INTRODUCTION to the ARITHMETIC THEORY of QUADRATIC FORMS Contents 1. Review of Linear Algebra and Tensors 1 2. Quadratic Forms A

INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS

SAM RASKIN

Contents 1. Review of linear algebra and tensors 1 2. Quadratic forms and spaces 10 3. 푝-adic ﬁelds 25 4. The Hasse principle 44 References 57

1. Review of linear algebra and tensors 1.1. Linear algebra is assumed as a prerequisite to these notes. However, this section serves to review the language of abstract linear algebra (particularly tensor products) for use in the remainder of these notes. We assume the reader familiar with these notions, and therefore do not provide a complete and detailed treatment. In particular, we assume the reader knows what a homomorphism of algebraic objects is.

1.2. Rings and fields. Recall that a ring is a set 퐴 equipped with associative binary operations ` and ¨ called “addition” and “multiplication” respectively, such that p푘, `q is an abelian group with identity element 0, such that multiplication distributes over addition, and such that there is a multiplicative identity 1. The ring 퐴 is said to be commutative if its multiplication is a commutative operation. Homomorphisms of rings are maps are always assumed to preserve 1. We say that a commutative ring 퐴 is a field if every non-zero element 푎 P 퐴 admits a multiplicative inverse 푎´1. We usually denote ﬁelds by 푘 (this is derived from the German “K¨orper”).

Example 1.2.1. For example, the rational numbers Q, the real numbers R, and the complex numbers C are all fields. The integers Z form a commutative ring but not a field. However, for 푝 a prime number, the integers modulo 푝 form a field that we denote by F푝, and refer to as the field with 푝 elements.

For 퐴 a ring, we let 퐴ˆ denote the group of invertible elements in 퐴.

ˆ ˆ Exercise 1.1. If 푘 is a ﬁeld and 퐺 Ď 푘 is a ﬁnite subgroup, show that 퐺 is cyclic. Deduce that F푝 is a cyclic group of order 푝 ´ 1.

Date: Spring, 2014. 1 2 SAM RASKIN

1.3. Characteristic. For a ring 퐴, we have a canonical (and unique) homomorphism Z Ñ 퐴:

푛 ÞÑ 1 ` ... ` 1 P 퐴 푛 times and we abuse notation in letting 푛 P 퐴 denotelooooomooooon the image of 푛 P Z under this map. For a field 푘, this homomorphism has a kernel of the form 푝 ¨ Z for 푝 either a prime number or 0; we say that 푘 has characteristic 푝 accordingly. We see that 푘 has characteristic 0 if Q Ď 푘, and 푘 has characteristic 푝 ą 0 if F푝 Ď 푘. We denote the characteristic of 푘 as charp푘q. 1.4. Vector spaces. Let 푘 be a field fixed for the remainder of this section. A vector space over 푘 (alias: 푘-vector space) is an abelian group p푉, `q equipped with an action of 푘 by scaling. That is, for 휆 P 푘 and 푣 P 푉 , we have a new element 휆 ¨ 푣 P 푉 such that this operation satisfies the obvious axioms. We frequently refer to elements of 푉 as vectors. We sometimes refer to homomorphisms of vector spaces sometimes as linear transformations, or as 푘-linear maps. Remark 1.4.1. Note that the totality of linear transformations 푇 : 푉 Ñ 푊 can itself be organized into a vector space Hom푘p푉, 푊 q, also denoted Homp푉, 푊 q if the field 푘 is unambiguous. The vector space operations are defined point-wise:

p푇1 ` 푇2qp푣q :“ 푇1p푣q ` 푇2p푣q p휆 ¨ 푇 qp푣q :“ 휆 ¨ p푇 p푣qq. 1.5. Direct sums and bases. We will primarily be interested in finite-dimensional vector spaces, but for the construction of tensor products it will be convenient to allow infinite direct sums. Definition 1.5.1. Given two vector spaces 푉 and 푊 , we define their direct sum 푉 ‘ 푊 to be the set-theoretic product 푉 ˆ 푊 equipped with the structure of termwise addition and scalar multiplication. More generally, given a set 퐼 and a collection t푉푖u푖P퐼 of vector spaces indexed by elements of 퐼, we define the direct sum ‘푖P퐼 푉푖 as the subset of 푖P퐼 푉푖 consisting of elements 푣 “ p푣푖q푖P퐼 (푣푖 P 푉푖) such that 푣푖 “ 0 for all but finitely many 푖 P 퐼. Note that this does indeed generalize the pairwise example above. ś

Given t푉푖u푖P퐼 as above, let 휀푖 : 푉푖 Ñ‘푗P퐼 푉푗 denote the linear transformation sending 푣푖 P 푉푖 to the vector that is 푣푖 in the 푖th coordinate and 0 in all others.

Proposition 1.5.2. Given t푉푖u푖P퐼 as above and 푊 an auxiliary vector space, the map:

t푇 : ‘푖P퐼 푉푖 Ñ 푊 a linear transformationu Ñ t푇푖 : 푉푖 Ñ 푊 linear transformationsu푖P퐼 (1.1) 푇 ÞÑ t푇 ˝ 휀푖u푖P퐼 is a bijection.

Proof. Suppose that we are given a collection 푇푖 : 푉푖 Ñ 푊 of linear transformations. Note that for every vector 푣 P‘푖P퐼 푉푖, there exists a unique ﬁnite subset 푆 Ď 퐼 and collection of non-zero vectors t푣푠 P 푉푠u푠P푆 such that:

푣 “ 휀푠p푣푠q 푠P푆 ÿ (if 푆 is empty, we understand the empty sum to mean the zero vector). We then deﬁne: INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 3

푇 p푣q “ 푇푠p푣푠q P 푊. 푠P푆 ÿ It is easy to verify that 푇 is a linear transformation, and that this operation is inverse to the map (1.1). Remark 1.5.3. The above proposition characterizes the direct sum by a universal property. This means that we have characterized the vector space ‘푖P퐼 푉푖 by describing how it maps to other vector spaces (it would be also be a proper use of the terminology universal property to describe how to map into a given vector space).

‘퐼 Notation 1.5.4. For 퐼 a set, we let 푉 denote the direct sum ‘푖P퐼 푉푖, i.e., the direct sum indexed by 퐼 with each “푉푖” equal to 푉 . 0 푛 For 퐼 “ t1, . . . , 푛u, 푛 P Zą , we use the notation 푉 ‘ instead. Example 1.5.5. The vector space 푘‘푛 consists of 푛-tuples of elements of 푘, considered as a vector space under termwise addition and scaling given by termwise multiplication. We represent an 푛 ‘푛 element 푣 “ p휆푖q푖“1 of 푘 by the vector notation:

휆1 휆2 ¨ . ˛ . (1.2) . ˚ ‹ ˚휆푛‹ ˚ ‹ ˝ ‚ Definition 1.5.6. For 푉 a vector space, we say that 푉 is finite-dimensional (of dimension 푛) if there exists an isomorphism 푉 ÝÑ» 푘‘푛. A choice of such isomorphism is called a basis for 푉 . The vectors 푒1, . . . , 푒푛 in 푉 corresponding under this isomorphism to the vectors:

1 0 0 0 1 0 ¨.˛ , ¨.˛ ,..., ¨.˛ . . . ˚ ‹ ˚ ‹ ˚ ‹ ˚0‹ ˚0‹ ˚1‹ ˚ ‹ ˚ ‹ ˚ ‹ are the basis vectors of 푉 corresponding˝ ‚ to˝ this‚ basis.˝ We‚ say that 푛 is the dimension of 푉 and write 푛 “ dim 푉 .

Remark 1.5.7. To say that 푒1, . . . , 푒푛 are basis vectors of a basis of 푉 is to say that every vector 푣 P 푉 can be written uniquely as:

푛 푣 “ 휆푖푒푖 푖“1 ÿ for 휆푖 P 푘. The expression of 푣 as such corresponds to the expression (1.2) for 푣. Notation 1.5.8. Any linear transformation 푇 : 푘‘푛 Ñ 푘‘푚 determines a p푚 ˆ 푛q-matrix 퐴, i.e., a matrix with 푚 rows and 푛 columns. The 푖th column of the matrix is the vector 푇 p푒푖q, written as in (1.2). 4 SAM RASKIN

1.6. Linear transformations. A subspace 푈 Ď 푉 is a subset closed under addition and scalar multiplication. Subspaces are the same as injective linear transformations 푇 : 푈 Ñ 푉 . Dually, we define quotient spaces as surjective linear transformations 푇 : 푉 Ñ 푊 . Notation 1.6.1. By abuse, we let 0 denote the subspace of any vector space consisting only of the element 0. Given a subspace 푈 of 푉 , we can form the quotient space 푊 :“ 푉 {푈 whose elements are cosets of 푈 in 푉 , i.e., an element 푤 P 푉 {푈 is a subset of 푉 of the form 푣 ` 푈 “ t푣 ` 푢 | 푢 P 푈u for some 푣 P 푉 . Addition and scalar multiplication on 푉 {푈 are defined in the obvious ways, as is the homomorphism 푉 Ñ 푉 {푈. The quotient space satisfies a universal property: giving a homomorphism 푊 “ 푉 {푈 Ñ 푊 1 is equivalent (by means of restriction along 푉 Ñ 푉 {푈) to giving a homomorphism 푉 Ñ 푊 1 with the property that 푈 maps to 0 under this transformation. Construction 1.6.2. Given a linear transformation 푇 : 푉 Ñ 푊 , we can associate various other vector spaces: ‚ We define the kernel Kerp푇 q of 푇 as the subspace of 푉 consisting of vectors 푣 P 푉 such that 푇 p푣q “ 0 P 푊 . ‚ We define the image Imagep푇 q of 푇 as the subspace of 푊 consisting of vectors 푤 P 푊 of the form 푇 p푣q for some 푣 P 푉 . ‚ We define the cokernel Cokerp푇 q as the quotient 푊 { Imagep푇 q of 푊 . Notation 1.6.3. A diagram:

푇 푆 0 Ñ 푈 ÝÑ 푉 ÝÑ 푊 Ñ 0 (1.3) is a short exact sequence of vector spaces if Kerp푆q “ 푈 and Cokerp푇 q “ 푊 . In other words, we say that (1.3) is a short exact sequence if 푈 injects into 푉 , 푉 surjects onto 푊 , the composition 푆 ˝ 푇 is equal to 0, and the induced maps:

푈 Ñ Kerp푆q Cokerp푆q Ñ 푊 are equivalences (it suffices to check this for either of the two maps). Remark 1.6.4. There is clearly no mathematical content here. Still, the notion of short exact sequence is regarded as a useful organizing principle by many mathematicians, and we will use it as such below. _ 1.7. Duality. The dual vector space 푉 to a vector space 푉 is defined as Hom푘p푉, 푘q. Elements of 푉 _ are referred to as linear functionals. Example 1.7.1. For 푉 “ 푘‘푛, the dual is canonically identified with 푉 itself in the obvious way: however, the corresponding isomorphism 푉 » 푉 _ depends on the choice of basis. We have a canonical map:

푉 Ñ p푉 _q_ 푣 ÞÑ 휆 ÞÑ 휆p푣q that is an isomorphism for 푉 ﬁnite-dimensional´ (since we¯ need only verify it for 푉 “ 푘‘푛, where it is obvious). Given a map 푇 : 푉 Ñ 푊 , we obtain the dual (aliases: transpose or adjoint) map 푇 _ : 푊 _ Ñ 푉 _ by restriction. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 5

Exercise 1.2. For 푉 “ 푘‘푛 and 푊 “ 푘‘푚, show that the matrix of the dual transformation is the transpose of the matrix of the original linear transformation. 1.8. Let 푉 , 푊 , and 푈 be 푘-vector spaces. Definition 1.8.1. A function 퐵p´, ´q : 푉 ˆ 푊 Ñ 푈 is (푘-)bilinear if for every 푣 P 푉 the map 퐵p푣, ´q : 푊 Ñ 푈 is linear, and for every 푤 P 푊 the map 퐵p´, 푤q : 푉 Ñ 푈 is linear. Remark 1.8.2. We emphasize that bilinear maps are not linear (except in degenerate situations).

Proposition 1.8.3. For every pair of vector spaces 푉 and 푊 , there is a 푘-vector space 푉 b푘 푊 defined up to unique isomorphism and equipped with a bilinear pairing:

푉 ˆ 푊 Ñ 푉 b 푊 (1.4) 푘 such that every 푘-bilinear map 푉 ˆ 푊 Ñ 푈 factors uniquely as:

푉 ˆ 푊 / 푉 b 푊 푘

" 푈 | with the right map being 푘-linear.

‘퐼 Proof. Define the set 퐼 as 푉 ˆ 푊 . Define the vector space 푉 b푘 푊 as 푘 . For each 푖 “ p푣, 푤q P 퐼, we abuse notation in letting p푣, 푤q denote the element 휀푖p1q P 푉 b푘 푊 , i.e., the element of the direct sum that is 1 is the entry 푖 and 0 in all other entries.Č Define the subspace 퐾 of 푉 b푘 푊 to be spanned by vectors of theČ form: ‚ p푣1 ` 푣2, 푤q ´ p푣1, 푤q ´ p푣2, 푤q for 푣푖 P 푉 , 푤 P 푊 , 휆 P 푘 ‚ p푣, 푤1 ` 푤2q ´ p푣, 푤1q ´Č p푣, 푤2q for 푣 P 푉 , 푤푖 P 푊 , 휆 P 푘 ‚ p휆 ¨ 푣, 푤q ´ 휆 ¨ p푣, 푤q for 푣 P 푉 , 푤 P 푊 , 휆 P 푘 ‚ p푣, 휆 ¨ 푤q ´ 휆 ¨ p푣, 푤q for 푣 P 푉 , 푤 P 푊 , 휆 P 푘.

Define 푉 b푘 푊 to be the quotient of 푉 b푘 푊 by 퐾. We need to verify the universal property. By the universal property of direct sums, giving a linear transformation 퐵 : 푉 b푘 푊 Ñ 푈Čis equivalent to giving a function 퐵 : 푉 ˆ 푊 Ñ 푈: indeed, 퐵p푣, 푤q is the image of p푣, 푤q under 퐵. Moreover, by the universal property of quotients, this map will factor through 푉 br푘 푊Čif and only if 퐵 sends each element of 퐾 to 0 P 푈. It suffices to verify this for the spanning set of vectors definingr 퐾, where it translates to the relations: r 퐵p푣1 ` 푣2, 푤q ´ 퐵p푣1, 푤q ´ 퐵p푣2, 푤q “ 0

퐵p푣, 푤1 ` 푤2q ´ 퐵p푣, 푤1q ´ 퐵p푣, 푤2q “ 0 퐵p휆 ¨ 푣, 푤q ´ 휆 ¨ 퐵p푣, 푤q “ 0 퐵p푣, 휆 ¨ 푤q ´ 휆 ¨ 퐵p푣, 푤q “ 0 as relations in 푈. But these are exactly the conditions of bilinearity of the map 퐵.

Notation 1.8.4. Where the ﬁeld 푘 is unambiguous, we use the notation 푉 b 푊 “ 푉 b푘 푊 . Notation 1.8.5. For p푣, 푤q P 푉 ˆ 푊 , let 푣 b 푤 denote the corresponding vector in 푉 b 푊 , i.e., the image of p푣, 푤q under the map (1.4). 6 SAM RASKIN

Notation 1.8.6. For an integer 푛 ě 0, we let 푉 b푛 denote the 푛-fold tensor product of 푉 with itself, where for the case 푛 “ 0 we understand the empty tensor product as giving 푘. Exercise 1.3. (1) Show using the proof of Proposition 1.8.3 that vectors of the form 푣 b 푤 for 푣 P 푉 and 푤 P 푊 span 푉 b 푊 . (2) Show the same using the universal property of tensor products alone (hint: consider the span of the image of the map 푉 ˆ 푊 Ñ 푉 b푘 푊 and show that this subspace also satisﬁes the universal property of tensor products). (3) If dim 푉 ‰ 0, 1, show that not every vector in 푉 b 푊 is of the form 푣 b 푤. Remark 1.8.7. The construction appearing in the proof of Proposition 1.8.3 appears very complicated: we take a huge vector space (with a basis indexed by elements of 푉 ˆ 푊 ), and quotient by a huge subspace. However, things are not as bad as they look! For 푉 “ 푘‘푛 and 푊 “ 푘‘푚, we claim that there is a canonical identiﬁcation:

푉 b 푊 » 푘‘p푛푚q. In particular, the tensor product of finite-dimensional vector spaces is again finite-dimensional. 1 1 We denote the given bases of 푉 and 푊 by 푒1, . . . , 푒푛 and 푒1, . . . , 푒푚 respectively. It suffices to 1 1 show that the vectors 푒푖 b 푒푗 form a basis of 푉 b 푊 . Given 푣 “ 푖 휆푖푒푖 and 푤 “ 푗 휂푗푒푗, by bilinearity of 푉 ˆ 푊 Ñ 푉 b 푊 , we have: ř ř 1 푣 b 푤 “ 휆푖휂푗 ¨ 푒푖 b 푒푗. 푖,푗 ÿ 1 By Exercise 1.3, this shows that the vectors 푒푖 b 푒푗 span 푉 b 푊 . In particular, the dimension of 푉 b 푊 is ď 푛 ¨ 푚. ‘p푛푚q Denote the given basis of 푘 by 푓1, . . . , 푓푛¨푚. Define the bilinear map:

푉 ˆ 푊 Ñ 푘‘p푛푚q by the formula:

1 p 휆푖푒푖, 휂푗푒푗q ÞÑ 휆푖휂푗 ¨ 푓p푖´1q¨푚`푗. 푖 푗 푖,푗 ÿ ÿ ÿ (Note that as p푖, 푗q varies subject to the conditions 1 ď 푖 ď 푛, 1 ď 푗 ď 푚, the value p푖 ´ 1q ¨ 푚 ` 푗 obtains each integral value between 1 and 푛푚 exactly once). One immediately veriﬁes that this map actually is bilinear. The induced map:

푉 b 푊 Ñ 푘‘p푛푚q 1 sends 푒푖 b 푒푗 to 푓p푖´1q¨푚`푗, and therefore spans the target. Therefore, since dim 푉 b 푊 ď 푛 ¨ 푚, this map must be an isomorphism, as desired. As a corollary, one may try to think of tensor product of vector spaces as an animation of the usual multiplication of non-negative integers, since dimp푉 b 푊 q “ dimp푉 q ¨ dimp푊 q. Exercise 1.4. Show that the map:

Homp푉 b 푊, 푈q Ñ Homp푉, Homp푊, 푈qq

푇 ÞÑ 푣 ÞÑ 푤 ÞÑ 푇 p푣, 푤q ˆ ´ ¯˙ INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 7 is an isomorphism. Exercise 1.5. For 푉 and 푊 vector spaces, show that the map:

푉 _ b 푊 Ñ Homp푉, 푊 q 휆 b 푤 ÞÑ p푣 ÞÑ 휆p푣q ¨ 푤q is an isomorphism if 푉 is ﬁnite-dimensional. In the case when 푊 is also ﬁnite-dimensional, understand the relationship to the matrix representation of linear transformations in the presence of bases (hint: note that the right hand side is then a vector space of matrices). Exercise 1.6. Show that the map:

푉 _ b 푊 _ Ñ p푉 b 푊 q_ induced by the map:

휆 b 휂 ÞÑ 푣 b 푤 ÞÑ 휆p푣q ¨ 휂p푤q is an isomorphism for 푉 and 푊 ﬁnite-dimensional.´ ¯ 1.9. Suppose that 푉 is a vector space. Deﬁne the tensor algebra 푇 p푉 q of 푉 as the direct sum b푛 ‘푛PZě0 푉 . Note that 푇 p푉 q can indeed be made into a 푘-algebra in a natural way: the multiplication is given component-wise by the maps:

푉 b푛 b 푉 b푚 ÝÑ» 푉 bp푛`푚q. Proposition 1.9.1. Let 퐴 be a 푘-algebra. Restriction along the map 푉 Ñ 푇 p푉 q induces an isomorphism:

Hom 푇 푉 , 퐴 » Hom 푉, 퐴 . Alg{푘 p p q q ÝÑ 푘p q Here Hom indicates the set of maps of 푘-algebras. Alg{푘 Remark 1.9.2. This is a universal property of the tensor algebra, typically summarized by saying that 푇 p푉 q is the free associative algebra on the vector space 푉 . Proof. Given 푇 : 푉 Ñ 퐴 a map of vector spaces, we obtain a map 푉 b푛 Ñ 퐴 as the composition:

푇 ... 푇 푉 b ... b 푉 ÝÝÝÝÝÑb b 퐴 b . . . 퐴 Ñ 퐴 where the last map is induced by the 푛-fold multiplication on 퐴. b푛 One easily veriﬁes that the corresponding map 푇 p푉 q “ ‘푛푉 Ñ 퐴 is a map of 푘-algebras and provides an inverse to the restriction map.

