Quadratic Forms and Elliptic Curves Ernst Kani Queen’s University Fall 2008 Revised: January 2010 Contents I Quadratic Forms and Lattices 1 1 Binary Quadratic Forms 3 1.1 Introduction . 3 1.2 Basic Concepts . 5 1.3 Lagrange’s Method: Equivalence and Reduction . 8 1.3.1 Equivalence . 8 1.3.2 Reduction . 13 1.3.3 Reduction of indefinite forms (overview) . 22 1.3.4 Applications to representation numbers . 26 1.3.5 Applications to the representation problem . 31 1.4 Gauss: The Theory of Genera and of Composition . 35 1.4.1 Genera . 35 1.4.2 Composition . 42 2 Lattices and Quadratic Modules 55 2.1 Introduction . 55 2.2 Quadratic Modules . 56 2.3 Lattices and Orders . 58 2.4 Quadratic Orders and Lattices . 64 2.4.1 Quadratic Fields . 64 2.4.2 Quadratic Orders . 67 2.4.3 Quadratic Lattices . 69 2.4.4 Dedekind’s Main Result . 75 2.4.5 Reinterpretation of the representation problem . 79 2.4.6 The Homomorphismρ ¯ : Pic(O∆) → Pic(OK ) . 83 2.4.7 Genus theory . 93 i Part I Quadratic Forms and Lattices 1 Chapter 1 Binary Quadratic Forms 1.1 Introduction In this chapter we shall study the elementary theory of (integral) binary quadratic forms f(x, y) = ax2 + bxy + cy2, where a, b, c are integers. This theory was founded by Fermat, Euler, Lagrange, Legendre and Gauss, and its development is synonymous with the early development of number theory.1 However, problems involving binary quadratic forms were already studied in antiquity. For example,√ around 400BC people in India and Greece found successive approximations a b to 2 which satisfied the equation a2 − 2b2 = 1, (cf. [Di], II, p. 341), and a Greek epigram which is attributed to Archimedes (ca. 150BC) (but which was only discovered in 1773) leads to the equation x2 − ay2 = 1 in which a ≈ 4 × 1014; cf. [Di] II, p. 342, [We], p. 19. It is not known if Archimedes knew how to solve such equations. As is well-known, Fermat’s “birth of number theory” was inspired by Bachet’s trans- lation (1621) of the Arithmetica of Diophantus (ca. 250AD). In that text one finds many problems involving sums of squares, often in connection with triangles and the Theorem of Pythagoras. For example, in Book V, Problem 12, Diophantus poses a problem which is equivalent to solving the equation (1.1) x2 + y2 = n (n odd), 1According to Weil[We], p. 1-2, (modern) number theory was born first around 1630 by Fermat and then reborn in 1730 by Euler. 3 and remarks that we must have n 6= 4k + 3; cf. [Di], II, p. 225, [We], p. 30. (He himself considers the case n = 13.) This led readers of Diophantus study the following problem. Problem A. When can a given number n be written as a sum of two squares? This problem was studied by a number of people during the middle ages and some solutions (often incorrect) were proposed (cf. [Di], II, p. 225-7). However, the correct answer only found in 1625 by Girard (but without proof). Because of the identity (1.2) (x2 + y2)(z2 + t2) = (xz − yt)2 + (xt + yz)2, which was probably known to Diophantus (cf. [We], p. 11) but which was first written down explictly by Fibonacci (1225) (cf. [Di] II, p. 226), one can reduce Problem A to case that n is a prime number. Fermat proved in 1640 the following result ([We], p. 67): Theorem 1.1 (Fermat) If p is a prime, then 2 2 p = x + y with x, y, ∈ Z ⇔ p ≡ 1 (mod 4) or p = 2. He did not write down a proof of this result, but mentioned in a letter that he proved it by his “method of infinite descent”. Such a proof was found by Euler around 1745: if p ≡ 1 (mod 4), then 1) There exist x, y ∈ Z such that x2 + y2 ≡ 0 (mod p), so x2 + y2 = mp for some m ∈ Z; 2 2 2) If x + y = mp for some m > 1, then there are x1, y1, m1 ∈ Z with 1 ≤ m1 < m 2 2 such that x1 + y1 = m1p. (“Method of descent”) It is clear that Fermat’s theorem follows from these two steps. We shall later see in §1.3.4 how to prove this result by a related but slightly different method. Throughout his life, Euler studied the following generalization of the above problem and/or Fermat’s theorem (cf. [We], p. 204): Problem B. Given a number N 6= 0, when does the equation x2 + Ny2 = p, p a prime, have a solution (in integers)? Can these primes p be described by congruence conditions on p? Euler solved this problem (positively) for N = 1, ±2, 3,2 and obtained some partial results for other N’s. For example, he observed: x2 ≡ −N (mod p) has a solution x2 + Ny2 = p, p 2N ⇒ - x2 ≡ p (mod N) has a solution 2In fact, the cases N = 1, 2, 3 were already done by Fermat; cf. [We], p. 205. 