theorems of Conway and Schneeberger, and ends with a short sketch of the proof of the 290-Theorem by Bhargava and Hanke. This survey is self-contained in the sense that all the basic definitions and concepts are provided.Let R be a commutative ring with 1. A quadratic form in n variables over R is a homogeneous polynomial q = q(X 1 , . . . , X n ) = 1≤i≤j≤n a i,j X i X j of degree two with coefficients a i,j in R. An element s ∈ R is a value of q if there is a vector (r 1 , . . . , r n ) ∈ R n such that q(r 1 , . . . , r n ) = s. In this case q is said to represent s. If q represents 0 non-trivially then q is isotropic, and q is anisotropic if 0 is represented only by the zero vector.The Babylonians in the 18th century B.C. already knew about triples (a, b, c) of positive integers that satisfy the equation a 2 + b 2 = c 2 . So they had insight into the ways that the quadratic form X 2 1 + X 2 2 − X 2 3 over the integers Z represents 0. In the 6th century B.C., the Pythagoreans appear to have been unsettled by the fact that the quadratic form 2X 2 1 − X 2 2 over Z is anisotropic. However, the story of quadratic forms, in any reasonably systematic sense, begins with Diophantus and his Arithmetica [11] in the 3rd century A.D. For example, wanting to find two squares of rational numbers having a given difference, Diophantus saw that 60 = (x + 3) 2 − x 2 with x = 8 1 2 . In particular, the quadratic form X 2 1 − X 2 2 over 1 This article is dedicated to a wonderful and exceptional mathematician, human being, and friend: to Timothy O'Meara on the occasion of his 80-th birthday.

Minimal universality criterion sets on the representations of quadratic forms

“…Remark 2.3. As pointed out by [6], Bhargava also proved the above result. However, no proof of this has appeared in the literature to the author's knowledge.…”

Prime-Universal Quadratic Forms And

“…This was proved in more generality by Kim et al [7]. Bhargava asserted (see [5,Theorem C,page 674]) that if T is the set of prime numbers, then the corresponding smaller finite set S is given by S = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73}, but no proof of this has appeared in the literature to the authors' knowledge. Should this set S be proven to be correct, then it is straightforward to verify that the unproven forms in Table 1 represent all primes in S, and Theorems 1.6 and 1.7 would be unconditional.…”

The holography of non-invertible self-duality symmetries

“…. , t r ) using an invertible matrix U (see, e.g., [75]). The TQFT is then characterized by the number of +1 and −1 Legendre symbols of the t i 's.…”