1.10. Let 푛 ě 0 be an integer and let 푉 be a vector space. We consider quotients Sym푛 and Λ푛 of 푉 uniquely characterized by the following universal properties: 8 SAM RASKIN

‚ A map 푇 : 푉 b푛 Ñ 푊 factors as:

푉 b푛 푇 $ Sym푛p푉 q / 푊

if and only if 푇 is symmetric, i.e., for every collection 푣1, . . . , 푣푛 of vectors of 푉 and every permutation (i.e., bijection) 휎 : t1, . . . , 푛u ÝÑ» t1, . . . , 푛u, we have:

푇 p푣1 b ... b 푣푛q “ 푇 p푣휎p1q b ... b 푣휎p푛qq. ‚ A map 푇 : 푉 b푛 Ñ 푊 factors as:

푉 b푛 푇

# Λ푛p푉 q / 푊

if and only if 푇 is alternating, i.e., for every collection 푣1, . . . , 푣푛 of vectors of 푉 and every permutation (i.e., bijection) 휎 : t1, . . . , 푛u ÝÑ» t1, . . . , 푛u, we have:

푇 p푣1 b ... b 푣푛q “ sgnp휎q ¨ 푇 p푣휎p1q b ... b 푣휎p푛qq. Here sgnp휎q is the sign of the permutation 휎. Definition 1.10.1. We say that Sym푛p푉 q (resp. Λ푛p푉 q) is the 푛th symmetric power (resp. 푛th alternating power) of 푉 .

Notation 1.10.2. For 푣1, . . . , 푣푛 P 푉 , we let 푣1 . . . 푣푛 or 푣1 ¨ ... ¨ 푣푛 denote the corresponding element 푛 b푛 푛 of Sym p푉 q, i.e., the image of 푣1 b ... b 푣푛 under the structure map 푉 Ñ Sym p푉 q. We let 푛 푣1 ^ ... ^ 푣푛 denote the corresponding vector of Λ p푉 q. Note that e.g. for 푛 “ 2 we have 푣1푣2 “ 푣2푣1 and 푣1 ^ 푣2 “ ´푣2 ^ 푣1.

Exercise 1.7. Suppose that 푉 is finite-dimensional with basis 푒1, . . . , 푒푚. (1) Show that Sym푛p푉 q has a basis indexed by the set of non-decreasing functions 푓 : t1, . . . , 푛u Ñ t1, . . . , 푚u, where the corresponding basis vector is 푒푓p1q . . . 푒푓p푛q. 푛 ` 푚 ´ 1 Deduce that dimpSym푛p푉 qq is the binomial coefficient . 푚 (2) Show that Λ푛p푉 q has a basis indexed by the subsets 퐼 ofˆt1, . . . , 푚u ˙of order 푛, where the corresponding basis vector is 푒퐼 :“ ^푖P퐼 푒푖 (i.e., the iterated wedge product of the elements 푒푖 for 푖 P 퐼, where we write the wedge product in the increasing order according to size of 푖 P r1, 푛s). 푚 Deduce that dimpΛ푛p푉 qq is the binomial coefficient for 0 ď 푛 ď 푚, and dimension 푛 0 for 푛 ą 푚. ˆ ˙

푛 We deﬁne the symmetric algebra Symp푉 q as ‘푛ě0 Sym p푉 q. Note that Symp푉 q is a commutative algebra in an obvious way. We then have the following analogue of Proposition 1.9.1. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 9

Proposition 1.10.3. Let 퐴 be a commutative 푘-algebra. Restriction along the map 푉 Ñ Symp푉 q induces an isomorphism:

Hom Sym 푉 , 퐴 » Hom Sym 푉 , 퐴 . Alg{푘 p p q q ÝÑ 푘p p q q Remark 1.10.4. This is a universal property of the symmetric algebra, typically summarized by saying that Symp푉 q is the free commutative algebra on the vector space 푉 . Construction 1.10.5. Elements of Sym푛p푉 _q can be considered as 푘-valued functions on 푉 , where _ for 휆1, . . . , 휆푛 P 푉 the function on 푉 corresponding to 휆1 . . . 휆푛 is 푣 ÞÑ 휆1p푣q . . . 휆푛p푣q. Functions on 푉 arising in this manner are by deﬁnition polynomial functions of degree 푛 on 푉 . Sums of such functions are called polynomial functions on 푉 . 1.11. Bilinear forms. Let 푉 be a 푘-vector space. Definition 1.11.1. A (푘-valued) bilinear form on 푉 is a bilinear map 퐵 : 푉 ˆ 푉 Ñ 푘. Suppose that 푉 is ﬁnite-dimensional. Remark 1.11.2. Note that Homp푉 b 푉, 푘q “ p푉 b2q_ » p푉 _qb2, so that bilinear forms may be considered as elements of 푉 _,b2. Given a bilinear form 퐵, we obtain a second bilinear form 퐵푇 by setting 퐵푇 p푣, 푤q :“ 퐵p푤, 푣q. We refer to 퐵푇 as the transpose or adjoint bilinear form. Remark 1.11.3. By Exercise 1.5, we have an isomorphism:

푉 _,b2 ÝÑ» Homp푉, 푉 _q That is, bilinear forms are equivalent to linear transformations 푉 Ñ 푉 _. To normalize the con- ventions: under this dictionary, the bilinear form 퐵 maps to the linear transformation 푉 Ñ 푉 _ associating to a vector 푣 the linear functional 푤 ÞÑ 퐵p푣, 푤q. Exercise 1.8. Show that if 푇 : 푉 Ñ 푉 _ corresponds to a bilinear form 퐵, the linear transformation corresponding to 퐵푇 is the dual:

푇 _ : p푉 _q_ “ 푉 Ñ 푉 _ of 푇 . Definition 1.11.4. A bilinear form 퐵 is non-degenerate if the corresponding linear transformation 푉 Ñ 푉 _ is an isomorphism. Exercise 1.9. (1) Show that 퐵 is non-degenerate if and only if for every 0 ‰ 푣 P 푉 the corresponding functional 퐵p푣, ´q is non-zero, i.e., there exists 푤 P 푉 with 퐵p푣, 푤q ‰ 0. (2) Suppose that 푒1, . . . , 푒푛 is a basis for 푉 and that 퐵p푒푖, ´q is a non-zero functional for each 푖. Is 퐵 necessarily non-degenerate? (3) Show that 퐵 is non-degenerate if and only if 퐵푇 is. 1.12. Symmetric bilinear forms. We now focus on the special case where the bilinear form is symmetric. Definition 1.12.1. A bilinear form 퐵 is symmetric if 퐵 “ 퐵푇 . The following result records some equivalent perspectives on the symmetry conditions. Proposition 1.12.2. The following are equivalent: 10 SAM RASKIN

(1) The bilinear form 퐵 is symmetric. (2) 퐵p푣, 푤q “ 퐵p푤, 푣q for all 푣, 푤 P 푉 . (3) 퐵 P 푉 _,b2 is left invariant under the automorphism:

휆 b 휂 ÞÑ 휂 b 휆 of 푉 _,b2. (4) The linear transformation 푉 Ñ 푉 _ corresponding to 퐵 is self-dual.

2. Quadratic forms and spaces 2.1. Quadratic spaces. Let 푘 be a ﬁeld.

Definition 2.1.1. A quadratic space over 푘 is a pair p푉, 푞q where 푉 is a finite-dimensional 푘-vector space and 푞 P Sym2p푉 _q. Remark 2.1.2. Given a quadratic space p푉, 푞q, we obtain a function 푉 Ñ 푘 by Construction 1.10.5. By abuse of notation, we denote this function also by 푞. Note that 푞 satisfies the properties: ‚ 푞p휆푣q “ 휆2푞p푣q for 푣 P 푉 and 휆 P 푘. ‚ The map 푉 ˆ 푉 Ñ 푘 defined by p푣, 푤q ÞÑ 푞p푣 ` 푤q ´ 푞p푣q ´ 푞p푤q is bilinear. Definition 2.1.3. A function 푞 : 푉 Ñ 푘 satisfying the two conditions of Remark 2.1.2 is said to be a quadratic form on 푉 .

Exercise 2.1. If 푞 : 푉 Ñ 푘 is a quadratic form, show that 푞 arises from a unique quadratic space structure on 푉 .

Example 2.1.4. Suppose that p푉, 푞q is a quadratic space with a chosen basis 푒1, . . . , 푒푛 of 푉 . Letting _ 푥1, . . . , 푥푛 denote the corresponding dual basis of 푉 , we see that 푞 is an expression of the form:

2 푎푖,푖푥푖 ` 푎푖,푗푥푖푥푗 (2.1) 푖 푖ă푗 ÿ ÿ for 푎푖,푗 P 푘. The corresponding function 푞 : 푘푛 Ñ 푘 is given by substitution in the above expression:

휆1 . 2 ¨ . ˛ ÞÑ 푎푖,푖휆푖 ` 푎푖,푗휆푖휆푗. 푖 푖ă푗 휆푛 ˚ ‹ ÿ ÿ In this case, we will write 푞 as˝ a function‚ of 푛-variables in the usual way. Definition 2.1.5. A quadratic space structure on 푘푛 is a quadratic form in 푛 variables. That is, a quadratic form in 푛-variables is an expression (2.1). Terminology 2.1.6. When referencing quadratic forms without mentioning an ambient vector space, we will typically mean a quadratic form in some variables, i.e., a quadratic form on some 푘‘푛. Terminology 2.1.7. We follow classical usage is referring to a quadratic form in one (resp. two, resp. three, resp. four) variables as a unary (resp. binary, resp. ternary, resp. quaternary) form. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 11

Remark 2.1.8. By definition, a quadratic form is a quadratic space with chosen coordinates. His- torically, quadratic forms were the original objects of study. However, we will see that for some questions (e.g., representability by the form—see below), the perspective of quadratic spaces is much more flexible. In what follows, we will first develop the theory in the abstract setting of quadratic spaces, and then explain how to translate the questions of the theory into problems of the matrix calculus of linear algebra. This approach will not be ideal for all readers, some of whom will prefer to practice computing before developing the abstract (and simpler) setting. Such readers are advised to skip ahead to S2.6 and refer back as necessary. Remark 2.1.9. The definition of quadratic space allows to consider the case 푉 “ 0 (and 푞 “ 0). We refer to this as the zero quadratic space. This should not be confused with the zero quadratic form on a vector space 푉 , which is defined as 푞p푣q “ 0 for all 푣 P 푉 .

2.2. Morphisms of quadratic spaces. A morphism 푇 : p푉, 푞푉 q Ñ p푊, 푞푊 q of quadratic spaces is a linear transformation 푇 : 푉 Ñ 푊 such that 푞푊 p푇 p푣qq “ 푞푉 p푣q for every 푣 P 푉 . Remark 2.2.1. Such linear transformations are often called isometries in the literature (this phrase also sometimes refers more specifically to injective or bijective morphisms of quadratic spaces). An isomorphism of quadratic spaces is a morphism that is a morphism inducing an isomorphism 푉 ÝÑ» 푊 of vector spaces. We say that two quadratic forms in 푛-variables are equivalent if the corresponding quadratic spaces are isomorphic. 2.3. Operations with quadratic spaces. We give some basic constructions of new quadratic spaces from old: direct sum, restriction, and extension of scalars. Suppose that p푉, 푞푉 q and p푊, 푞푊 q are quadratic spaces. Then 푉 ‘푊 carries a canonical quadratic form 푞푉 ‘ 푞푊 defined by p푞푉 ‘ 푞푊 qp푣, 푤q “ 푞푉 p푣q ` 푞푊 p푤q. Next, suppose that 푇 : 푉 Ñ 푊 is a linear transformation and 푞푊 is a quadratic form on 푊 . Define 푞푉 by 푞푉 p푣q :“ 푞푊 p푇 p푣qq. Obviously 푞푉 is a quadratic form on 푉 ; we say that it is obtained by restriction and sometimes denote it by 푞푊 |푉 .

푞p푣 ` 푤q “ 푞p푣q ` 푞p푤q ` 퐵푞p푣, 푤q Finally, let p푉, 푞q be a quadratic space over 푘, and let 푘 ãÑ 푘1 be an embedding of fields. Then 1 1 1 푉 b푘 푘 is naturally a 푘 -vector space and 푞 extends to a quadratic form 푞푘1 on 푉 b푘 푘 . For example, ‘푛 1 if we fix a basis 푉 » 푘 of 푉 , then 푉 b푘 푘 acquires a natural basis, and the quadratic form (2.1) defines the “same” quadratic form, where we consider elements of 푘 as elements of 푘1 through the embedding.

Remark 2.3.1. It would be better to refer to p푉 ‘ 푊, 푞푉 ‘푊 q as the orthogonal direct sum, as in [Ser]. Indeed, morphisms p푉 ‘ 푊, 푞푉 ‘푊 q Ñ p푈, 푞푈 q are equivalent to giving a pair of morphisms 푇 : p푉, 푞푉 q Ñ p푈, 푞푈 q and 푆 : p푊, 푞푊 q Ñ p푈, 푞푈 q with the additional conditions that their images be orthogonal, i.e., 퐵푞푈 p푇 p푣q, 푆p푤qq “ 0 for all 푣 P 푉 and 푤 P 푊 . 2.4. Representability. Suppose that p푉, 푞q is a quadratic space. Definition 2.4.1. We say that 푞 represents 휆 P 푘 if there exists a vector 푣 P 푉 such that 푞p푣q “ 휆 (i.e., if 휆 is in the image of the function 푞). Remark 2.4.2. If 푞 represents 휆, then 푞 represents 휂2휆 for any 휂 P 푘: indeed, if 푞p푣q “ 휆 then 푞p휂푣q “ 휂2휆. Therefore, we may also speak about 푞 representing classes in 푘ˆ{p푘ˆq2. 12 SAM RASKIN

Example 2.4.3. Suppose that 푉 “ 푘 is the trivial one-dimensional vector space. Then by Example 2.1.4, 푞 must be of the form 푞p푥q “ 푎푥2 for some 푎 P 푘, and we assume 푎 ‰ 0 for conveninence. We see that 푞 represents 휆 if and only if 휆{푎 is a square in 푘. We specialize this to particular ﬁelds: (1) For example, if 푘 is algebraically closed, 푞 represents all values 휆 P 푘 so long as 푎 ‰ 0. (2) If 푘 “ R, 푞 represents 휆 if and only if 푎 and 휆 have the same sign. (3) If 푘 “ F푝, then 푞 represents 휆 if and only if 휆{푎 is a quadratic residue in F푝. Indeed, here “quadratic residue” simply means “is a square.” (4) Suppose 푘 “ Q. We introduce the following notation: for 푝 a prime number and 푛 ‰ 0 ě0 an integer, let 푣푝p푛q P Z be the order to which 푝 divides 푛 (alias: the valuation of 푛 at 푝), i.e., it is the unique integer such that 푛 P and 푛 is not. More generally, 푝푣푝p푛q Z 푝푣푝p푛q`1 푛 ˆ 2 for 푟 “ 푚 P Q , let 푣푝p푟q :“ 푣푝p푛q ´ 푣푝p푚q. (For example: 푣푝p푝q “ 1, 푣푝p푝 q “ 2, and 1 푣푝p 푝 q “ ´1). Then 푞 as above represents 휆 if and only if 푎 and 휆 have the same sign, and moreover, 푣푝p푎q ´ 푣푝p휆q is even for each prime 푝 (if this is confusing, it is instructive here to work out the case 푎 “ 1). We notice the following feature: 푞 represents a particular value if and only if certain conditions are veriﬁed at each prime 푝, and there is an additional condition involving R (here that 푎 and 휆 have the same sign). This is the most elementary instance of the Hasse principle. (5) In general, if 푎 and 휆 are non-zero (the case where either is zero being trivial), then 푞 obviously represents 휆 if and only if 푎 and 휆 project to the same element in the (2-torsion) 2 2 abelian group 푘ˆ{p푘ˆq . The above analysis amounts to saying that Rˆ{pRˆq “ Z{2Z 2 (according to sign), and Qˆ{pQˆq is isomorphic to the direct sum:

Z{2Z ‘ ‘ Z{2Z 푝 prime given by the sign homomorphism on the` ﬁrst factor˘ and the parity of the valuation at 푝 on the factor corresponding to a prime 푝. Exercise 2.2. This exercise is meant to ﬂesh out (3) above. ˆ ˆ 2 (1) Show that there is a (necessarily unique) isomorphism F푝 {pF푝 q » Z{2Z as long as 푝 ‰ 2. ˆ 푎 (2) For 푎 P F푝 , we use the Legendre symbol 푝 P t1, ´1u for the resulting map:

ˆ ˆ ˆ 2 ´ ¯ F푝 Ñ F푝 {pF푝 q » Z{2Z “ t1, ´1u. Here we use multiplicative notation for Z{2Z. ´ ˆ Show that 푝 is a homomorphism F푝 Ñ t1, ´1u. 푝´1 ˆ (3) Show that 2 ´ elements¯ of F푝 are quadratic residues and an equal number are not. 2 2 2 Exercise 2.3. Suppose that 푘 “ Q, 푉 “ 푘 and 푞p푥, 푦q “ 푥 ` 푦 . Show that 푞 represents 휆 P Z implies that 휆 ‰ 3 modulo 4. Of course, every quadratic form represents 0, since 푞p0q “ 0. However, the following condition is less trivial. Definition 2.4.4. An isotropic vector in p푉, 푞q is a nonzero 푣 P 푉 such that 푞p푣q “ 0. The quadratic form 푞 : 푉 Ñ 푘 is isotropic if there exists an isotropic vector in p푉, 푞q. We say that 푞 is anisotropic if it is not isotropic. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 13

Exercise 2.4. Let 푘 be a subfield of R (e.g., 푘 “ Q or 푘 “ R). 푛 2 (1) show that 푞p푥1, . . . , 푥푛q “ 푖“1 푥푖 is anisotropic. Show that for 0 ‰ 푎푖 P 푘 “ R, the form: ř 푛 2 푞p푥1, . . . , 푥푛q “ 푎푖푥푖 푖“1 ÿ is anisotropic if and only if all the 푎푖 have the same sign (i.e., are all positive or all negative). Show that for 푘 “ Q this condition is no longer sufficient. (2) More generally, we say that a form p푉, 푞q is positive-definite (resp. negative-definite) if 푞p푣q ą 0 (resp. 푞p푣q ă 0) for all 0 ‰ 푣 P 푉 b푘 R, and merely definite if it is either positive or negative definite.1 Show that any definite form is anisotropic. (3) Show that a form 푞p푥, 푦q “ 푎푥2 ` 푏푥푦 ` 푐푦2 is definite if and only if the discriminant ∆ “ 푏2 ´ 4푎푐 is negative. Exercise 2.5. Let 푘 be algebraically closed. Show that every quadratic space p푉, 푞q over 푘 with dimp푉 q ą 1 is isotropic. Note that the property of being isotropic is preserved under extension of scalars, although the property of being anisotropic is not necessarily. Suppose that 휆 P 푘 and 푞 : 푉 Ñ 푘 is a quadratic form. Define 푞휆 : 푉 ‘ 푘 Ñ 푘 by 푞휆p푣, 휂q “ 2 푞p푣q ´ 휆 ¨ 휂 : obviously 푞휆 is a quadratic form.

Proposition 2.4.5. If 푞 represents 휆, then 푞휆 is isotropic. Moreover, if 푞 is itself anisotropic, then this condition is necessary and sufficient. Proof. If 푞p푣q “ 휆, then:

푞휆p푣, 1q “ 푞p푣q ´ 휆 ¨ 1 “ 휆 ´ 휆 “ 0 so that 푞휆 is isotropic. Conversely, if 푞휆p푣, 휂q “ 0, then either 휂 “ 0, in which case 푞p푣q “ 0, or else:

´1 ´2 ´2 2 푞p휂 푣q “ 휂 푞p푣q “ 휂 푞휆p푣, 휂q ` 휆 ¨ 휂 q “ 휆 as desired. ` Remark 2.4.6. In this manner, representability questions can be reduced to the question of a form being isotropic, which can be easier to treat. We will develop these methods further below, proving in particular the strengthening Corollary 2.10.3 of Proposition 2.4.5.