4 and for a while thought that the converse of the second implication might be true ([We], p. 214). However, this is already false for N = 5, 6 as he later realized, and so he was very far from establishing a general theory of such equations. In 1773 Lagrange was able to greatly clarify the piecemeal results of Euler. The main idea of Lagrange was that one should not only study a fixed form such as x2 + Ny2, but also certain “related” binary quadratic forms ax2 + bxy + cy2. More precisely, he proved Theorem 1.2 (Lagrange) If −N is not a square, then there is an (explicitly com- putable) finite list of binary quadratic forms 2 2 f1(x, y) = x + Ny , f2(x, y), . , fh(x, y) such that for every prime number p - 2N we have: 2 fk(x, y) = p has a solution for some k, 1 ≤ k ≤ h, ⇔ x ≡ −N (mod p) has a solution. From this theorem the aforementioned results of Fermat and Euler follow (almost) immediately because one has h = 1 when N = 1, ±2, 3. On the other hand, for N = 5, 6 we have h > 1, so this explains why Euler’s “converse” failed in those cases. Note that the above congruence condition x2 ≡ −N (mod p) is not a priori a congru- ence condition on p, so further work is necessary to analyze this condition. It turns out, however, that by the famous Law of Quadratic Reciprocity 3 this condition can be re- written as a list of congruences mod 4N, and so Theorem 2 does give a partial resolution of Problem 2. Lagrange’s theory was further refined and developed by Legendre (1785) and particu- larly by Gauss (1801), who introduced the theory of genera and the composition of forms. Later Dirichlet (around 1850) and Dedekind (1860) further simplified and generalized the theory and embedded it in a general theory of algebraic number fields. 1.2 Basic Concepts As was already mentioned in §1.1, an (integral) binary quadratic form is a polynomial f(x, y) of the form 2 2 f(x, y) = ax + bxy + cy , where a, b, c ∈ Z. We shall usually abbreviate this formula by writing f = [a, b, c]. Definition. The form f = [a, b, c] is said to represent an integer n if there exist x, y ∈ Z such that f(x, y) = n. If, in addition, x and y can be chosen such that gcd(x, y) = 1, then we say that f primitively represents the integer n. 3This law was discovered by Euler in 1772 and was published in 1783 ([We], p. 187), and Legendre attempted to give a proof of it in 1785. Gauss (1801) gave the first correct proof and, in fact, gave 8 different proofs of it. 5 Example 1.1 (a) If f = [a, b, c], then a = f(1, 0), c = f(0, 1) and a ± b + c = f(1, ±1) are primitively represented by f. n x y (b) If n = f(x, y) is represented by f and if g = gcd(x, y), then g2 = f( g , g ) is primitively represented by f. In particular, if n = p is a prime number, then f represents p if and only if f represents p primitively. From the discussion in §1.1 we see that a natural (but extremely difficult) question about binary quadratic forms is the following. Problem 1.1 For a given form f = [a, b, c], determine (or describe) the set R(f) := {f(x, y): x, y ∈ Z, gcd(x, y) = 1} of integers which are primitively represented by f. As was explained in §1.1, this problem does not have a satisfactory answer except in special cases. Two related but easier questions are the following. Problem 1.2 For a given form f = [a, b, c] and integer n, determine the set S(f, n) = {(x, y): f(x, y) = n} of all integer solutions of the equation (1.3) f(x, y) = n. Alternately, determine the set P (f, n) = {(x, y) ∈ S(f, n) : gcd(x, y) = 1} of all primitive solutions of this equation. Problem 1.3 For a given form f = [a, b, c], determine its minimum min(f) := min{|f(x, y)| : x, y ∈ Z, (x, y) 6= (0, 0)} = min{|n| : n ∈ R(f)}. As we shall see, the nature (and method) of the solutions of these problems depends heavily on whether the form is definite or indefinite.