2.5. By Remark 2.1.2, to every quadratic space p푉, 푞q there is an associated bilinear form 퐵푞:

퐵푞 : 푉 ˆ 푉 Ñ 푘

퐵푞p푣, 푤q :“ 푞p푣 ` 푤q ´ 푞p푣q ´ 푞p푤q. Obviously 퐵푞 is symmetric. Likewise, given 퐵 a bilinear form on 푉 , the function 푞퐵 deﬁned by 푞퐵p푣q :“ 퐵p푣, 푣q is a quadratic form, since:

2 2 푞퐵p휆푣q “ 퐵p휆푣, 휆푣q “ 휆 퐵p푣, 푣q “ 휆 푞퐵p푣q

1 We remark explicitly that definiteness depends only on the form obtained by extending scalars from 푘 to R. 14 SAM RASKIN and:

푞퐵p푣 ` 푤q ´ 푞퐵p푣q ´ 푞퐵p푤q “ 퐵p푣 ` 푤, 푣 ` 푤q ´ 퐵p푣, 푣q ´ 퐵p푤, 푤q “ 퐵p푣, 푣q ` 퐵p푣, 푤q ` 퐵p푤, 푣q ` 퐵p푤, 푤q ´ 퐵p푣, 푣q ´ 퐵p푤, 푤q “ 퐵p푣, 푤q ` 퐵p푤, 푣q “ p퐵 ` 퐵푇 qp푣, 푤q and the latter expression is manifestly bilinear as the sum of two bilinear forms. Observe that these constructions are almost, but not quite, inverses to each other: the quadratic form 푞퐵푞 (associated with the bilinear form associated with the quadratic from 푞) is 2 ¨ 푞:

푞퐵푞 p푣q “ 퐵푞p푣, 푣q “ 푞p푣 ` 푣q ´ 푞p푣q ´ 푞p푣q “ 푞p2 ¨ 푣q ´ 2 ¨ 푞p푣q “ 4푞p푣q ´ 2푞p푣q “ 2푞p푣q. 푇 Likewise, 퐵푞퐵 “ 2 ¨ 퐵 if 퐵 is symmetric bilinear (generally, it is 퐵 ` 퐵 , as we have already seen). Therefore, we deduce that there is a bijection between quadratic forms and symmetric bilinear forms as long as the characteristic of 푘 is not 2, with each of the maps above being bijections (though not quite mutually inverse bijections due to the factor of 2). Exercise 2.6. Show that the bilinear form associated with the quadratic form:

푛 2 푎푖,푖푥푖 ` 푎푖,푗푥푖푥푗 푖“1 푖ă푗 ÿ ÿ is given by the bilinear form:

휆1 휂1 푛 푛 . . ¨ . ˛ , ¨ . ˛ ÞÑ 2푎푖,푖휆푖휂푗 ` 푎푖,푗p휆푖휂푗 ` 휆푗휂푖q. ˜ ¸ 푖“1 푖ă푗 휆푛 휂푛 ˚ ‹ ˚ ‹ ÿ ÿ Deduce that the assignment˝ ‚ of˝ a symmetric‚ bilinear form to a quadratic form is neither injective nor surjective in characteristic 2.

Exercise 2.7. If 푇 : p푉, 푞푉 q Ñ p푊, 푞푊 q is a morphism of quadratic spaces, show that for every 푣1, 푣2 P 푉 we have:

퐵푞푉 p푣1, 푣2q “ 퐵푞푊 p푇 p푣1q, 푇 p푣2qq. Remark 2.5.1. The equivalence between quadratic forms and symmetric bilinear forms for charp푘q ‰ 2 may be considered as an instance of Maschke’s theorem from the representation theory of ﬁnite 2 groups, considering the representation of Z{2Z on 푉 b given by switching factors. 2.6. Matrices. Suppose that 푉 is equipped with a bilinear form 퐵 : 푉 b 푉 Ñ 푘 and a basis _ 푒1, . . . , 푒푛. As in S1.11, we associate to 퐵 a linear transformation 푉 Ñ 푉 , and using our basis we may view this as a linear transformation from 푘‘푛 to itself, i.e., as a matrix. This evidently deﬁnes a bijection between p푛 ˆ 푛q-matrices and bilinear forms on 푘‘푛. Exercise 2.8. (1) Show that the bilinear form associated to a p푛 ˆ 푛q-matrix 퐴 is given by:

p푣, 푤q ÞÑ 푤푇 퐴푣 where we consider 푣 and 푤 as column vectors, and the superscript 푇 indicates the transpose of a matrix (so 푤푇 is the row vector associated to 푤); note that the result of the above matrix multiplication is a p1 ˆ 1q-matrix, i.e., an element of 푘. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 15

(2) Show that the matrix associated to the bilinear form 퐵 above is:

퐵p푒1, 푒1q 퐵p푒2, 푒1q . . . 퐵p푒푛, 푒1q 퐵p푒1, 푒2q 퐵p푒2, 푒2q . . . 퐵p푒푛, 푒2q 퐴 “ ¨ . . . . ˛ . . . . . ˚ ‹ ˚퐵p푒1, 푒푛q 퐵p푒2, 푒푛q . . . 퐵p푒푛, 푒푛q‹ ˚ ‹ In particular, the bilinear˝ form is symmetric if and only if‚ its associated matrix is symmetric (meaning that it is equal to its own transpose).

Exercise 2.9. Suppose that 푓1, . . . , 푓푛 is a second choice of basis. Let 푆 be the (invertible) matrix ‘푛 whose 푖th column is the vector 푓푖 P 푘 (written in coordinates relative to the basis t푒푗u). Suppose 1 1 that we ﬁnd that matrix 퐴 of a bilinear form 퐵 with respect to the basis 푓1, . . . , 푓푛. Show that 퐴 is obtained from 퐴 as:

퐴1 “ 푆푇 퐴푆. Conclude that if charp푘q ‰ 2, then equivalence classes of quadratic spaces over 푘 are in bijection 푇 with the quotient set of symmetric matrices modulo the action of 퐺퐿푛p푘q by 푆 ¨ 퐴 :“ 푆 퐴푆. We will also speak of the matrix of a quadratic form, meaning the matrix of the associated symmetric bilinear form.

2.7. A remark regarding characteristic 2. Quadratic forms in characteristic 2 behave different from in other characteristics. In many respects, they can be regarded as pathological. We note that they do arise in nature as the reduction modulo 2 of integral quadratic forms, and for some purposes they cannot be avoided. However, for the purposes of these notes, we will not need the theory. Still, the author has opted to include some finer material regarding characteristic 2 than is needed. The reader may safely ignore all of it, ignoring in particular the difference between the radical and the reduced radical, and the difference between various notions of non-degeneracy. The only loss to the reader would be a few fun exercises.

2.8. Radical. Let p푉, 푞q be a quadratic space. We define the radical Radp푉 q of 푉 as the subset of vectors 푣 P 푉 such that 퐵푞p푣, 푤q “ 0 for every 푤 P 푉 . We define the reduced radical Radp푉 q as the subset of isotropic vectors in the radical of 푉 . Clearly Radp푉 q is a subspace, and one immediately finds that Radp푉 q is as well: indeed, if 푣, 푤 P Radp푉 q, then 푞p푣q “ 푞p푤q “ 0 and 퐵푞p푣, 푤q “ 0 so that 푞p푣`푤q “ 푞p푣q`푞p푤q`퐵푞p푣, 푤q “ 0. If charp푘q ‰ 2, then Radp푉 q “ Radp푉 q: indeed, for any 푣 P Radp푉 q we have 0 “ 퐵푞p푣, 푣q “ 2 ¨ 푞p푣q. Construction 2.8.1. For every 푣 P Radp푉 q and every 푤 P 푉 , we have:

푞p푣 ` 푤q “ 푞p푣q ` 푞p푤q ` 퐵푞p푣, 푤q “ 푞p푤q and therefore we obtain an induced quadratic form on the quotient space 푉 {Radp푉 q. Of course, more generally, we obtain such a form on the quotient by any subspace of the reduced radical. We deﬁne the rank rankp푞q of a quadratic space p푉, 푞q as dimp푉 q´dimpRadp푉 qq and the reduced rank of a quadratic space as dimp푉 q ´ dimpRadp푉 qq. 16 SAM RASKIN

Exercise 2.10. Let 푘 be a perfect2 ﬁeld of characteristic 2. Show that dimpRadp푉 qq ´ dimpRadp푉 qq must equal 0 or 1. 2.9. Non-degenerate quadratic forms. We say that p푉, 푞q is non-degenerate if Radp푉 q “ 0, and we say that p푉, 푞q is strongly non-degenerate if Radp푉 q “ 0 (equivalently: 퐵푞 is non-degenerate). These notions coincide if charp푘q ‰ 2. Remark 2.9.1. Unwinding the deﬁnitions: p푉, 푞q is strongly non-degenerate if for every 0 ‰ 푣 P 푉 there exists 푤 P 푉 with 퐵푞p푣, 푤q ‰ 0, and non-degenerate if for every 0 ‰ 푣 P 푉 , either there exists such a 푤 P 푉 or else 푞p푣q ‰ 0. Remark 2.9.2. Clearly a form is (resp. strongly) non-degenerate if and only if its rank (resp. reduced rank) is equal to its dimension. Example 2.9.3. The 1-variable quadratic form 푞p푥q “ 푎푥2 for 푎 P 푘 is non-degenerate in the above sense if and only if 푎 ‰ 0 (for 푘 of arbitrary characteristic). However, for charp푘q “ 2, this form is not strongly non-degenerate.

Exercise 2.11. Suppose that 푇 : p푉, 푞푉 q Ñ p푊, 푞푊 q is a morphism of quadratic spaces with p푉, 푞푉 q non-degenerate. Show that 푇 is injective. Exercise 2.12. Show that 푉 {Radp푉 q is non-degenerate when equipped with the quadratic space structure of Construction 2.8.1. Definition 2.9.4. Suppose that p푉, 푞q is a quadratic space and 푊 is a subspace of 푉 . The orthogonal K subspace 푊 to 푊 is the subspace of vectors 푣 P 푉 such that 퐵푞p푣, 푤q “ 0 for all 푤 P 푊 . Remark 2.9.5. If p푉, 푞q is strongly non-degenerate, then dim 푊 ` dim 푊 K “ dim 푉 , since 푊 K :“ Kerp푉 Ñ 푊 _q and the map 푉 Ñ 푊 _ is surjective by strong non-degeneracy. Remark 2.9.6. If charp푘q ‰ 2, then any vector in 푊 X 푊 K is isotropic.

Proposition 2.9.7. Suppose that p푉, 푞q is a quadratic space and 푊 Ď 푉 with p푊, 푞|푊 q strongly K non-degenerate. Then p푉, 푞q » p푊, 푞|푊 q ‘ p푊 , 푞|푊 K q. K p푊 , 푞|푊 K q is strongly non-degenerate if and only if p푉, 푞q is. K Proof. First, we claim that 푊 X 푊 “ 0. Suppose that 0 ‰ 푤 P 푊 . Then, because 푞|푊 is strongly 1 1 K non-degenerate, there exists 푤 P 푊 with 퐵푞p푤, 푤 q ‰ 0, meaning that 푤 R 푊 . 1 1 Next, for any 푣 P 푉 , we claim that there exists 푤 P 푊 with 퐵푞p푣, 푤 q “ 퐵푞p푤, 푤 q for every 1 푤 P 푊 . Indeed, 퐵푞p푣, ´q : 푊 Ñ 푘 is a linear functional, and therefore, by strong non-degeneracy of 푊 , it is of the form 퐵푞p푤, ´q for a unique choice of 푤 P 푊 . Observing that 푣 ´ 푤 is obviously in 푊 K, we obtain that 푉 “ 푊 ‘ 푊 K. We then note that for any 푤 P 푊 , 푤1 P 푊 K, we have:

1 1 1 1 푞p푤 ` 푤 q “ 푞p푤q ` 푞p푤 q ` 퐵푞p푤, 푤 q “ 푞p푤q ` 푞p푤 q so that the quadratic form 푞 is obtained as the direct sum of its restriction to these subspaces. The second part follows from Exercise 2.13 below. Exercise 2.13. Show that the direct sum of two quadratic spaces is non-degenerate if and only if each of the summands is.

2Recall that a field of characteristic 푝 is said to be perfect if the map 푥 ÞÑ 푥푝 is an isomorphism. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 17

The following lemma is useful for verifying non-degeneracy computationally. Lemma 2.9.8. Suppose that 푞 is a quadratic form in 푛-variables with associated bilinear form having matrix 퐴, as in S2.6. Then 푞 is strongly non-degenerate if and only if detp퐴q ‰ 0. Proof. Indeed, for 푉 “ 푘푛, 퐴 is the matrix associated to the map 푘푛 “ 푉 Ñ 푉 _ “ 푘푛, so it is an equivalence if and only if its determinant is non-zero. Remark 2.9.9. Some authors use the term “regular” instead of non-degenerate. Remark 2.9.10. The reader may safely skip this remark. A more sophisticated variant of the above in characteristic 2: we say that p푉, 푞q is geometrically 1 1 non-degenerate if Radp푉 b푘 푘 q “ 0 for every field extension 푘 ãÑ 푘 . Note that formation of the usual radical Rad commutes with extension of scalars, so this definition still gives the same answer in characteristic not equal to 2. Note that by Exercise 2.10 this implies that the dimension of the radical is ď 1. We note that if 휆 P 푘 is not a square, then 푥2 `휆푦2 is an example of a non-degenerate but not geometrically non-degenerate quadratic form. The reader may take as an exercise to show that geometric non-degeneracy is equivalent to strong non-degeneracy if dimp푉 q is even, and equivalent to dimpRadq “ 1 if dimp푉 q is odd. We remark that geometric non-degeneracy is equivalent to algebro-geometric conditions: essentially just smoothness of the associated (projective) quadric hypersurface 푞 “ 0. 2.10. The hyperbolic plane. The most important example of an isotropic quadratic space is the hyperbolic plane.3 By definition, this is the quadratic space 푘‘2 with the quadratic form 푞p푥, 푦q “ 푥푦. We denote the hyperbolic plane as p퐻, 푞퐻 q. The matrix for the associated symmetric bilinear form is:

0 1 . 1 0 ˆ ˙ Proposition 2.10.1. For every non-degenerate isotropic quadratic space p푉, 푞q is isomorphic to 1 1 1 1 1 1 p푉 , 푞 q ‘ p퐻, 푞퐻 q for an appropriate choice of p푉 , 푞 q. The quadratic space p푉 , 푞 q is necessarily non-degenerate as well. Proof. Let 푣 be an isotropic vector in 푉 . We claim that there is an isotropic vector 푤 P 푉 such that 퐵푞p푣, 푤q “ 1. Indeed, by non-degeneracy of 푞, there exists 푤0 P 푉 with 퐵푞p푣, 푤0q ‰ 0. Therefore, we can deﬁne:

1 푞p푤0q 푤 :“ ¨ 푤0 ´ ¨ 푣 . 퐵푞p푣, 푤0q 퐵푞p푣, 푤0q We then readily verify: ` ˘

1 푞p푤0q 1 푞p푤0q 푞p푤q “ 2 ¨ 푞p푤0 ´ ¨ 푣q “ 2 ¨ 푞p푤0q ´ 퐵푞p 푣, 푤0q “ 0 퐵푞p푣, 푤0q 퐵푞p푣, 푤0q 퐵푞p푣, 푤0q 퐵푞p푣, 푤0q and: ` ˘

3We note that there’s no substantive relationship here to the hyperbolic plane in geometry. Rather, 퐻 is a plane because it is 2-dimensional, and hyperbolic because its non-zero level sets are hyperbolas. 18 SAM RASKIN

1 푞p푤0q 1 퐵푞p푣, 푤q “ ¨ 퐵푞p푣, 푤0 ´ 푣q “ 퐵푞p푣, 푤0q “ 1 퐵푞p푣, 푤0q 퐵p푣, 푤0q 퐵푞p푣, 푤0q as desired. This gives an embedding of the hyperbolic plane into p푉, 푞q, and then we obtain the result by Proposition 2.9.7. Corollary 2.10.2. If 푞 is non-degenerate and isotropic, then every 휆 P 푘 is represented by 푞. Proof. This follows from Proposition 2.10.1 upon noting that the hyperbolic plane represents every 휆 P 푘.

Corollary 2.10.3. A non-degenerate quadratic space p푉, 푞q represents 휆 P 푘 if and only if p푉 ‘푘, 푞휆q 2 is isotropic (recall that 푞휆p푥, 휂q :“ 푞p푥q ´ 휆휂 ). Proof. The case that p푉, 푞q is anisotropic is treated in Proposition 2.4.5. If p푉, 푞q is isotropic, then by Corollary 2.10.2 p푉, 푞q represents every value. Exercise 2.14. Suppose in (slightly) more generality that p푉, 푞q is an isotropic non-degenerate quadratic space. Does p푉, 푞q necessarily contain a copy of the hyperbolic plane? 2.11. Diagonalization. Let p푉, 푞q be a quadratic space.

Definition 2.11.1. A diagonalization of p푉, 푞q is a decomposition p푉, 푞q “ p푉1, 푞1q ‘ ... ‘ p푉푛, 푞푛q with dimp푉푖q “ 1 for all 푖. Remark 2.11.2. In terms of quadratic forms in 푛-variables, a diagonalization of a quadratic form 1 푞p푥1, . . . , 푥푛q is an equivalence of 푞 with a form 푞 of the type:

푛 1 2 푞 p푦1, . . . , 푦푛q “ 푎푖푦푖 . 푖“1 ÿ Proposition 2.11.3. Let p푉, 푞q be a quadratic space over 푘 with charp푘q ‰ 2. Then there exists a diagonalization of p푉, 푞q. Proof. First, suppose that 푉 is non-degenerate. We proceed by induction: the result is tautological if dimp푉 q “ 0. If dimp푉 q ą 0 there exists 푣 P 푉 with 푞p푣q ‰ 0, since otherwise 푞 is identically zero and therefore 퐵푞 could not be non-degenerate. Let 푊 Ď 푉 be the subspace 푘 ¨ 푣 spanned by 푣. Then 푊 K X 푊 “ 0 by assumption on 푣, and we obviously have a decomposition direct sum decomposition 푉 “ 푊 ‘ 푊 K. Moreover, for 푤 P 푊 and 푤1 P 푊 K, we compute:

1 1 1 1 푞p푤 ` 푤 q “ 푞p푤q ` 푞p푤 q ` 퐵푞p푤, 푤 q “ 푞p푤q ` 푞p푤 q and therefore we obtain:

K 푉 “ p푊, 푞|푊 q ‘ p푊 , 푞|푊 K q and obtain the result in this case by induction. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 19

1 1 1 In general, let 푉 Ď 푉 be a subspace so that Radp푉 q‘푉 » 푉 . Note that p푉 , 푞|푉 1 q maps isomor- phically onto the quotient 푉 { Radp푉 q equipped with its canonical quadratic space structure, and 1 therefore is non-degenerate by Exercise 2.12. Moreover, we see that p푉, 푞q » p푉 , 푞|푉 1 q‘pRadp푉 q, 0q, each of which can be diagonalized. Corollary 2.11.4. Over an algebraically closed field 푘 with charp푘q ‰ 2, all quadratic spaces of the same rank and dimension are equivalent. Exercise 2.15. Show that the hyperbolic plane cannot be diagonalized in characteristic 2. Deduce that Proposition 2.11.3 fails for strongly non-degenerate quadratic spaces in characteristic 2. Where does the argument in Proposition 2.11.3 fail? Exercise 2.16. Deduce Corollary 2.11.4 from Proposition 2.10.1 and Exercise 2.5. 2.12. Transposes. Let p푉, 푞q be a strongly non-degenerate quadratic space and let 푆 : 푉 Ñ 푉 be any linear transformation (not necessarily a morphism of quadratic spaces). We deﬁne the 푞-transpose 푆푇푞 : 푉 Ñ 푉 of 푆 as the unique linear transformation such that the diagram:

푆푇푞 푉 / 푉

» » (2.2) 푆_ 푉 _ / 푉 _.

푇푞 commutes. Here the vertical maps are deﬁned by 퐵푞. Note that 푆 is uniquely deﬁned because these vertical maps are isomorphisms by the assumption of strong non-degeneracy.

Proposition 2.12.1. 푆푇푞 is the unique linear transformation 푉 Ñ 푉 satisfying:

푇푞 퐵푞p푆 p푣q, 푤q “ 퐵푞p푣, 푆p푤qq for all 푣, 푤 P 푉 . Proof. The commutation of the diagram (2.2) says that for every 푣 P 푉 , two certain functionals 푉 Ñ 푘 should be equal. Evaluating these two functionals on 푤 P 푉 , one ﬁnds that the bottom leg 푇푞 of the diagram gives 퐵푞p푣, 푆p푤qq, while the top leg of the diagram gives 퐵푞p푆 p푣q, 푤q. Lemma 2.12.2. In the above setting, detp푆q “ detp푆푇푞 q. Proof. Immediate from the commutation of the diagram (2.3), since this gives detp푆푇푞 q “ detp푆_q, and detp푆_q “ detp푆q. 푛 Exercise 2.17. Suppose that 푉 “ 푘 , so that 퐵푞 is given by a symmetric matrix 퐴 (that is invertible by assumption) and 푆 can be considered as a matrix. Show that the matrix 푆푇푞 is computed as:

푆푇푞 “ 퐴´1푆푇 퐴. 푛 2 Deduce that the 푞-transpose associated with the quadratic form 푖“1 푥푖 is the usual transpose. ř 20 SAM RASKIN

2.13. Orthogonal groups. Let p푉, 푞q be a quadratic space. We deﬁne 푂p푞q as the group of auto- morphisms of the quadratic space p푉, 푞q, i.e., elements of 푂p푞q are isomorphisms p푉, 푞q ÝÑ» p푉, 푞q with multiplication in 푂p푞q deﬁned by composition. Note that 푂p푞q acts on 푉 .4

Example 2.13.1. Let 푣 P 푉 with 푞p푣q ‰ 0. Deﬁne 푠푣 P 푂p푞q the reflection through 푣 by the formula:

퐵푞p푣, 푤q 푠푣p푤q “ 푤 ´ ¨ 푣. 푞p푣q Clearly 푠푣 is a linear transformation. To see that it lies in 푂p푞q, we compute:

퐵푞p푣, 푤q 퐵푞p푣, 푤q 2 퐵푞p푣, 푤q 푞p푠푣p푤qq “ 푞p푤 ´ ¨ 푣q “ 푞p푤q ` p q 푞p푣q ´ 퐵푞p푣, 푤q “ 푞p푤q. 푞p푣q 푞p푣q 푞p푣q

Note that 푠푣p푣q “ ´푣, and if 퐵푞p푣, 푤q “ 0 then 푠푣p푤q “ 0. Therefore, if charp푘q ‰ 2 we see that 푠푣 is a non-identity linear transformation with eigenvalue -1 of multiplicity 1 and eigenvalue 1 of multiplicity dimp푉 q ´ 1. Lemma 2.13.2. Let p푉, 푞q be a strongly non-degenerate quadratic space. 푇푞 (1) A linear transformation 푔 : 푉 Ñ 푉 lies in 푂p푞q satisfies 푔 푔 “ id푉 . If charp푘q ‰ 2, then the converse holds as well. (2) For every 푔 P 푂p푞q, detp푔q “ ˘1. Proof. For (1), we use Proposition 2.12.1 to compute:

푇푞 퐵푞p푔 푔p푣q, 푤q “ 퐵푞p푔p푣q, 푔p푤qq.

푇푞 We see that the identity 퐵푞p푔p푣q, 푔p푤qq “ 퐵푞p푣, 푤q is equivalent to 푔 푔 “ id푉 , giving the desired result. For (2), we compute:

푇푞 푇푞 2 1 “ detpid푉 q “ detp푔 푔qq “ detp푔 q ¨ detp푔q “ detp푔q using Lemma 2.12.2. det Remark 2.13.3. By Example 2.13.1, if charp푘q ‰ 2 then the map 푂p푞q ÝÑ t1, ´1u is surjective. We let 푆푂p푞q denote its kernel, the special orthogonal group. Exercise 2.18. The purpose of this exercise is to show that Lemma 2.13.2 (2) holds even when 푞 is merely assumed to be non-degenerate. Let p푉, 푞q be a quadratic space. (1) Show that any 푔 P 푂p푞q ﬁxes Radp푉 q and Radp푉 q. 5 (2) Suppose that detp푔|Radp푉 qq P t1, ´1u. Show that detp푔q P t1, ´1u. (3) Suppose that p푉, 푞q is a non-degenerate quadratic space with 푉 “ Radp푉 q. Show that 푂p푞q consists only of the identity element. (4) Deduce that (2) holds for 푞 non-degenerate.

4In fact, 푂p푞q acts in a way that commutes appropriately with addition and scaling operations in 푉 , i.e., its action on 푉 arises through a homomorphism 푂p푞q Ñ 퐺퐿p푉 q. In this situation, one says that 푉 is a representation of 푂p푞q. 5The notation is a bit lazy. For clarity: in characteristic 2 we regard t1, ´1u as consisting of one element. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 21

2.14. Spheres. Let p푉, 푞q be a quadratic space and ﬁx 푟 P 푘. We deﬁne the associated sphere S푟p푞q as t푥 P 푉 | 푞p푥q “ 푟u. 푛`1 2 ? Example 2.14.1. Let 푘 “ R and let 푞 “ 푖“1 푥푖 . Then S푟p푞q is the usual 푛-sphere of radius 푟.

Note that the action of 푂p푞q on 푉 preservesř S푟p푞q. Proposition 2.14.2. Let p푉, 푞q be a non-degenerate quadratic space and suppose charp푘q ‰ 2. Then for every 푟 ‰ 0, the action of 푂p푞q on S푟p푞q is transitive.

Proof. Let 푣, 푤 P S푟p푞q. We claim that either 푣 ` 푤 or 푣 ´ 푤 is not isotropic. Indeed, we otherwise have:

0 “ 푞p푣 ` 푤q ´ 푞p푣 ´ 푤q “ 푞p푣q ` 푞p푤q ` 퐵푞p푣, 푤q ´ 푞p푣q ´ 푞p푤q ` 퐵푞p푣, 푤q “ 2퐵푞p푣, 푤q so that 퐵푞p푣, 푤q “ 0. Then 푞p푣 ` 푤q “ 푞p푣q ` 푞p푤q “ 2푟 ‰ 0. Suppose that 푣 ´ 푤 is not isotropic. Then we can make sense of the reﬂection 푠푣´푤 P 푂p푞q. We claim that 푠푣´푤p푣q “ 푤. First, note that 퐵푞p푣 ` 푤, 푣 ´ 푤q “ 0 since:

퐵푞p푣 ` 푤, 푣 ´ 푤q “ 2푞p푣q ´ 2푞p푤q ` 퐵푞p푣, 푤q ´ 퐵푞p푣, 푤q “ 0. Therefore, we have:

푣 ` 푤 “ 푠푣´푤p푣 ` 푤q “ 푠푣´푤p푣q ` 푠푣´푤p푤q

´푣 ` 푤 “ 푠푣´푤p푣 ´ 푤q “ 푠푣´푤p푣q ´ 푠푣´푤p푤q.

Adding these equations, we obtain 푠푣´푤p푣q “ 푤 as desired. Otherwise, 푣 ` 푤 is anisotropic, and the above gives that 푠푣`푤p푣q “ ´푤; composing 푠푣`푤 with ´ id푉 P 푂p푞q, we obtain the result. Exercise 2.19. In the above setting, suppose that rankp푞q ą 1. Does 푆푂p푞q act transitively on S푟p푞q? 2.15. Witt’s cancellation theorem. Suppose that we have an identiﬁcation:

p푉, 푞푉 q ‘ p푊1, 푞푊1 q » p푉, 푞푉 q ‘ p푊2, 푞푊2 q of quadratic spaces. What can we deduce about the relationship between p푊1, 푞푊1 q and p푊2, 푞푊2 q? Witt’s theorem addresses this problem completely.

Theorem 2.15.1 (Witt). Suppose that charp푘q ‰ 2. Then in the above situation, p푊1, 푞푊1 q and p푊2, 푞푊2 q are isomorphic. Proof. We proceed by steps.

Step 1. First, we reduce to the case that 푞푉 is non-degenerate. Indeed, choosing a complementary subspace in 푉 to Radp푉 q, we see as in the proof of Proposition 2.11.3 that 푉 is the direct sum of its radical and a non-degenerate quadratic space, giving the reduction.

Step 2. Next, we reduce to the case where 푞푊1 and 푞푊2 are non-degenerate. One immediately ﬁnds (supposing that 푞푉 is non-degenerate) that Radp푉 ‘푊1q “ Radp푊1q and similarly for 푊2. Therefore, quotienting out by the radical of 푉 ‘ 푊1 “ 푉 ‘ 푊2, we ﬁnd that: 22 SAM RASKIN

p푉, 푞푉 q ‘ p푊1{ Radp푊1q, 푞푊1{ Radp푊1qq » p푉, 푞푉 q ‘ p푊2{ Radp푊2q, 푞푊2{ Radp푊2qq Therefore, if we know the result for triples of non-degenerate spaces, we can deduce that:

p푊1{ Radp푊1q, 푞푊1{ Radp푊1qq » p푊2{ Radp푊2q, 푞푊2{ Radp푊2qq.

Since e.g p푊1, 푞푊1 q is the direct sum of its radical and its quotient by its radical, and since the radicals of 푊1 and 푊2 have been identiﬁed, we obtain the result in the general case. Step 3. Choose 푣 P 푉 with 푞p푣q ‰ 0, and let 푉 1 be the orthogonal complement to 푘 ¨ 푣, so that 푉 “ 푉 1 ‘ spanp푣q as a quadratic space. » Under the isomorphism 푇 : 푉 ‘푊1 ÝÑ 푉 ‘푊2, we have 푞푉 ‘푊2 p푇 p푣, 0qq “ 푞푉 p푣q “ 푞푉 ‘푊2 p푣, 0q.

Therefore, by Proposition 2.14.2, there is an automorphism of p푉 ‘ 푊2, 푞푉 ‘푊2 q taking 푇 p푣q to p푣, 0q. This automorphism induces an equivalence between the orthogonal complement to (the span of) 1 푇 p푣q and p푣, 0q. The latter is obviously 푉 ‘ 푊2. Using 푇 , we see that the former is isomorphic to 1 푉 ‘ 푊1. We now obtain the result by induction.

Corollary 2.15.2 (Sylvester’s theorem). Every quadratic form 푞p푥1, . . . , 푥푛q over R of rank 푟 is 푗 2 푟 2 equivalent to 푖“1 푥푖 ´ 푖“푗`1 푥푖 for a unique choice of 1 ď 푗 ď 푟. Proof. Diagonalizingř 푞,ř it is clear that 푞 is equivalent to such a form. Uniqueness then follows by induction from Witt’s theorem. 푗 2 푟 2 Definition 2.15.3. For 푞 a quadratic form over R equivalent to 푖“1 푥푖 ´ 푖“푗`1 푥푖 , the pair p푗, 푟 ´ 푗q P 2 is called the signature of 푞. Z ř ř 2.16. Tensor products. We suppose in S2.16-2.18 that charp푘q ‰ 2.

Construction 2.16.1. Suppose that p푉, 푞푉 q and p푊, 푞푊 q are quadratic spaces. Recall that 푞 1 “ 2 퐵푞푉 푞푉 , and similarly for 푞푊 . We deﬁne a quadratic space structure 푞푉 b푊 on 푉 b 푊 by associating it to the bilinear form:

푉 b 푊 Ñ p푉 b 푊 q_ “ 푉 _ b 푊 _ _ _ 1 1 obtained by tensor product of the maps 푉 Ñ 푉 and 푊 Ñ 푊 deﬁned by 2 퐵푞푉 and 2 퐵푞푊 .

Exercise 2.20. Show that the bilinear form 퐵푞푉 b푊 satisﬁes:

1 퐵 p푣 b 푤 , 푣 b 푤 q “ 퐵 p푣 , 푣 q퐵 p푤 , 푤 q. 푞푉 b푊 1 1 2 2 2 푞푉 1 2 푞푊 1 2

Deduce that 푞푉 b푊 p푣 b 푤q “ 푞푉 p푣q푞푊 p푤q. 2 Exercise 2.21. Show that p푉, 푞푉 q b p푘, 푞p푥q :“ 푥 q » p푉, 푞푉 q. Exercise 2.22. Go to the Wikipedia page for “Kronecker product” and ﬁgure out the relation to this construction. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 23

2.17. Alternating and symmetric powers. Let 푛 ě 0 be an integer and let p푉, 푞푉 q be a quadratic space. We claim that there are natural quadratic space structures on Sym푛p푉 q and Λ푛p푉 q. (We remind that we have assumed that charp푘q ‰ 2). First, note that (in any characteristic) we have canonical “norm” maps:

Sym푛p푉 q Ñ 푉 b푛 and Λ푛p푉 q Ñ 푉 b푛. The former is induced by the (unnormalized) symmetrization map:

푉 b푛 Ñ 푉 b푛

푣1 b ... b 푣푛 ÞÑ 푣휎p1q b ... b 푣휎p푛q 휎P푆 ÿ푛 and the latter is induced similarly by:

푉 b푛 Ñ 푉 b푛

푣1 b ... b 푣푛 ÞÑ sgnp휎q ¨ 푣휎p1q b ... b 푣휎p푛q. 휎P푆 ÿ푛 We remark that each composition:

Sym푛p푉 q Ñ 푉 b푛 Ñ Sym푛p푉 q Λ푛p푉 q Ñ 푉 b푛 Ñ Λ푛p푉 q is multiplication by 푛! “ |푆푛|. Exercise 2.23. What is the relationship between the above constructions and the connection between symmetric bilinear forms and quadratic forms?

푛 푛 We now obtain a quadratic form 푞Sym푛p푉 q (resp. 푞Λ푛p푉 q) on Sym p푉 q and Λ p푉 q by restriction of 푞푉 b푛 (as deﬁned by Construction 2.16.1) along these homomorphisms. The following important exercise gives the interaction between symmetric and alternating powers and duality, and then describes the relation to the above constructions. Exercise 2.24. The following exercises (except (3)) work for both Sym푛 and Λ푛: we write them for Sym푛 for deﬁniteness. (1) Show that the map Sym푛p푉 q_ Ñ 푉 b,_ dual to the structure map 푉 b푛 Ñ Sym푛p푉 q is injective and has image the subspace of functionals on 푉 b푛 invariant under the action of the symmetric group. (2) Show that the norm map Sym푛p푉 _q Ñ 푉 _,b푛 factors through the subspace Sym푛p푉 q_. (3) Show that the maps Sym푛p푉 _q Ñ Sym푛p푉 q_ are isomorphisms if and only if charp푘q “ 0 or 푛 ă charp푘q, but that the maps Λ푛p푉 _q Ñ Λ푛p푉 q_ are always isomorphisms whenever charp푘q ‰ 2. (4) Suppose now that 푉 Ñ 푉 _ is a symmetric bilinear form. Consider the induced map:

Sym푛p푉 q Ñ Sym푛p푉 q_ (2.3) deﬁned as the composition:

Sym푛p푉 q Ñ Sym푛p푉 _q Ñ Sym푛p푉 q_ 24 SAM RASKIN

with the ﬁrst map given by functoriality6 of Sym푛 and the second map is the one constructed in (2) above. Show that if 푞푉 is the induced quadratic form on 푉 , then 푞Sym푛p푉 q is induced by (2.3).

2.18. The discriminant. Suppose that p푉, 푞푉 q is a quadratic space and charp푘q ‰ 2. Recall that detp푉 q :“ Λdimp푉 qp푉 q is 1-dimensional. By S2.17, we obtain a quadratic space structure on this 1-dimensional vector space. Lemma 2.18.1. The quadratic form on detp푉 q is non-zero if and only if p푉, 푞q is non-degenerate. _ Proof. Let 퐵 : 푉 Ñ 푉 be defined by the symmetric bilinear structure on 푉 giving rise to 푞푉 . By Exercise 2.24, the induced map detp퐵q : detp푉 q Ñ detp푉 _q “ detp푉 q_ is the symmetric bilinear 1 form giving rise to 푞detp푉 q. But 퐵 is an isomorphism if and only if detp퐵q is, and since 퐵 “ 2 퐵푞 we obtain the result. Suppose now that p푉, 푞q is a non-degenerate quadratic space. By Lemma 2.18.1, we obtain that pdetp푉 q, 푞detp푉 qq is a non-degenerate 1-dimensional quadratic space. Obviously isomorphism classes of non-degenerate 1-dimensional quadratic forms are in bijection with the set 푘ˆ{p푘ˆq2, where the quadratic form 푞p푥q “ 푎푥2, 푎 ‰ 0 has invariant the class of 푎 in 푘ˆ{p푘ˆq2. Definition 2.18.2. For p푉, 푞q non-degenerate as above, the discriminant discp푞q P 푘ˆ{p푘ˆq2 of 푉 is the invariant of pdetp푉 q, 푞detp푉 qq as defined above. Now let us explain how to concretely compute the discriminant. Proposition 2.18.3. Suppose that 퐴 is an invertible symmetric p푛 ˆ 푛q-matrix and let 푞 be the associated quadratic form 푞p푥q “ 푥푇 퐴푥 in 푛-variables. Then discp푞q “ detp퐴q modp푘ˆq2. Proof. By construction the map detp푉 q Ñ detp푉 q_ defined by 푞 is given as multiplication by detp퐴q. Therefore, the induced quadratic form in 1-variable is 푥 ÞÑ detp퐴q ¨ 푥2 as desired. Remark 2.18.4. Recall from Exercise 2.9 that matrices 퐴 and 퐴1 define equivalent quadratic forms if and only if 퐴1 “ 푆푇 퐴푆, in which case:

detp퐴1q “ detp푆푇 퐴푆q “ detp푆푇 q detp퐴q detp푆q “ detp푆q2 detp퐴q showing directly that detp퐴q considered modulo p푘ˆq2 is a well-deﬁned invariant of the underlying quadratic space. Remark 2.18.5. Suppose that 푞 has been diagonalized and we realize 푞 as equivalent to a form 푛 2 ˆ 2 푖“1 푎푖푥푖 . We see that discp푞q “ 푎푖 modp푘 q . 2 2 řExercise 2.25. Let 푘 “ F푝 with 푝 ś‰ 2 and let 푞1 and 푞2 be the binary forms 푞1p푥, 푦q “ 푥 ` 푦 and 2 2 ˆ 푟 푞2p푥, 푦q “ 푥 ` 푟푦 for 푟 P F푝 a quadratic non-residue (i.e., 푝 “ ´1). (1) Show that each of 푞1 and 푞2 represents every element´ of¯ F푝. (2) Show that 푞1 and 푞2 are inequivalent quadratic forms.

6Functorial is a scientific word that here refers to the obvious푛 mapSym p푉 q Ñ Sym푛p푊 q induced by a given linear transformation 푇 : 푉 Ñ 푊 (similarly, Λ푛 is functorial.) INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 25

Exercise 2.26. Use discriminants to prove Witt’s theorem Theorem 2.15.1 in the case that p푉, 푞q is non-degenerate and dim 푊1 “ dim 푊2 “ 1. Exercise 2.27. (1) Show that the discriminant of the hyperbolic plane is ´1. (2) Show that if p푉, 푞q is a non-degenerate quadratic space with dimp푉 q “ 2 and discp푞q “ ´1, then 푉 is equivalent to the hyperbolic plane.

3. 푝-adic fields 2 3.1. Recall from S2.4 that 1-dimensional quadratic forms over Q are of the form 푞p푥q “ 푎푥 , 푎 P Q, where the equivalence class of 푞 is determined by the sign of 푎 and the parity of 푣푝p푎q for each prime 푝, where 푣푝p푎q measures the degree to which 푝 divides 푎. We want to formulate this information in a more systematic way so that we can generalize it to forms 푞 of higher rank. To generalize the sign of 푎, we use 푞R the extension of scalars of 푞 along Q ãÑ R. For each prime 푝, we will introduce the ﬁeld Q푝 Ą Q of 푝-adic numbers such that the extension of scalars 푞Q푝 generalizes 푣푝p푎q from the rank 1 case.

3.2. We will give several perspectives on Q푝 below: as a metric completion of Q, as a (field of 푛 fractions of) a kind of limit of the rings Z{푝 Z, and as infinite base 푝 exansions of numbers. 3.3. Analytic approach. Recall that for 푎 P Qˆ, we have defined in S2.4 the 푝-adic valuation 푣푝p푎q 푛 푣푝p푎q as the unique integer such that 푎 “ 푝 푚 with 푛 and 푚 not divisible by 푝. We extend this definition to Q by setting 푣푝p0q “ 8.

Lemma 3.3.1. (1) For 푎, 푏 P Q, we have 푣푝p푎푏q “ 푣푝p푎q ` 푣푝p푏q. (2) For 푎, 푏 P Q, we have mint푣푝p푎q, 푣푝p푏qu ď 푣푝p푎 ` 푏q. 푣 푎 푣 푏 min 푎,푏 Proof. The ﬁrst part is obvious. If 푎, 푏 P Z with 푎 “ 푝 푝p q푎1, 푏 “ 푝 푝p q푏1, then clearly 푝 t u divides 푎 ` 푏 giving the second part in this case. We can deduce it in general by either the same method, or by clearing denominators and applying the ﬁrst part.

Example 3.3.2. We have 푣푝 1 ` p푝 ´ 1q “ 1 ą 0 “ maxt푣푝p1q, 푣푝p푝 ´ 1qu, giving an example in which the inequality in Lemma 3.3.1 (2) is strict. ` ˘ ´푣푝p푎q We then deﬁne the 푝-adic absolute norm |푎|푝 P R of 푎 P Q as 푝 , where for 푎 “ 0 we interpret 푝´8 as 0.

Z 1 1 2 Remark 3.3.3. The function |¨|푝 : Q Ñ R can only take the values in t푝 uYt0u “ t..., 푝2 , 푝 , 1, 푝, 푝 ,...uY t0u. ´1 Warning 3.3.4. We have |푝|푝 “ 푝 . From Lemma 3.3.1, we immediately deduce:

Lemma 3.3.5. We have |푎푏|푝 “ |푎|푝|푏|푝 and |푎 ` 푏|푝 ď maxt|푎|푝, |푏|푝u.

Note that 푑p푥, 푦q :“ |푥 ´ 푦|푝 deﬁnes a metric on Q. Indeed, by the lemma we have:

|푥 ´ 푦|푝 ` |푦 ´ 푧|푝 ě maxt|푥 ´ 푦|푝, |푦 ´ 푧|푝u ě |푥 ´ 푧|푝 so that 푑p푥, 푦q satisﬁes a strong version of the triangle inequality. We therefore refer to the inequality |푎 ` 푏|푝 ď maxt|푎|푝, |푏|푝u as the ultrametric inequality. We deﬁne Q푝 to be the completion of Q with respect to this metric. 26 SAM RASKIN

Exercise 3.1. (1) Show that addition and multiplication on Q are continuous with respect to the 푝-adic metric. Deduce that Q푝 is a commutative ring with Q a dense subring. 1 (2) Show that the inversion map Qˆ Ñ Qˆ, 푥 ÞÑ 푥´ is a continuous homomorphism with respect to the 푝-adic metric. Deduce that Q푝 is a ﬁeld.

We see that 푣푝 and | ¨ |푝 extend to Q푝 by continuity; we denote these extensions in the same manner. Since on Q these functions took values in the closed subsets Z Ď R and 푝Z Y t0u Ď R respectively, their extensions to Q푝 do as well. 푛 We will also use the notation 푝 |푎 to say that 푣푝p푎q ě 푛.

Notation 3.3.6. By Lemma 3.3.5, we see that t푎 P Q푝 | |푎|푝 ď 1u forms a subring of Q푝. We deduce this subring by Z푝 and call it the ring of 푝-adic integers. Note that 푥 P Z푝 if and only if 푣푝p푥q ě 0.

Exercise 3.2. (1) Show that Z푝 is open and closed in Q푝. 푛 (2) Show that a rational number 푚 P Q Ď Q푝 lies in Z푝 if and only if when 푛 and 푚 are chosen relatively prime 푝 does not divide 푚. (3) Show that the image of Z ãÑ Z푝 has dense image. (4) Show that the ﬁeld of fractions of Z푝 is Q푝. More precisely, show that Q푝 is obtained from Z푝 by inverting the single element 푝. Exercise 3.3. Prove the product formula, which states that for 푎 P Q we have:

|푎|8 ¨ |푎|푝 “ 1 푝 a prime ź where | ¨ |8 is the “usual” absolute value on Q, whose completion is R.

Exercise 3.4. Show that if 푥 P Q푝 is a root of a monic polynomial:

푛 푛´1 푥 ` 푎1푥 ` ... ` 푎푛 with 푎푖 P Z푝, then 푥 P Z푝.

3.4. Infinite summation in Q푝. We brieﬂy digress to discuss inﬁnite sums in Q푝. 푛 We say that a sequence 푥1, 푥2,... has convergent sum if the sequence 푆푛 :“ 푖“1 푥푖 converges in 8 Q푝. In this case, we write 푖“1 for the limit of the sequence 푆푛. We will summarize the situation 8 ř informally by saying that the sum 푥푖 converges. ř 푖“1 8 Lemma 3.4.1. The sum 푖“1 푥푖 convergesř if and only if 푣푝p푥푖q Ñ 8 as 푖 Ñ 8.

Proof. To check if the sequenceř 푆푛 of partial sums is convergent, it suﬃces to check if it is Cauchy, which in turn translates to verifying that for every 휀 ą 0 there exists 푁 such that for every 푚 ą 푁 we have:

푚 | 푥푖|푝 ă 휀. 푖“푁 푚ÿ 푚 By the ultrametric inequality, we have | 푖“푁 푥푖|푝 ď maxt|푥푖|푝u푖“푁 , giving the result. ř Remark 3.4.2. This result stands in stark contrast to the case of R, where convergence of an inﬁnite series is a much more subtle question. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 27

3.5. Some computations. We’ll now do some computations to get a feeling for working with Q푝. 2 3 푛 ´푛 A ﬁrst observation is that 1, 푝, 푝 , 푝 ,... converges to 0 in Q푝. Indeed, we have |푝 |푝 “ 푝 , giving the result. More generally, we see that a sequence t푥푛u푛ě0 converges to 0 if and only if for every 푁 ą 0 푁 푁 there exists 푀 such that for 푛 ą 푀 we have 푥푛 P 푝 Z푝. Indeed, 푝 Z푝 is the open (and closed) ´푁 neighborhood of the identity consisting of all 푥 P Q푝 with |푥|푝 ď 푝 . 2 푛 Example 3.5.1. Note that the sequence 푥푛 :“ 1`푝`푝 `...`푝 is Cauchy and therefore converges in Q푝. The usual argument shows that this element is the inverse to 1 ´ 푝 in Q푝. Clearly each of 2 1 ´ 푝 and 1 ` 푝 ` 푝 ` ... lie in Z푝. 푛 Next, observe that 푝 Z푝 is an ideal of Z푝 with:

푛 » 푛 Z{푝 Z ÝÑ Z푝{푝 Z푝. 푛 Indeed, this follows since Z is dense in Z푝 and 푝 Z푝 is an open ideal.

Proposition 3.5.2. The topological space Z푝 is compact. ´푛 Remark 3.5.3. Note that this proposition does not hold for Q푝, since the sequence 푝 (푛 ě 0) does not converge to a limit.

Proof of Proposition 3.5.2. We need to show that every sequence 푥푛 contains a convergent subse- quence 푦푚. There must be some residue class 푎1 in Z{푝Z such that 푥푛 mod 푝Z푝 takes that value inﬁnitely often. Choose 푦1 “ 푥푛1 to be some element in our sequence that reduces to this element. 2 Of the classes in Z{푝 Z, there must be at least one that reduces to 푎1 inﬁnitely often. Call one 2 such as 푎2. Then there must be an index 푛2 ą 푛1 such that 푦2 :“ 푥푛2 reduces to 푎2 modulo 푝 Z푝. 3 Repeating this for Z{푝 Z, etc., we obtain a sequence 푦푛 that is obviously Cauchy and therefore convergent. 3.6. The 푝-adic numbers as a limit. Suppose we have a sequence:

휙푛 휙푛´1 휙1 ... Ñ 퐴푛`1 Ñ 퐴푛 Ñ ... Ñ 퐴1 of commutative rings 퐴푛. We call such a datum a projective system. Deﬁne the projective limit lim푛 퐴푛 as the subset of the product 푛 퐴푛 consisting of elements p푎푛q P 퐴푛 such that for each 푖 ą 1 we have 휙푛´1p푎푛q “ 푎푛´1. That is, elements of lim푛 퐴푛 are 푛 ś elements of 퐴1 with a chosen lift to 퐴2, and a chosen lift of that element of 퐴2 to 퐴3, etc. The ś commutative algebra structure on lim푛 퐴푛 is deﬁned by termwise addition and multiplication. 푛 푛`1 Example 3.6.1. We have a projective system with 퐴푛 “ Z{푝 Z, where the structure map Z{푝 Z Ñ 푛 푛 푛 1 Z{푝 Z is given by modding out by the ideal 푝 Z{푝 ` Z. Remark 3.6.2. We can also make sense of projective limits of groups, projective limit of abelian groups, projective limit of rings, etc. Proposition 3.6.3. The canonical map:

푛 푛 푝 Ñ lim 푝{푝 푝 “ lim {푝 Z 푛 Z Z 푛 Z Z is an isomorphism. 28 SAM RASKIN

Proof. First, note that the resulting map is injective: an element 푎 P Z푝 goes to zero if and only if 푛 ´푛 it lies in 푝 Z푝 for each 푛, but this implies that the |푎|푝 ď 푝 for each 푛, so |푎|푝 “ 0 which implies that 푎 “ 0. 푛 To see that this map is surjective, let p푎푛q be an element of the right hand side, we 푎푛 P Z{푝 Z 푛 a compatible sequence. Choose 푥푛 P Z reducing to 푎푛 modulo 푝 . Note that for 푛, 푚 ě 푁, we have 푁 ´푁 푝 | 푥푛 ´푥푚, so that |푥푛 ´푥푚|푝 ď 푝 . Therefore, this sequence is Cauchy and therefore converges to some 푥 P Z푝. 푛 푛 푛 Note that 푥 ´ 푥푛 P 푝 Z푝 since 푥푚 ´ 푥푛 P 푝 Z for all 푚 ě 푛. Therefore, 푥 ´ 푥푛 P 푝 Z푝, so that 푛 푥 “ 푎푛 mod 푝 Z푝, as desired.

Remark 3.6.4. One upshot of this construction of Z푝 is that it is algebraic, i.e., it makes no reference to the real numbers (whereas the very notion of metric space does). Note that we can obtain Q푝 from Z푝 by inverting 푝, so this construction gives an algebraic perspective on Q푝 as well 3.7. Infinite base 푝 expansion. We have the following lemma that gives a concrete way of thinking about 푝-adic numbers.

Proposition 3.7.1. For every 푥 P Q푝, there exist unique integers 0 ď 푎푖 ă 푝 defined for 푖 ě 푣푝p푥q such that:

푖 푥 “ 푎푖푝 . 푖 푣 푥 “ÿ푝p q The coefficient 푎푣푝p푥q is non-zero. We have 푎푖 “ 0 for all 푖 sufficiently large if and only if 푣푝p푥q 푝 푥 P Z Ď Z푝.

푣푝p푥q Proof. First, by multiplying by 푝 , we reduce to the case when 푥 P Z푝. 푛 푛 푛 Reduce 푥 modulo 푝 ; there is a unique integer 0 ď 푥푛 ă 푝 with 푥푛 “ 푥 mod 푝 Z푝. Take the base 푝 expansion of 푥푛:

푛´1 푖 푥푛 “ 푎푖푝 . 푖“0 ÿ It is clear that the coefficients 푎푖, 0 ď 푖 ă 푛 are the same when we work with 푥푚 for 푚 ě 0 instead. Moreover, the infinite sum obviously converges to 푥, giving the existence of such an expression. The remaining properties are easily verified.

Remark 3.7.2. The proof shows that we could replace the condition that 푎푖 are integers between 0 and 푝 ´ 1 by the condition that the 푎푖 P 푆 where 푆 Ď Z푝 is a subset of coset representatives of Z{푝Z. A common choice one ﬁnds in the literature instead of 푆 “ t0, 1, . . . , 푝 ´ 1u is that 푆 is the set of Teichmuller lifts of elements of F푝: these are deﬁned in Remark 3.10.6. The Teichmuller normalization is important in the theory of Witt vectors.

3.8. Invertibility in Z푝. Here we will give several perspectives on the following important result.

Proposition 3.8.1. An element 푥 P Z푝 is invertible in Z푝 if and only if 푝 - 푥. ´1 First proof of Proposition 3.8.1. Suppose 푥 P Z푝 is non-zero, so we can make sense of 푥 P Q푝. Then we have: INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 29

´1 ´1 0 “ 푣푝p1q “ 푣푝p푥푥 q “ 푣푝p푥q ` 푣푝p푥 q. ´1 Since by hypothesis 푣푝p푥q ě 0, we see that 푥 P Z푝 if and only if:

´1 푣푝p푥q “ 푣푝p푥 q “ 0 as desired. We will give two additional and more explicit proofs below modeled on Example 3.5.1. Though one argument will be analytic and the other algebraic, we note from the onset that they are two essentially identical arguments packaged in diﬀerent ways.

Second proof of Proposition 3.8.1. First, suppose that 푥 “ 1 mod 푝Z푝. Write 푥 “ 1 ´ p1 ´ 푥q so that we heuristically expect:

1 1 8 “ “ p1 ´ 푥q푖 푥 1 ´ p1 ´ 푥q 푖“0 ÿ Now observe that 푝푖 | p1 ´ 푥q푖 by assumption on 푥, so that the series on the right converges. We then easily compute:

8 8 8 p1 ´ 푥q ¨ p1 ´ 푥q푖 “ p1 ´ 푥q푖 “ p1 ´ 푥q푖 ´ 1 푖“0 푖“1 푖“0 8 ÿ 푖 ÿ ` ÿ ˘ so that on subtracting 푖“0p1 ´ 푥q from each side we ﬁnd:

ř 8 푥 ¨ p1 ´ 푥q푖 “ 1 푖“0 ÿ as desired. In general, choose 푦 P Z such that 푥푦 “ 1 mod 푝Z푝: such 푦 exists because 푥 ‰ 0 mod 푝Z푝. Then 1 푦 P Q Ď Q푝 obviously lies in Z푝, so we see that 푦 is invertible on the one hand and 푥푦 is too, giving the result. Remark 3.8.2. Here’s another presentation of the above argument: as in the proof, we reduce to the case where 푥 “ 1 mod 푝Z푝. Then the operator 푇 : Z푝 Ñ Z푝 deﬁned as 푇 p푦q “ p1 ´ 푥q푦 ` 1 1 multiplication by 1 ´ 푥 is contracting, since |푇 p푦1q ´ 푇 p푦2q|푝 ď 푝 ¨ |푦1 ´ 푦2|푝. Therefore, by Banach’s ﬁxed point theorem, there exists 푥0 P Z푝 with 푥0 “ 푇 p푥0q “ 푥0p1 ´ 푥q ` 1 meaning that 푥0푥 “ 1 as desired. For the third proof, we use the following result. Lemma 3.8.3. (1) Let 퐴 be a commutative ring, let 푥 P 퐴 be invertible, and let 푦 P 퐴 be nilpotent (i.e., 푦푁 “ 0 for 푁 large enough). Then 푥 ` 푦 is invertible. (2) Let 퐴 be a commutative ring and let 퐼 be an ideal consisting only of nilpotent elements of 퐴. Then 푥 P 퐴 is invertible if and only if its reduction 푥 P 퐴{퐼 is invertible. 30 SAM RASKIN

Proof. For (1), we explicitly compute:

´1 8 1 푥 1 푦 푖 “ “ p´1q푖 푥 ` 푦 1 ` 푦 푥 푥 푥 푖“0 ÿ ` ˘ where the infinite sum makes sense because 푦 is nilpotent, i.e., it is really only a finite sum. For (2): first, note that if 푥 is invertible in 퐴, then it certainly is modulo 퐼 as well. For the converse, choose some 푦 P 퐴 such that 푥푦 “ 1 mod 퐼. Then 푥푦 is invertible by (1), since it is of the form 1`pan element of 퐼q, and every element of 퐼 is nilpotent. We we find that p푥푦q´1 ¨푦 is the inverse to 푥.

푛 Third proof of Proposition 3.8.1. To show that 푥 is invertible, it suffices to show that 푥 mod 푝 Z푝 is invertible for each 푛: indeed, the inverses clearly form a compatible system and therefore define 푛 an element of Z푝 “ lim Z{푝 Z. 푛 1 푛 But for each 푛 ą 1, the ideal 푝 ´ Z{푝 Z consists only of nilpotent elements, so to test if an 푛 element in Z{푝 Z is invertible, it suffices by Lemma 3.8.3 (2) to check that its reduction modulo 푝 is.

ˆ 3.9. Algebraic structure of the unit group. We want to describe Q푝 in more detail. First, note that we have an isomorphism:

ˆ » ˆ Q푝 ÝÑ Z ˆ Z푝 (3.1) ´푣푝p푥q 푥 ÞÑ 푣푝p푥q, 푝 ¨ 푥q.

ˆ Therefore, it suﬃces to understand Z푝 . ` For each 푛 ą 0, note that:

ˆ 푛 ˆ Ker Z푝 Ñ pZ푝{푝 Z푝q 푛 ˆ is the subgroup 1 ` 푝 Z푝, noting that subgroup` indeed lies in˘ Z푝 by Proposition 3.8.1. Moreover, the resulting map is surjective because

Proposition 3.9.1. The map:

ˆ 푛 ˆ Ñ lim p 푝{푝 푝q Z푝 푛 Z Z is an isomorphism.

ˆ 푛 Proof. If 푥 P Z푝 is in the kernel of this map, then 푥 P 1 ` 푝 Z푝 for all 푛 ą 0, meaning that 푥 is arbitrarily close to 1 and therefore equals 1 as desired. Surjectivity is clear from Proposition 3.6.3: ˆ any element in the projective limit arises from an element of Z푝 that obviously lies in Z푝 . Remark 3.9.2. The 푝-adic exponential and logarithm functions introduced below shed further light ˆ on Z푝 . INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 31

3.10. Hensel’s lemma. Hensel’s lemma is a generalization Proposition 3.8.1, giving solutions to equations in Z푝 by checking for their reduction modulo 푝. 푛 푖 Definition 3.10.1. Let 퐴 be a ring and let 푓p푡q “ 푖“0 푎푖 ¨ 푡 P 퐴r푡s be a polynomial. We deﬁne the formal derivative of 푓 as: ř 푛 1 푖´1 푓 p푡q “ 푖푎푖푡 P 퐴r푡s 푖“1 ÿ Proposition 3.10.2 (Hensel’s lemma). Let 푓p푡q P Z푝r푡s be a polynomial with coefficients in Z푝. Let 푓p푡q P F푝r푡s be the polynomial induced by reducing the coefficients of 푓 modulo 푝. 1 1 7 Suppose that 푓p푡q has a root 푥 with 푓 p푡qp푥q ‰ 0 P F푝, where 푓 is the formal derivative of 푓. Then there exists a unique root 푥 P Z푝 of 푓 that reduces to 푥. Remark 3.10.3. The hypothesis on the derivative of 푓 is necessary: otherwise, consider 푓p푡q “ 푡2 ´푝. 2 Since 푓p푡q “ 푡 , there exists a solution modulo 푝, but Q푝 does not contain a square root of 푝: indeed, ? 1 1 we would have 푣푝p 푝q “ 2 푣푝p푝q “ 2 , and 푣푝 can only take integral values.

Proof of Proposition 3.10.2. We use the notation 푥1 in place of 푥. For each 푛 ą 0, we let 푓푛p푡q P 푛 Z{푝 Zr푡s denote the corresponding reduction of 푓. We will prove the following statement by induction: 푛 p˚q푛: There exists a unique root 푥푛 P Z{푝 Z of 푓푛 lifting 푥1. Clearly this statement suffices, since the 푥푛 obviously form a compatible system and therefore define the desired 푥 in Z푝. Observe that the inductive hypothesis p˚q1 is obvious; therefore, it suffices to perform the inductive step, deducing p˚q푛`1 from p˚q푛. 푛`1 푛 Let 푥푛`1 P Z{푝 Z be some element lifting 푥푛. Define the polynomial 푔p푡q as 푓푛`1p푥푛`1 `푝 ¨푡q. 푛 푛 It suffices to show that 푔p푡q has a exactly 푝 roots 푦, since then 푥푛`1 ` 푝 푦 “ 푥푛`1 obviously is well-definedr and is the unique solution to 푓푛`1 lifting 푥푛. r 푛 푟 푟 푛 푛 2 푛`1 Note that for each 푟 ě 0, p푥푛`1 ` 푝 ¨ 푡q “ 푥푛`1 ` 푟푝 푥푛`1 ¨ 푡rsince p푝 q “ 0 P Z{푝 Z. We 푛 1 deduce by linearity that 푔p푡q “ 푓푛`1p푥푛`1q ` 푝 푓푛`1p푥푛`1q ¨ 푡. 1 푛`1 We see that 푓푛`1p푥푛`1q isr a unit in Z{푝 rZ by assumptionr on 푓 and by Lemma 3.8.3 (2). 푛 1 Therefore, 푔p푡q is a linear polynomialr over Z{푝 ` Zr with leading and constant terms lying in 푛 푛`1 푝 Z{푝 Z, giving ther claim. ˆ Corollary 3.10.4. Let 푝 be an odd prime. Then 푥 P Z푝 is a square if and only if 푥 mod 푝Z푝 is a quadratic residue. ´푣푝p푥q More generally, 푥 P Q푝 is a square if and only if 푣푝p푥q is even and 푝 푥 is a quadratic residue modulo 푝Z푝. 2 Proof. For 푥 P Z푝, the polynomial 푓p푡q “ 푡 ´ 푥 has derivative 2푡, which is non-vanishing derivative ˆ at any element of F푝 as long as 푝 is prime. We deduce the first part then from Hensel’s lemma. For the more general case, we appeal to the isomorphism (3.1). Corollary 3.10.5. For 푝 odd, the map:

7This condition is equivalent to saying that 푥 is a simple root of 푓. 32 SAM RASKIN

ˆ Q푝 Ñ t1, ´1u ˆ t1, ´1u 푝´푣푝p푥q푥 푥 ÞÑ p´1q푣푝p푥q, ˜ 푝 ¸ ´ ¯ induces a group isomorphism:

ˆ ˆ 2 » Q푝 {pQ푝 q ÝÑ Z{2Z ˆ Z{2Z.

´푣푝p푥q ˆ Here the second coordinate of this map indicates the Legendre symbol of 푝 푥 P F푝 , this ´푣푝p푥q ˆ element being the reduction modulo 푝 of 푝 푥 P Z푝 .

Exercise 3.5. (1) Show that Q푝 contains all p푝 ´ 1qst roots of unity. (2) Show that there is a unique multiplicative (but not additive) map F푝 Ñ Z푝 such that the induced map F푝 Ñ Z푝 Ñ F푝 is the identity.

Remark 3.10.6. The map F푝 Ñ Z푝 is called “Teichmuller lifting.” 3.11. 푝-adic exponential and logarithm. We now introduce 푝-adic analogues of the usual expo- ˆ nential and logarithm functions in order to better understand the structure of the unit groups Q푝 . Note that this is in analogy with the case of the real and complex numbers, where these functions are crucial for giving a complete description of the relevant multiplicative groups. We will deﬁne exp and log in Notation 3.11.3 by the usual power series deﬁnitions after discussing the convergence of these series. 1 Lemma 3.11.1. (1) Suppose that 푥 P Q푝 with 푣푝p푥q ą 푝´1 . Then the series:

8 1 ¨ 푥푖 (3.2) 푖! 푖“0 ÿ converges in Q푝. (2) For |푥|푝 ă 1, the series:

8 1 ´ 푥푖 (3.3) 푖 푖“1 converges. ÿ 1 Remark 3.11.2. We note that in (1), 푣푝p푥q ą 푝´1 is equivalent to 푥 P 푝Z푝 for 푝 odd, and is equivalent 2 to 푥 P 푝 Z푝 for 푝 “ 2. We formulate the result in terms of the apparently too precise estimate of 1 푝´1 since it is what emerges from the proof, and is relevant for generalizations of this lemma to ﬁnite extensions of Q푝. Therefore, we will continue to use this estimate below to indicate the radius of convergence of the 푝-adic exponential function, silly though it might appear given the above. 1 푖 Proof of Lemma 3.11.1. For (1): it suﬃces to see that 푣푝p 푖! 푥 q Ñ 8. A standard sieving argument gives:

8 푖 푣 p푖!q “ t u 푝 푝푖 푖“1 ÿ (note that this sum is actually ﬁnite). Therefore, we estimate: INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 33

8 푖 1 1 1 푣푝p푖!q ă “ 푖 ¨ ¨ “ 푖 ¨ . 푝푖 푝 1 1 푝 ´ 1 푖“1 ´ 푝 푖 ÿ On the other hand, 푣푝p푥 q “ 푖 ¨ 푣푝p푥q, so we ﬁnd:

푖 푥 푖 1 푣푝p q “ 푣푝p푥 q ´ 푣푝p푖!q ą 푖 ¨ p푣푝p푥q ´ q 푖! 푝 ´ 1 1 which goes to 8 with 푖 if 푣푝p푥q ą 푝´1 . 1 푖 푣푝p푖q logp푖q For (2): we expand 푣푝p 푖 푥 q as 푖 ¨ 푣푝p푥q ´ 푣푝p푖q. As 푝 ď 푖, we can bound 푣푝p푖q by logp푝q , so that:

1 푖 logp푖q 푣푝p 푥 q ě 푖푣푝p푥q ´ . 푖 logp푝q As long as |푥|푝 ă 1 (equivalently: 푣푝p푥q ą 0), the linear leading term grows faster than the loga- rithmic second term, and we obtain the result. 1 Notation 3.11.3. For 푣푝p푥q ą 푝´1 , we let expp푥q denote the result of evaluating the series (3.2). For |푥|푝 ă 1, let logp1 ´ 푥q denote the result of evaluating the series (3.3) at 푥. 1 Note that expp푥q lies in the subset 1 ` 푝Z푝 of Q푝 (for 푣푝p푥q ą 푝´1 ), since the ﬁrst term in the series (3.2) is 1 and the higher order terms are divisible by 푝 (we see that for 푝 “ 2, we even have expp푥q P 1 ` 4Z2). Note that 푥 P 1 ` 푝Z푝 is equivalent to requiring |1 ´ 푥|푝 ă 1, so these are exactly the elements for which we can make sense of the 푝-adic logarithm logp푥q. 1 Proposition 3.11.4. (1) For 푥 and 푦 of valuation greater than 푝´1 , we have:

expp푥 ` 푦q “ expp푥q ¨ expp푦q. (2) For 푥, 푦 P 1 ` 푝Z푝, we have:

logp푥푦q “ logp푥q ` logp푦q. 1 1 1 (3) For 푣푝p푥q ą 푝´1 , we have logpexpp푥qq “ 푥, and for 푣푝p1´푥q ą 푝´1 we have 푣푝plogp푥qq ą 푝´1 and expplogp푥qq “ 푥. Proof. These all follow from standard series manipulations. For (1):

8 1 8 푖 1 푖! 8 푖 1 1 expp푥 ` 푦q :“ p푥 ` 푦q푖 “ 푥푗푦푖´푗 “ 푥푗푦푖´푗 “ 푖! 푖! 푗!p푖 ´ 푗q! 푗! p푖 ´ 푗q! 푖“0 푖“0 푗“0 푖“0 푗“0 ÿ ÿ ÿ ÿ ÿ 8 1 1 8 1 8 1 푥푗푦푘 “ 푥푗 ¨ 푦푘 “ expp푥q ¨ expp푦q. 푗! 푘! 푗! 푘! 푖“0 푗,푘ě0 ´ 푗“0 ¯ ´ 푘“0 ¯ ÿ 푗`ÿ푘“푖 ÿ ÿ For (2), we ﬁrst claim that for a general ﬁeld 푘 of characteristic 0, in the ring 푘rr푡, 푠ss of formal power series in two variables we have an identity:

8 1 8 1 ´ p푠 ` 푡 ´ 푠푡q푖 “ ´ p푠푖 ` 푡푖q. (3.4) 푖 푖 푖“1 푖“1 ÿ ÿ 34 SAM RASKIN

Note that in the left hand side the coefficient of 푠푛푡푚 obviously has only finitely many contributions, so this is a well-defined formal power series. To verify this identity, we first apply the formal partial B derivative B푡 to the left hand side and compute:

B 8 1 8 1 ´ 푠 ´ p푠 ` 푡 ´ 푠푡q푖 “ ´p1 ´ 푠qp푠 ` 푡 ´ 푠푡q푖´1 “ ´ “ B푡 푖 p1 ´ 푠 ´ 푡 ` 푠푡q 푖“1 푖“1 ´ ÿ ¯ ÿ 1 ´ 푠 1 B 8 1 ´ “ ´ “ ´ 푡푖 ` 푠푖 . p1 ´ 푠qp1 ´ 푡q 1 ´ 푡 B푡 푖 푖“1 ´ ÿ ¯ Therefore, it suﬃces to verify the equality (3.4) after setting 푡`“ 0, since˘ the formal derivative computation implies that the coeﬃcients of 푠푛푡푚 of the left and right hand sides of (3.4) are equal whenever 푛 ą 0. But we obviously have the desired equality after setting 푡 “ 0. We then immediately deduce (2) from the equality of the formal power series (3.4), since by convergence we can substitute 1 ´ 푥 for 푠 and 1 ´ 푦 for 푡 into (3.4). Finally, for (3), we similarly use formal power series in a variable 푡. First, note that:

8 1 8 1 푗 ´ 푡푖 푗! 푖 푗“0 푖“1 ÿ ´ ÿ ¯ is defined in 푘rr푡ss (for 푘 as above), since one easily sees that for each coefficient 푡푖 only finitely many terms contribute. Let 푔p푡q denote the resulting power series; we wish to show that 푔p푡q “ 1´푡. One computes:

푑 푑 8 1 8 1 푗 8 1 8 1 푗´1 ´1 ´1 푔p푡q “ ´ 푡푖 “ ´ 푡푖 ¨ “ ¨ 푔p푡q. 푑푡 푑푡 푗! 푖 p푗 ´ 1q! 푖 1 ´ 푡 1 ´ 푡 푗“0 푖“1 푗“1 푖“1 ˆ ÿ ´ ÿ ¯ ˙ ÿ ´ ÿ ¯ Multiplying by 1 ´ 푡, we see that this equation defines a recursion on the coefficients of 푔p푡q, and one finds that it characterizes 푔p푡q up to scaling. Moreover, 1 ´ 푡 is a solution to this equation. Setting 푡 “ 0, we then deduce that 푔p푡q “ 1 ´ 푡. One argues similarly that:

8 ´1 8 1 ´ 푡푗q푖 “ 푡. 푖 푗! 푖“0 푗“1 ÿ ÿ ` 1 ˘ Next, we claim that for 푦 P Q푝 with 푣푝p푦q ą 푝´1 , we have: 1 푖 ¨ 푣푝p푦q ´ 푣푝p푖q ą . (3.5) 푝 ´ 1 Supposing this inequality for a moment, then applying this to 푦 “ 1 ´ 푥 we see that logp푥q “ 8 1 푖 1 푖“1 ´ 푖 p1 ´ 푥q is a sum of terms of valuation at least 푝´1 , giving that logp푥q has the same property, as desired. From here, the rest of (3) follows from convergence and the properties of formalř power series noted above. It remains to verify the claim (3.5). First, observe that it is true for 푖 ă 푝: in this case, we have 푣푝p푖q “ 0, so the result is clear. To treat 푖 ě 푝, we ﬁrst bound the left hand side as:

푖 logp푖q 푖 ¨ 푣푝p푦q ´ 푣푝p푖q ą ´ . (3.6) 푝 ´ 1 logp푝q We can now regard 푖 as a continuous variable in R. The derivative of this function is then: INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 35

1 1 ´ . 푝 ´ 1 푖 logp푝q 푝´1 This derivative is zero exactly for 푖 “ logp푝q , which is obviously the minimum for the function. 푝´1 : Note that logp푝q ă 푝 for all 푝. Indeed, for 푝 ‰ 2 (i.e., 푝 ą 푒 “ the base of the natural logarithm) this is clear since then the left hand side is less than 푝 ´ 1. For 푝 “ 2, this says that 1 ă logp2q, ? 2 i.e., 푒 ă 2, which is clear. Moreover, substituting the value 푖 “ 푝 into the right hand side of (3.6) above gives: 푝 1 ´ 1 “ . 푝 ´ 1 푝 ´ 1 Since this 푝 is past the minimum of the function, this gives the desired result for all 푖 ě 푝. Corollary 3.11.5. The exponential and logarithm maps define mutually inverse equivalences of 1 1 abelian groups between t푥 P Z푝 | 푣푝p푥q ą 푝´1 u and t푥 P Z푝 | 푣푝p1 ´ 푥q ą 푝´1 u, the former being considered as an abelian group under addition, and the latter as an abelian group under multiplication.

3.12. Squares in Q2. We now discuss the consequences of Corollary 3.11.5 for the structure of squares in Q푝, generalizing Corollary 3.10.4. ˆ Proposition 3.12.1. 푥 P Z2 is a square if and only if 푥 “ 1 mod 8. Proof. To see necessity, note that invertibility of 푥 implies that 푥 P t1, 3, 5, 7u mod 8, and the only 2 2 2 square in Z{8Z among these is 1 (since p2푘 ` 1q “ 4p푘 ` 푘q ` 1 and 푘 ` 푘 is always even). 1 For the converse, first note that for 푝 “ 2, the inequality 푣푝p푥q ą 푝´1 translates to 푣푝p푥q ě 2. log Therefore, by Corollary 3.11.5 we have an isomorphism 1 ` 4Z2 » 4Z2, the former group being considered with multiplication and the latter with addition. Since the image of multiplication by 2 in 4Z2 is 8Z2, it suffices to show that this equivalence identifies 8Z2 with 1 ` 8Z2. Recall from the proof of Lemma 3.11.1 that we proved in general that:

푥푖 1 푣푝p q ą 푖 ¨ p푣푝p푥q ´ q. 푖! 푝 ´ 1 푥푖 For 푝 “ 2, we obtain 푣2p 푖! q ą 푖 ¨ p푣2p푥q ´ 1q. For 푣2p푥q ě 2, this implies that for 푖 ě 2 we have: 푥푖 푣 p q ą 2p푣 p푥q ´ 1q “ 푣 p푥q ` p푣 p푥q ´ 2q ě 푣 p푥q 2 푖! 2 2 2 2 so that:

|1 ´ expp푥q|푝 “ |푥|푝 for all 푥 with 푣푝p푥q ě 2, verifying the claim. Corollary 3.12.2. We have an isomorphism:

ˆ ˆ 2 » ˆ Q2 {pQ2 q ÝÑ t1, ´1u ˆ pZ{8Zq 푣 푥 푥 ÞÑ p´1q 2p q, 2´푣2p푥q푥 ´ ¯ 36 SAM RASKIN

´푣2p푥q ˆ ´푣2p푥q ˆ where 2 푥 P pZ{8Zq denotes the reduction of 2 ¨ 푥 P Z2 . ˆ ˆ 2 Remark 3.12.3. In particular, we see that |Q2 {pQ2 q | “ 8. ˆ ˆ 2 Exercise 3.6. Show that Q2 {pQ2 q is generated by the images of 2, 3 and 5.

3.13. Quadratic forms over finite fields. Before proceeding to discuss quadratic forms over Q푝, we need to digress to treat quadratic forms over ﬁnite ﬁelds. The main result on the former subject is the following. Proposition 3.13.1. Let 푝 be an odd prime. (1) Any form of rank ě 2 represents every value in F푝. (2) Two non-degenerate forms over F푝 are equivalent if and only if their discriminants are equal. (3) Let 푟 be a quadratic non-residue modulo 푝. Then every non-degenerate rank 푛 form 푞 over F푝 is equivalent to exactly one of the forms:

2 2 푥1 ` ... ` 푥푛 or 2 2 푥1 ` ... ` 푟푥푛. Proof. For (1), by diagonalization it suﬃces to show any form 푎푥2 ` 푏푦2 with 푎, 푏 ‰ 0 represents every value 휆 P F푝. 2 푝`1 푝´1 Note that 푥 represents exactly 2 values in F푝: the 2 quadratic residues and 0. Therefore, the same counts hold for 푎푥2 and 푏푦2, since 푎, 푏 ‰ 0. Therefore, the intersection:

2 2 t푎훼 | 훼 P F푝u X t휆 ´ 푏훽 | 훽 P F푝u 푝 is non-empty, since each of these sets has order greater than 2 . Choosing 훼, 훽 P F푝 deﬁning an element of the intersection, this gives 푎훼2 “ 휆 ´ 푏훽2 as desired. For (3), we proceed by induction on 푛. The result is clear for 푛 “ 1. Otherwise, by (1), 푞 represents 1 P F푝. Choosing a vector 푣 in the underlying quadratic space 푉 with 푞p푣q “ 1 and decomposing according to orthogonal complements, we obtain the result by induction. Finally, (2) obviously follows from (2).

3.14. Hilbert symbol. Let 푘 “ Q푝 for some 푝 or 푘 “ R. For 푎, 푏 P 푘ˆ, we deﬁne the Hilbert symbol of 푎 and 푏 as 1 if the binary quadratic form 푎푥2 ` 푏푦2 represents 1, and ´1 otherwise. We denote the Hilbert symbol of 푎 and 푏 by:

p푎, 푏q푝 P t1, ´1u if 푘 “ Q푝, and p푎, 푏q8 P t1, ´1u if 푘 “ R. Remark 3.14.1. Of course, this definition makes sense over any field, but it does not have good properties: Theorem 3.14.5 will use specific facts about the choice 푘 “ Q푝 and 푘 “ R. This failure represents the fact that our definition is somewhat ad hoc. Remark 3.14.2. The role the Hilbert symbol plays for us is that it is clearly an invariant of the quadratic form 푞p푥, 푦q “ 푎푥2 ` 푏푦2, i.e., it only depends on the underlying quadratic space, not on 푎 and 푏. Eventually, we will denote it by 휀p푞q, call it the Hasse-Minkowski invariant of 푞, and we will generalize it to higher rank quadratic forms. Here we record the most obvious properties of the Hilbert symbol. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 37

Proposition 3.14.3. The following identities are satisfied by the Hilbert symbol for 푝 a prime or 푝 “ 8.

(1) p푎, 푏q푝 “ p푏, 푎q푝. (2) p푎, 1 ´ 푎q푝 “ 1. (3) p1, 푎q푝 “ 1. (4) p푎, ´푎q푝 “ 1. 2 (5) p푎휆 , 푏q푝 “ p푎, 푏q푝 for all non-zero 휆. (6) p푎, 푏q푝 “ p푎, ´푎푏q푝. Proof. In words, (1) says that 푎푥2 ` 푏푦2 represents 1 if and only if 푏푥2 ` 푎푦2 does, which is clear. Then (2) says that 푎푥2 ` p1 ´ 푎q푦2 represents 1, which it does with p푥, 푦q “ p1, 1q, while (3) says that 푥2 ` 푎푦2 represents 1, which it does with p1, 0q. Next, (4) says that 푎푥2 ´ 푎푦2 represents 1, and this is clear because the form is isotropic and non-degenerate, therefore equivalent to the hyberbolic plane, and therefore represents all values. 2 2 (5) follows because p푎, 푏q푝 is an invariant of the equivalence class of the quadratic form 푎푥 `푏푦 . 2 2 2 Finally, for (6), we note that p푎, 푏q푝 “ 1 if and only if 푞1p푥, 푦, 푧q :“ 푎푥 ` 푏푦 ´ 푧 is isotropic by Corollary 2.10.3. Similarly, p푎, ´푎푏q푝 “ 1 if and only if:

´1 1 푞 p푥, 푦, 푧q :“ p푎푥2 ´ 푎푏푦2 ´ 푧2q “ ´푥2 ` 푏푦2 ` 푧2 2 푎 푎 is isotropic. Therefore, it suﬃces to see that 푞1 and 푞2 are equivalent quadratic forms. But this is 1 2 clear: swap the variables 푥 and 푧 and change 푎 to 푎 by multiplication by 푎 . Remark 3.14.4. By (5), we the Hilbert symbol deﬁnes a symmetric pairing:

ˆ ˆ 2 ˆ ˆ 2 Q푝 {pQ푝 q Ñ Q푝 {pQ푝 q Ñ t1, ´1u (3.7) and similarly for R. We will prove the following less trivial results below. It is here that we will really need that our ﬁeld is Q푝 or R. Theorem 3.14.5. Let 푝 be a prime or 8.

(1) p푎, 푏푐q푝 “ p푎, 푏q푝 ¨ p푎, 푐q푝. (2) If p푎, 푏q푝 “ 1 for all invertible 푏, then 푎 is a square. ˆ ˆ 2 Remark 3.14.6. We may view 푘 {p푘 q as an F2-vector space, where we change from multiplicative notation in 푘ˆ{p푘ˆq2 to additive notation for vector addition. Then Theorem 3.14.5 exactly says that for 푘 “ Q푝 or 푘 “ R, p푎, 푏q ÞÑ p푎, 푏q푝 P t1, ´1u » F2 deﬁnes a non-degenerate symmetric bilinear form on the vector space 푘ˆ{p푘ˆq2 (which we note is ﬁnite-dimensional over F2, of dimension 1 for 푘 “ R, 2 for 푘 “ Q푝 and 푝 ‰ 2, and 3 for Q2). 3.15. Consequences for quadratic forms. For this subsection, we will assume Theorem 3.14.5, leaving its proof (and a detailed description of how to compute with the Hilbert symbol) to S3.16. Let 푘 “ Q푝 or 푘 “ R; in the latter case, the reader should understand 푝 below as a stand-in for 8. As in Remark 3.14.2 binary8 quadratic form 푞, we let 휀p푞q P t1, ´1u be 1 if 푞 represents 1 and 2 2 ´1 otherwise. So if 푞 “ 푎푥 ` 푏푦 , then 휀p푞q “ p푎, 푏q푝.

8 We remind that this means that its underlying quadratic space has dimension 2. 38 SAM RASKIN

Proposition 3.15.1. Let 푞 be a non-degenerate binary quadratic form with discriminant discp푞q. Then 푞 represents 휆 P 푘ˆ if and only if:

p´ discp푞q, 휆q푝 “ 휀p푞q. 1 Proof. That 푞 represents 휆 is equivalent to saying that 휆 푞 represents 1. Diagonalizing 푞 so that 2 2 푎 푏 푞p푥, 푦q “ 푎푥 `푏푦 , we see that this is equivalent to having p 휆 , 휆 q푝 “ 1. For convenience, we replace 1 ˆ ˆ 2 this expression by p휆푎, 휆푏q푝, which is justiﬁed by the equality 휆 “ 휆 P 푘 {p푘 q . Applying the bilinearity Theorem 3.14.5 (1) of the Hilbert symbol repeatedly, we compute:

2 p휆푎, 휆푏q푝 “ p휆, 휆q푝p휆, 푏q푝p푎, 휆q푝p푎, 푏q푝 “ p´1, 휆q푝p휆, 휆q푝p푏, 휆q푝p푎, 휆q푝p푎, 푏q푝 “

p´1, 휆q푝p휆, 휆q푝 p´1, 휆q푝p푏, 휆q푝p푎, 휆q푝 p푎, 푏q푝 “ p´휆, 휆q푝p´푎푏, 휆q푝p푎, 푏q푝.

Observing that´ p´휆, 휆q푝 “ 1¯´ by Proposition 3.14.3 (4),¯ we see that:

p휆푎, 휆푏q푝 “ p´ discp푞q, 휆q푝 ¨ 휀p푞q.

This exactly means that 휀p푞q “ p´ discp푞q, 휆q푝 if and only if 푞 represents 휆, as desired. Remark 3.15.2. Here are some sanity tests to make sure that Proposition 3.15.1 accords with reality. First of all, for 휆 “ 1, this formula says that 푞 represents 1 if and only if p´ discp푞q, 1q푝 “ 휀p푞q, but since 1 is a square, we see that p´ discp푞q, 1q푝 “ 1, so this is equivalent to saying 휀p푞q “ 1, as desired. Next, if 푞 is the hyperbolic plane, we have discp푞q “ ´1 and 휀p푞q “ 1, so Proposition 3.15.1 says that 푞 represents 휆 if and only if p1, 휆q푝 “ 1. As above p1, 휆q푝 “ 1 for all 휆, so this says that the hyperbolic plane represents every value, which is obviously true.

Corollary 3.15.3. Non-degenerate binary quadratic forms 푞1 and 푞2 are equivalent if and only if discp푞1q “ discp푞2q and 휀p푞1q “ 휀p푞2q. 2 2 Proof. Diagonalize 푞1 as 푎푥 ` 푏푦 . By Proposition 2.12.1, 푞1 and 푞2 represent the same values, and 2 1 2 therefore 푞2 represents 푎 as well. By Proposition 2.9.7, 푞2 can be diagonalized as 푎푥 ` 푏 푦 . We now see that:

1 1 2 푎푏 “ discp푞2q “ discp푞1q “ 푎푏 modp푘 q so that changing 푏1 by a square we can take 푏1 “ 푏 as desired.9 Corollary 3.15.4. If 푞 is a non-degenerate binary quadratic form other than the hyperbolic plane, then 푞 represents exactly half of the elements of the finite set 푘ˆ{p푘ˆq2. Proof. By Exercise 2.27, ´ discp푞q P p푘ˆq2 if and only 푞 is the hyperbolic plane. Therefore, if 푞 is not the hyperbolic plane, then p´ discp푞q, ´q : p푘ˆ{p푘ˆq2q Ñ t1, ´1u is a non-trivial homomorphism by Theorem 3.14.5 (2), and 휆 is represented by 푞 if and only if it lies in the appropriate coset of this homomorphism.

9 This argument implies over a general field of characteristic ‰ 2 that two non-degenerate binary quadratic forms of the same discriminant and representing the same values are equivalent. Note that this is not true in rank ě 3. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 39

Exercise 3.7. For 푘 “ R, verify each of the above statements by hand (note that the Hilbert symbol for R is explicitly described in Proposition 3.16.1 below). Corollary 3.15.5. Any non-degenerate ternary quadratic form 푞 represents every 휆 P 푘ˆ{p푘ˆq2 except possibly 휆 “ ´ discp푞q. Proof. We diagonalize 푞 as 푞p푥, 푦, 푧q “ 푎푥2 ` 푏푦2 ` 푐푧2 for 푎, 푏, 푐 P 푘ˆ. 2 2 2 2 Fix 휆 ‰ ´ discp푞q. By Corollary 2.10.3, it suffices to show that 푞휆 :“ 푞1´푞2 “ 푎푥 `푏푦 `푐푧 ´휆푤 is isotropic. 2 2 2 2 Observe that the forms 푞1p푥, 푦q :“ 푎푥 ` 푏푦 and 푞2p푧, 푤q “ ´푐푧 ` 휆푤 have different discriminants by assumption on 휆. Note that we can assume that discp푞1q and discp푞2q are each not ´1, since otherwise one of these forms is the hyperbolic plane, which obviously would imply that 푞휆 is isotropic. Therefore, ´ discp푞1q and ´ discp푞2q are linearly independent when regarded as elements of the ˆ ˆ 2 F2-vector space 푘 {p푘 q , since they are distinct non-zero vectors in a F2-vector space. Therefore, ˆ by Theorem 3.14.5 (2), there exists 휂 P 푘 with p´ discp푞1q, 휂q푝 “ 휀p푞1q and p´ discp푞2q, 휂q푝 “ 휀p푞2q. Indeed, in the perspective of Remark 3.14.6, we are given linearly independent vectors and a non- degenerate bilinear form, so we can find a vector that pairs to these vectors in any specified way. 2 2 2 2 But then 푞1p휂q ´ 푞2p휂q “ 0, so the form 푎푥 ` 푏푦 ` 푐푧 ´ 휆푤 is isotropic, giving the claim.

In the special case that 푘 “ Q푝 (i.e., 푘 ‰ R), we moreover obtain the following results.

Corollary 3.15.6. Any quadratic form 푞 over 푘 “ Q푝 of rank ě 5 is isotropic.

Proof. We can write 푞 “ 푞1 `푞2 `푞3 where 푞1 has rank 3, 푞2 has rank 2 and 푞3 is otherwise arbitrary. ˆ ˆ 2 By Corollary 3.15.5, 푞1 represents every value in 푘 {p푘 q except possibly 1 (namely, ´ discp푞1q). 2 Therefore, since 푘 ‰ R so that |푘ˆ{p푘ˆq | ą 2, by Corollary 3.15.4 there exists 휂 in 푘ˆ that is represented by both 푞1 and ´푞2, meaning that 푞1 ` 푞2–and therefore 푞 itself–is isotropic.

Corollary 3.15.7. For 푘 “ Q푝, any form of rank ě 4 represents every 휆 P 푘. Proof. Immediate from Corollary 3.15.6 and Corollary 2.10.3. 3.16. We now will give explicit formulae for the Hilbert symbol. We will deduce Theorem 3.14.5 from these explicit formulae. We separate into cases of increasing diﬃculty: 푘 “ R, 푘 “ Q푝 with 푝 ‰ 2, and Q2. ˆ Proposition 3.16.1. For 푎, 푏 P R , p푎, 푏q8 “ 1 unless 푎 and 푏 are both negative. Proof. This says that 푎푥2 ` 푏푦2 represents 1 unless 푎 and 푏 are both negative, which is clear.

In order to compute the Hilbert symbol for Q푝, it is convenient to pass through the tame symbol. ˆ ˆ This is a rule that takes 푎, 푏 P Q푝 and produces Tame푝p푎, 푏q P F푝 . 푣 p푏q To deﬁne the tame symbol, note that for 푎, 푏 P ˆ, we have 푎 푝 P ˆ, since its valuation is 0. Q푝 푏푣푝p푎q Z푝 푣 p푏q Therefore, we may reduce it to p 푎 푝 q P ˆ. Then we deﬁne: 푏푣푝p푎q F푝

푣푝p푏q 푣푝p푎q¨푣푝p푏q 푎 ˆ Tame푝p푎, 푏q :“ p´1q ¨ p q P F푝 . 푏푣푝p푎q 40 SAM RASKIN

Observe that the tame symbol is bilinear, i.e., we have:

1 1 ˆ Tame푝p푎푎 , 푏q “ Tame푝p푎, 푏q ¨ Tame푝p푎 , 푏q P F푝 and 1 1 ˆ Tame푝p푎, 푏푏 q “ Tame푝p푎, 푏q ¨ Tame푝p푎, 푏 q P F푝 . Proposition 3.16.2. If 푝 is an odd prime, then:

Tame푝p푎, 푏q p푎, 푏q “ . (3.8) 푝 푝 ˆ ˙ Proof. First, we analyze how the tame symbol of 푎 and 푏 changes when 푎 is multiplied by 푝2. By deﬁnition, this is computed as:

푝푣푝p푏q 2 ¨ Tame p푎, 푏q. 푏 푝 Therefore, when we apply the Legendre´` symbol,˘¯ this does not change the resulting value, i.e., the right hand side of (3.8) does not change under this operation. By Proposition 3.14.3 (5), the right hand side does not change under this operation either. Now observe that the right hand side of (3.8) is symmetric under switching 푎 and 푏: indeed, ˆ switching 푎 and 푏 has the eﬀect of inverting the tame symbol in F푝 , and the Legendre symbol is immune to this change. Therefore, we reduce to studying three cases: 1) 푣푝p푎q “ 푣푝p푏q “ 0, 2) 푣푝p푎q “ 0, 푣푝p푏q “ 1, and 3) 푣푝p푎q “ 푣푝p푏q “ 1.

Case 1. 푣푝p푎q “ 푣푝p푏q “ 0. In this case, we see that the tame symbol of 푎 and 푏 is 1. Therefore, we need to show that their Hilbert symbol is as well, i.e., that 푞p푥, 푦q “ 푎푥2 ` 푏푦2 represents 1. 2 2 By Proposition 3.13.1, we can solve 푎푥 ` 푏푦 “ 1 over F푝, say with p푥, 푦q “ p훼, 훽q. Choose a lift 2 2 훼 P Z푝 of 훼. We see that the polynomial 푏푡 ` p푎훼 ´ 1q satisﬁes the hypotheses of Hensel’s lemma and therefore there exists a unique lift 훽 of 훽 that is a solution to the equation 푏훽2 ` 푎훼2 ´ 1 “ 0, as desired.

Case 2. 푣푝p푎q “ 0, 푣푝p푏q “ 1. We ﬁnd Tame푝p푎, 푏q “ 푎, so the right hand side of (3.8) is the Legendre symbol of 푎. 2 2 2 2 2 To compare with the left hand side: note that if p푥, 푦q P Q푝 then 푣푝p푎푥 `푏푦 q “ mint푣푝p푎푥 q, 푣푝p푏푥 qu. 2 2 Indeed, since 푣푝p푎푥 q P 2Z and 푣푝p푏푥 q R 2Z, so these numbers are not equal. Therefore, if 2 2 2 2 푎푥 ` 푏푦 “ 1, then 0 “ mint푣푝p푎푥 q, 푣푝p푏푦 qu “ mint2푣푝p푥q, 1 ` 2푣푝p푦qu, meaning that 푥 and 푦 are 푝-adic integers with 푥 a 푝-adic unit. Therefore, we can reduce an equation 푎푥2 ` 푏푦2 “ 1 modulo 푝 to see that 푎 must be a quadratic residue. The converse direction follows from Corollary 3.10.4.

Case 3. 푣푝p푎q “ 푣푝p푏q “ 1. By Proposition 3.14.3 (6), we have p푎, 푏q푝 “ p푎, ´푎푏q푝. Noting that 푣푝p´푎푏q P 2Z, Case (2) applies and we see that:

Tame푝p푎, ´푎푏q p푎, ´푎푏q “ . 푝 푝 ˆ ˙ We then compute that: INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 41

푎2 푎 Tame푝p푎, ´푎푏q “ “ ´ “ Tame푝p푎, 푏q ´푎푏 푏 giving the desired result.

It remains to treat the case of Q2. ˆ p2´푣2p푎q푎q2´1 ˆ For 푎 P Q2 , let 휛p푎q :“ 8 P Z{2Z. That is, for 푎 P Z2 we have:

휛p푎q “ 0 if 푎 “ 1, ´1 mod 8Z2 #휛p푎q “ 1 if 푎 “ 3, 5 mod 8Z2

ˆ ´푣2p푎q and for general 푎 P Q2 we have 휛p푎q “ 휛p2 푎q. ´푣 p푎q 2 2 푎´1 ´푣2p푎q ˆ We also set 휗p푎q “ 2 P Z{2Z. So again, 휗p푎q “ 휗p2 푎q, and for 푎 P Z2 we have 휗p푎q “ 0 if 푎 “ 1 mod 4 and 휗p푎q “ 1 if 푎 “ 3 mod 4. ˆ Proposition 3.16.3. For all 푎, 푏 P Q2 , we have:

휛p푎q푣2p푏q`푣2p푎q휛p푏q`휗p푎q휗p푏q p푎, 푏q2 “ p´1q . (3.9)

Proof. As in the proof of Proposition 3.16.2, we reduce to the cases 1) 푣2p푎q “ 푣2p푏q “ 0, 2) 푣2p푎q “ 0, 푣2p푏q “ 1, and 3) 푣2p푎q “ 푣2p푏q “ 1.

Case 1. 푣2p푎q “ 푣2p푏q “ 0. In this case, we see that the right hand side of (3.9) is 1 unless 푎 “ 푏 “ 3 mod 4Z2. We treat separately the cases i) where 푎 (or 푏) is 1 modulo 8, ii) 푎 (or 푏) is 5 modulo 8, and iii) 푎 and 푏 are both 3 modulo 4.

Subcase 1. 푎 “ 1 mod 8Z2. 2 2 Then by Corollary 3.12.2, 푎 is a square in Q2, so 푎푥 ` 푏푦 represents 1.

Subcase 2. 푎 “ 5 mod 8Z2. 2 2 In this case, 푎 ` 4푏 “ 1 mod 8 and therefore is a square in Q2, meaning that 푎푥 ` 푏푦 represents a square and therefore represents 1.

Subcase 3. 푎 “ 푏 “ 3 mod 4Z2. 2 2 Suppose 푎푥 ` 푏푦 “ 1, 푥, 푦 P Q2. Let 푛 :“ maxt0, ´푣2p푥q, ´푣2p푦qu. Then we have:

푎p2푛푥q2 ` 푏p2푛푦q2 “ 4푛 as an equation in Z2. 2 2 If 푛 “ 0 so that 푥, 푦 P Z2, this equation gives 3푥 ` 3푦 “ 1 mod 4, which isn’t possible since 2 휆 P t0, 1u for 휆 P Z{4Z. 푛 ˆ If 푛 ą 0, then up to change of variables we can assume 푛 “ ´푣2p푥q, in which case 2 푥 P Z2 , so 푎p2푛푥q2 ` 푏p2푛푦q2 “ 3 ` 3p2푛푦q2 “ 0 mod 4. Now observe that p2푛푦q2 can only reduce to 0 or 1 modulo 4Z2, and neither of these provides a solution to the given equation.

Case 2. 푣2p푎q “ 0, 푣2p푏q “ 1. ˆ 휛p푎q`휗p푎q휗p푐q Let 푏 “ 2푐 with 푐 P Z2 . Then we need to show that p푎, 푏q2 “ p´1q . We note that, as in Case 2 in the proof of Proposition 3.16.2, any solution to 푎푥2 ` 푏푦2 “ 1 must have 푥, 푦 P Z2. 42 SAM RASKIN

2 2 Therefore, if 푎푥 ` 2푐푦 “ 1 does not have a solution in Z{8Z, then p푎, 푏q2 “ ´1. Note that any solution to this equation in Z{8Z must have 푥 “ 1 mod 2, so the ﬁrst term is 푎 ` 2푐푦2. From here, one readily sees that there are no solutions in the cases: ‚ 푎 “ 3 mod 8 and 푐 “ 1 mod 4 ‚ 푎 “ 5 mod 8 ‚ 푎 “ 7 mod 8 and 푐 “ 3 mod 4 We compute the left hand side of (3.9) in these cases as: ‚ p´1q1`1¨0 “ ´1 ‚ p´1q1`0¨1 “ ´1 ‚ p´1q0`1¨1 “ ´1 as desired. We now treat the remaining cases:

0`0¨휗p푐q Subcase 1. If 푎 “ 1 mod 8, then 푎 is a square so p푎, 푏q2 “ 1 “ p´1q . 1`1¨1 Subcase 2. If 푎 “ 3 mod 8 and 푐 “ 3 mod 4, then we need to show that p푎, 푏q2 “ 1 “ p´1q . We note that:

푎 ` 푏 “ 푎 ` 2푐 “ 3 ` 6 “ 1 mod 8 2 2 and therefore 푎 ` 푏 is a square in Z2, so that 푎푥 ` 푏푦 represents a square and therefore 1. 0`1¨0 Subcase 3. If 푎 “ 7 mod 8 and 푐 “ 1 mod 4, then we need to show that p푎, 푏q2 “ 1 “ p´1q . We see that 푎 ` 푏 “ 1 mod 8, so as above, we deduce that p푎, 푏q2 “ 1.

Case 3. 푣2p푎q “ 푣2p푏q “ 1. As in Case 3 for the proof of Proposition 3.16.2, we have p푎, 푏q2 “ p푎, ´푎푏q2, and we note that 푣2p´푎푏q is even so we can compute this Hilbert symbol using Case 2 as:

휛p´푎푏q`휗p푎q휗p´푎푏q p푎, ´푎푏q2 “ p´1q . Observe that p´1q휗p푎q휗p´푎푏q “ p´1q휗p푎q휗p푏q. Moreover, it is easy to check that we have:

p´1q휛p´푎푏q “ p´1q휛p푎q ¨ p´1q휛p푏q. Therefore, we obtain:

휛p푎q`휛p푏q`휗p푎q휗p푏q p푎, 푏q2 “ p푎, ´푎푏q2 “ p´1q . as desired. Exercise 3.8. Deduce Theorem 3.14.5 from Propositions 3.16.1, 3.16.2 and 3.16.3.

3.17. Hasse-Minkowski invariant. Let 푘 “ Q푝 or 푘 “ R. As always, if 푘 “ R, e.g. a Hilbert symbol p´, ´q푝 should be interpreted with 푝 “ 8.

Theorem 3.17.1. There is a unique invariant 휀푝p푞q P t1, ´1u defined for every non-degenerate quadratic form 푞 over 푘 such that:

‚ 휀푝p푞q “ 1 if 푞 has rank 1. ‚ 휀푝p푞1 ` 푞2q “ 휀푝p푞1q휀푝p푞2q ¨ pdiscp푞1q, discp푞2qq푝. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 43

Remark 3.17.2. This result is proved in elementary ways in [Ser] and [Cas]. We choose not to prove it here because a proper argument would require too much of a digression on Brauer groups. This is justiﬁed because Corollary 3.17.4 below gives an algorithm on how to compute it; the only question is well-deﬁnedness. Remark 3.17.3. We call this invariant the Hasse-Minkowski invariant of 푞.

2 Corollary 3.17.4. For 푞 diagonalized as 푞p푥1, . . . , 푥푛q “ 푖 푎푖푥푖 , we have: ř 휀푝p푞q “ p푎푖, 푎푗q푝. 푖ă푗 ź Exercise 3.9. Deduce the corollary from Theorem 3.17.1. 3.18. Using the Hasse-Minkowski invariant, we can complete the description of S3.15 describing when a quadratic form 푞 over 푘 “ Q푝 represents some 휆 P Q푝.

Proposition 3.18.1. A quadratic form 푞 over Q푝 represents 0 ‰ 휆 if and only if one of the following results holds: ‚ rankp푞q “ 1 and 휆 “ discp푞q P 푘ˆ{p푘ˆq2. ‚ rankp푞q “ 2 and p´ discp푞q, 휆q “ 휀푝p푞q. ‚ rankp푞q “ 3 and either 휆 ‰ ´ discp푞q or 휆 “ ´ discp푞q and p´ discp푞q, ´1q푝 “ 휀푝p푞q. ‚ rankp푞q ě 4. Proof. The case of rank 1 is obvious: it is true over any ﬁeld. The case of rank 2 is treated in Corollary 3.15.3, and the case of rank 4 is treated in Corollary 3.15.7. The case of rankp푞q “ 3 and 휆 ‰ ´ discp푞q is treated in Corollary 3.15.5. This leaves us to show that a non-degenerate ternary form 푞 respresents ´ discp푞q if and only if p´ discp푞q, ´1q푝 “ 휀푝p푞q. Let 푞p푥, 푦, 푧q “ 푎푥2 ` 푏푦2 ` 푐푧2. That 푞 represents ´ discp푞q “ ´푎푏푐 is equivalent to saying that 푞p푥, 푦, 푧, 푤q “ 푎푥2 ` 푏푦2 ` 푐푧2 ` 푎푏푐푤2 is isotropic, which in turn is equivalent to saying that 2 2 2 2 푞1p푥, 푦q “ 푎푥 ` 푏푦 and 푞2p푧, 푤q “ ´푐푧 ´ 푎푏푐푤 represent a common value. Note that discp푞1q “ discp푞2q. Therefore, by Corollary 3.15.3, saying that 푞1 and 푞2 represent a common value is equivalent to:

p푎, 푏q푝 “ 휀푝p푞1q “ 휀푝p푞2q “ p´푎푏푐, ´푐q푝. Using Theorem 3.14.5, we ﬁnd:

p´푎푏푐, ´푐q푝 “ p´푎푏푐, 푐q푝 ¨ p´푎푏푐, ´1q푝. Moreover, by Proposition 3.14.3 (6), we have p´푎푏푐, 푐q “ p푎푏, 푐q “ p푎, 푐q ¨ p푏, 푐q, so that the above is equivalent to:

휀푝p푞q “ p푎, 푏q푝p푎, 푐q푝p푏, 푐q푝 “ p´푎푏푐, ´1q푝 “ p´ discp푞q, ´1q푝 as desired.

Corollary 3.18.2. For 푘 “ Q푝, two non-degenerate quadratic forms 푞1 and 푞2 over 푘 are equivalent if and only if discp푞1q “ discp푞2q and 휀푝p푞1q “ 휀푝p푞2q. 44 SAM RASKIN

Proof. By Proposition 3.18.1, two quadratic forms 푞1 and 푞2 with the same discriminant and Hasse- Minkowski invariant represent the same values. Choose some 휆 that they both represent. 2 1 2 1 Then 푞1 “ 휆푥 ` 푞1 and 푞2 “ 휆푥 ` 푞2 by our usual method. Then we have:

1 1 ˆ ˆ 2 discp푞1q “ 휆 discp푞1q “ 휆 discp푞2q “ discp푞2q P 푘 {p푘 q 1 1 1 1 and 휀푝p푞1q “ 휀푝p푞1qpdiscp푞1q, 휆q푝 “ 휀푝p푞2qpdiscp푞2q, 휆q푝 “ 휀푝p푞2q so we obtain the result by induction. Remark 3.18.3. As a consequence, we obtain the following nice interpretation of the Hasse-Minkowski invariant: 푞 has 휀푝p푞q “ 1 if and only if 푞 is equivalent to a form:

2 2 2 푥1 ` ... ` 푥푛´1 ` 푑푥푛. Indeed, this is clearly only possible with 푑 “ discp푞q, and in that case, we see that this form has the same discriminant as 푞, and it obviously has 휀푝 “ 1.

4. The Hasse principle 4.1. The goal for this section is to prove the following theorem.

Theorem 4.1.1 (Hasse principle). A non-degenerate quadratic form 푞 over Q represents a value 휆 P Q if and only if the extension of scalars 푞Q푝 represents 휆 P Q Ď Q푝 for every prime 푝, and 푞R represents 휆. 4.2. We will prove Theorem 4.1.1 by treating different ranks 푛 of 푞 separately. Exercise 4.1. Deduce the 푛 “ 1 case of Theorem 4.1.1 from the discussion of Example 2.4.3 (4). We will give two proofs of the 푛 “ 2 case below, one from [Cas] using Minkowski’s geometry of numbers, and a second from [Ser] that is more direct. 4.3. Geometry of numbers. We now give a quick crash course in Minkowski’s geometry of numbers, following [Cas] Chapter 5. 푛 Definition 4.3.1. A lattice is an abelian group Λ isomorphic to Z for some integer 푛, i.e., it is a finitely generated torsion-free abelian group. The integer 푛 is called the rank of Λ. A metrized lattice is a pair pΛ, 푞Rq of a lattice Λ and a positive-definite quadratic form 푞R defined on ΛR Ď Λ b R. Z

Remark 4.3.2 (Discriminants). Let pΛ, 푞Rq be a metrized lattice. Let 푒1, . . . , 푒푛 be a basis of Λ. Then we can associate to this datum the matrix:

1 1 1 2 퐵푞 p푒1, 푒1q 2 퐵푞 p푒2, 푒1q ... 2 퐵푞 p푒푛, 푒1q 1 R 1 R 1 R 2 퐵푞 p푒1, 푒2q 2 퐵푞 p푒2, 푒2q ... 2 퐵푞 p푒푛, 푒2q 퐴 “ ¨ R . R . . R . ˛ . . . . . ˚ 1 1 1 ‹ ˚ 퐵푞 p푒1, 푒푛q 퐵푞 p푒2, 푒푛q ... 퐵푞 p푒푛, 푒푛q‹ ˚ 2 R 2 R 2 R ‹ As in Exercise 2.9, a change˝ of basis corresponding to a matrix 푆 P 퐺퐿푛‚pZq changes 퐴 by:

퐴 ÞÑ 푆푇 퐴푆. ˆ ˆ Because detp푆q P Z “ t1, ´1u, we see that disc pΛ, 푞Rq :“ detp퐴q P R is well-deﬁned. ` ˘ INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 45

Remark 4.3.3. Because 푞R is assumed to be a positive-definite quadratic space, pΛR, 푞Rq is isomor- 푛 2 2 phic to pR , 푥1 ` ... ` 푥푛q. We can therefore talk about open sets in ΛR and volumes of such open 푛 sets, since these notions do not depend on the given isomorphism (i.e., they are invariants of R 2 2 under the action of the orthogonal group for the form 푥1 ` ... ` 푥푛). ˆ ˆ 2 Remark 4.3.4. The discriminant in the sense above refines discp푞Rq P R {pR q “ t1, ´1u, and therefore we see that (because 푞R is assumed positive definite) that disc pΛ, 푞Rq ą 0. Remark 4.3.5. If 푉 is a finite-dimensional vector space over R, we say` that a˘ subset 푈 of 푉 is convex if 푥, 푦 P 푈 implies that 푡푥 ` p1 ´ 푡q푦 P 푈 for all 푡 P r0, 1s. We say that 푈 is symmetric if 푥 P 푈 implies that ´푥 P 푈.

Theorem 4.3.6 (Minkowski’s theorem). Let pΛ, 푞Rq be a metrized lattice of rank 푛 and let 푈 Ď ΛR be a convex symmetric open set with:

푛 volp푈q ą 2 ¨ disc pΛ, 푞Rq . Then 푈 contains a non-zero point of Λ. b ` ˘ Proof. For convenience, we use the language of measure theory. Note that 푇 :“ ΛR{Λ inherits a canonical measure 휇푇 , where the measure of a “small” connected subset 푇 is by deﬁnition the measure of a connected component of the inverse image in ΛR. Indeed, one can choose a fundamental domain 퐹 in ΛR for the action of Λ and restrict the usual measure on ΛR. E.g., choosing a basis 푒1, . . . , 푒푛 for Λ, we can choose 퐹 to be:

t푥 “ 푥푖푒푖 P ΛR, 푥푖 P R | 0 ď 푥푖 ă 1u. Clearly 휇 is an additive measureÿ on the torus 푇 , i.e., 휇p푥 ` 푉 q “ 휇p푉 q for every open set 푉 Ď 푇 and 푥 P 푇 . We claim:

휇p푇 q “ disc pΛ, 푞Rq . (4.1) b Indeed, choose an orthonormal basis 푓1, . . . , 푓푛 of Λ`R, and˘ deﬁne the matrix 퐵 P 푀푛pRq to have 푖th column the vectors 푒푖 written in the basis 푓1, . . . , 푓푛. By the usual characterization of volumes in terms of determinants, we have:

| detp퐵q| “ volp퐹 q “ 휇p푇 q. But for 퐴 as in Remark 4.3.2, we obviously have 퐵푇 퐵 “ 퐴, and therefore:

푇 2 disc pΛ, 푞Rq “ detp퐴q “ detp퐵 퐵q “ detp퐵q as was claimed in (4.1). ` ˘ Let 푝 denote the quotient map ΛR Ñ 푇 . Deﬁne the function:

휒 : 푇 Ñ R Y t8u 1 ´1 10 by letting 휒p푥q be the number of points in 2 푈 X 푝 p푥q. Note that 휒 is a measurable function. Then we have:

10One can prevent this function from possibly taking the value 8 by intersecting 푈 with a large disc around 0, but it’s not a big deal. 46 SAM RASKIN

1 1 휒p푥q푑휇p푥q “ volp 푈q “ volp푈q ą disc pΛ, 푞 q “ 휇p푇 q. 2 2푛 R ż푇 Therefore, 휒p푥q ą 1 for some 푥 P 푇 . Since 휒 takes integer values,` we˘ must have 휒p푥q ě 2, and 1 therefore there exist distinct points 푦, 푧 P 2 푈 with 푝p푦q “ 푝p푧q “ 푥. Then we have: 1 1 0 ‰ 푦 ´ 푧 “ p2푦q ` p´2푧q P 푈 2 2 by symmetry and convexity of 푈, and also 푦 ´ 푧 P Λ since 푝p푦 ´ 푧q “ 푝p푦q ´ 푝p푧q “ 0. Therefore, 푦 ´ 푧 is a non-zero point of Λ X 푈, as claimed. 1 Exercise 4.2. For pΛ, 푞Rq a metrized lattice and Λ Ď Λ a subgroup of index 푛 ă 8, prove that:

1 2 disc pΛ , 푞Rq “ 푛 disc pΛ, 푞Rq . 4.4. Application of geometry of numbers` ˘ to the Hasse` principle.˘ The above theorem can be used to prove Theorem 4.1.1 in the 푛 “ 2 case in general: we refer to [Cas] S6.4 where this is eﬀected. However, to simplify the explanation given in [Cas], we will only give the argument in a special case (that at least shows how things are done, and we’ll treat the general 푛 “ 2 case by a simpler method following [Ser] in S4.6). First, note that any binary quadratic form 푞p푥, 푦q “ 푎푥2 ` 푏푦2 is equivalent to one where 푣푝p푎q, 푣푝p푏q P t0, 1u for every prime 푝. Indeed, multiplying 푎 and 푏 by squares, we easily put them into this form. Let 푞p푥, 푦q “ 푎푥2 ` 푏푦2 be a binary quadratic form with 푎 and 푏 square-free as above, and, moreover assuming: (1) 푎, 푏 P Z relatively prime, (2) 푎, 푏 odd, and (3) 푎 ` 푏 “ 0 mod 4. We will prove that if 푞 represents 1 in every Q푝 then it does so in Q. For each prime 푝 dividing 푎, the fact that 푎푥2 ` 푏푦2 represents 1 means that we can choose some 2 푟푝 P Z with 푏푟푝 “ 1, and similarly for each prime 푝 dividing 푏, we can choose some 푟푝 P Z with 2 푎푟푝 “ 1. Indeed, this follows e.g. from Proposition 3.16.2. 3 3 Then deﬁne Λ Ď Z as consisting of the triples p푥, 푦, 푧q P Z with:

푦 “ 푟푝푧 if 푝 | 푎 푥 푟 푧 if 푝 푏 $ “ 푝 | ’푥 “ 푦 mod 2 &’ 2 | 푧. ’ 3 3 We consider Λ as a metrized lattice%’ by embedding Z into R in the usual way. By Exercise 4.2, we have discpΛq “ p2푎푏q4. Observe that for p푥, 푦, 푧q P Λ and 푝 a prime dividing 푎, we have:

2 2 2 2 2 푎푥 ` 푏푦 “ 푏푟푝푧 “ 푧 mod 푝 and similarly for primes dividing 푏. Moreover, we have:

푎푥2 ` 푏푦2 “ 0 “ 푧2 mod 4 where the ﬁrst equality follows because 푏 “ ´푎 mod 4, so 푎푥2 `푏푦2 “ 푎푥2 ´푎푦2 “ 푎p푥´푦qp푥`푦q “ 0 mod 4 because 푥 ´ 푦 “ 0 mod 4. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 47

Therefore, by the Chinese remainder theorem and the square-free assumption of 푎 and 푏, we have:

푎푥2 ` 푏푦2 “ 푧2 mod 4|푎푏|. (4.2) Now deﬁne:

3 2 2 2 푈 “ tp푥, 푦, 푧q P R | |푎|푥 ` |푏|푦 ` 푧 ă 4|푎푏|.u Then 푈 is obviously convex and symmetric, and moreover, we ﬁnd:

3 4 p4|푎푏|q 2 휋 volp푈q “ 휋 “ 23 ¨ 4|푎푏| ą 23 ¨ 4|푎푏| “ 23 discpΛq. 3 |푎| |푏| 3 Therefore, Minkowski’s theorem guarantees the existence of non-zero 푥, 푦, 푧 in Λ and 푈. Then a a p q we have:

2 2 2 푎푥 ` 푏푦 ´ 푧 P 4|푎푏|Z by (4.2), but lying in 푈 implies that:

|푎푥2 ` 푏푦2 ´ 푧2| ď |푎|푥2 ` |푏|푦2 ` 푧2 ă 4|푎푏| and therefore we must have 푎푥2 ` 푏푦2 “ 푧2 as desired. 4.5. Digression: Fermat and Legendre’s theorems. We now digress temporarily to prove two classical results. Theorem 4.5.1 (Fermat’s theorem). An odd prime 푝 can be written as the sum of two squares 2 2 푝 “ 푥 ` 푦 (푥, 푦 P Z) if and only if 푝 “ 1 mod 4. Proof. Necessity follows from congruences mod 4. For suﬃciency, suppose 푝 “ 1 mod 4. In this case, we can choose an integer 푟 with 푟2 “ ´1 mod 푝. 2 Deﬁne the lattice Λ Ď Z to consist of those p푥, 푦q with 푦 “ 푟푥 mod 푝. Note that for p푥, 푦q P Λ, we have:

푥2 ` 푦2 “ 푥2 ` 푟2푥2 “ 0 mod 푝. Moreover, observe that the discriminant of the metrized latticed Λ is 푝2. 2 Deﬁne 푈 Ď R to be the disc:

2 2 2 tp푥, 푦q P R | 푥 ` 푦 ă 2푝.u Then the area of 푈 is 2푝 ¨ 휋. Now observing that:

2푝 ¨ 휋 ą 4푝 because 휋 ą 2, Minkowski’s theorem applies, and we deduce the existence of p푥, 푦q ‰ p0, 0q in 푈 XΛ. But then 푝 | 푥2 ` 푦2 and 0 ă 푥2 ` 푦2 ă 2푝, so we must have 푥2 ` 푦2 “ 푝. Theorem 4.5.2 (Legendre’s theorem). Any integer 푛 ě 0 can be written as the sum of four squares:

2 2 2 2 푛 “ 푥 ` 푦 ` 푧 ` 푤 , 푥, 푦, 푧, 푤 P Z. 48 SAM RASKIN

Proof. First, observe that it suffices to treat the case where 푝 is a prime. Indeed, it is a easy, elementary identity to verify that the product of numbers that are each the sum of four squares is again of the same form (one uses the norm form on the quaternions to actually reconstruct this identity in practice). By Proposition 3.13.1, we can find integers 푟 and 푠 with 푟2 ` 푠2 “ ´1 mod 푝. We then define the lattice:

4 Λ “ tp푥, 푦, 푧, 푤q P Z | 푧 “ p푟푥 ` 푠푦q mod 푝, 푤 “ p푟푥 ´ 푠푦q mod 푝u. Then for p푥, 푦, 푧, 푤q P Λ, observe that:

푥2 ` 푦2 ` 푧2 ` 푤2 “ 푥2 ` 푦2 ` p푟푥 ` 푠푦q2 ` p푟푥 ´ 푠푦q2 “ 푥2 ` 푦2 ` p푟2 ` 푠2qp푥2 ` 푦2q “ 0 mod 푝. Observe that Λ has discriminant 푝4. 4 2 2 2 2 ? Now let 푈 Ď R be the disc tp푥, 푦, 푧, 푤q P R | 푥 ` 푦 ` 푧 ` 푤 ă 2푝u of radius 2푝. We recall 4 휋2 2 2 that in R , the disc of radius 1 has volume 2 . Therefore, 푈 has volume 2휋 푝 . Now, because 휋2 ą 8, we have:

2휋2푝2 ą 24푝2 and therefore Minkowski’s theorem applies. Clearly any non-zero vector in ΛX푈 solves our equation, giving the desired result. 4.6. Hasse-Minkowski for 푛 “ 2. We now prove the Hasse principle for binary quadratic forms following [Ser]. Proof of Theorem 4.1.1 for binary forms. Let 푞p푥, 푦q “ 푎푥2 ` 푏푦2. Up to rescaling, it suffices to show that 푞 represents 1 if and only if p푎, 푏q푝 “ 1 for all 푝 and p푎, 푏q8 “ 1. As in S4.4, we can assume 푎 and 푏 are (possibly negative) square-free integers. Without loss of generality, we can assume |푎| ě |푏|. We proceed by induction on |푎| ` |푏|. The base case is |푎| ` |푏| “ 2, i.e., |푎| “ |푏| “ 1. Then we must have 푎 “ 1 or 푏 “ 1, since otherwise 푎 “ 푏 “ ´1 and p푎, 푏q8 “ ´1, and the result is clear. We now perform the inductive step. We can assume |푎| ą 1. We first claim that 푏 is a square in Z{푎Z. By the Chinese remainder theorem and by the assumption that 푎 is square-free, it suffices to show that if 푝 is a prime dividing 푎, then 푏 is a square mod 푝. If 푝 “ 2, this is clear, since everything is a square in Z{2Z. Therefore, let 푝 be an odd prime dividing 푎. If 푝 divides 푏, then 푏 “ 0 mod 푝 and we’re done. Otherwise, 푏 is a unit in Z푝, and therefore p푎, 푏q푝 “ 1 implies that 푏 is a square by Proposition 3.16.2. That 푏 is a square in Z{푎Z means that we can find 푠, 푡 P Z such that:

푡2 ´ 푏 “ 푠푎. |푎| Certainly we can choose 푡 with 0 ď 푡 ď 2 . We claim that for 푝 “ 8 or 푝 a prime number,11 we have:

11 Not to be confused with our earlier use of 푝. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 49

p푏, 푠q푝 “ 1. (4.3) 2 Indeed, ﬁrst observe that p푏, 푡 ´ 푏q푝 “ 1 because:

1 1 푏 ¨ p q2 ` p푡2 ´ 푏q ¨ p q2 “ 1. 푡 푡 Then we observe that:

푡2 ´ 푏 p푏, 푠q “ p푏, q “ p푏, 푎p푡2 ´ 푏qq “ p푏, 푎q p푏, 푡2 ´ 푏q “ 1 ¨ 1 “ 1. 푝 푎 푝 푝 푝 푝 Moreover, we have:

푡2 ´ 푏 푡2 |푏| |푎| |푠| “ | | ď ` ď ` 1 ă |푎|. 푎 |푎| |푎| 4 |푎| where the last inequality holds because |푎| ě 2, and the ﬁrst inequality holds because 푡 ď 2 and |푏| ď |푎|. Then |푎|`|푏| ą |푠|`|푏|, and therefore, since p푏, 푠q푝 “ 1 for all 푝, by induction there exist rational numbers 푥 and 푦 such that:

푠푥2 ` 푏푦2 “ 1. (4.4) In this case, we claim:

푎p푠푥q2 ` 푏p1 ´ 푡푦q2 “ p푡 ´ 푏푦q2 (4.5) meaning that 푞 represents a square, giving the desired result. More precisely: if 푡 ´ 푏푦 ‰ 0, we’re clearly done, and if 푡´푏푦 “ 0, then we have shown that 푞 is a hyperbolic and therefore it represents 1.12 To verify this algebra, we compute:

p푎푠qp푠푥2q ` 푏p1 ´ 푡푦q2 “ p푡2 ´ 푏qp1 ´ 푏푦2q ` 푏 ´ 2푏푦푡 ` 푏푡2푦2 “ 푡2 ´ 푏 ´ 푏푡2푦2 ` 푏2푦2 ` 푏 ´ 2푏푦푡 ` 푏푡2푦2 “ 푡2 ` 푏2푦2 ´ 2푏푦푡 “ p푡 ´ 푏푦q2. Remark 4.6.1. The end of the proof could be explained in the following more conceptual way. First, ˆ 2 2 1 2 푦 2 for a field 푘 and fixed 푏 P 푘 not a square, to say that 푎푥 `푏푦 “ 1 is equivalent to 푎 “ p 푥 q ´푏p 푥 q , and from here one easily finds? that the solvability of this equation is equivalent to say that 푎 is a norm for the extension 푘r 푏s of 푘. In this way, using the multiplicativity of the norm map, one sees that the set of 푎 for which this equation is solvable forms a group under multiplication. Now, in the notation of the proof, we could have instead said that the identity 푎푠 “ 푡2 ´ 푏 says that 푎푠 is a norm, and the identity 푠푥2 ` 푏푦2 “ 1 says that 푠 is a norm, which in turn implies that 푠´1 is a norm, so this implies that 푎 “ 푎푠 ¨ 푠´1 is a norm, as desired. Indeed, the elementary but random identity verified at the end of the argument translates to this argument. By the same method, one could verify (4.3) more directly.

12To deduce that 푞 is hyperbolic, we need to be careful that in (4.5) we have 푠푥 and 1 ´ 푡푦 non-zero. But (4.4) and the square-free assumption on 푏 force 푠푥 to be non-zero, giving the claim. 50 SAM RASKIN

4.7. Quadratic reciprocity. We will need to appeal to Gauss’s quadratic reciprocity law below. We recall the statement and proof here. Theorem 4.7.1. (1) For 푝 and 푞 distinct odd primes, we have:

푝 푝´1 ¨ 푞´1 푞 “ p´1q 2 2 . 푞 푝 ˆ ˙ ˆ ˙ (2) For 푝 an odd prime, we have:

2 푝2´1 “ p´1q 8 . 푝 ˆ ˙ 푝´1 ¨ 푞´1 Remark 4.7.2. We note that the complicated-looking expression p´1q 2 2 produces the value 1 unless 푝 “ 푞 “ 3 mod 4, in which case it gives the value ´1. Proof of Theorem 4.7.1 (1). We follow the elementary proof of [Ser]. Let F푝 denote an algebraic closure of F푝 in what follows.

Step 1. Here is the strategy of the proof below. We will write down the Gauss sum G P F푝, and 푞´1 푝 2 verify ﬁrst G is a square root of p´1q ¨ 푞, and second that G lies in F푝 exactly when 푞 “ 1. 푞´1 푝 2 ´ ¯ This means that p´1q ¨ 푞 is a quadratic residue modulo 푝 exactly when 푞 “ 1, so that:

푞´1 푞´1 ´ ¯ p´1q 2 푞 푝 p´1q 2 푞 푝 1 “ ¨ “ . 푞 푝 푞 ˜ 푝 ¸ ˆ ˙ ˜ 푝 ¸ ˆ ˙ ˆ ˙ 1 푝´1 13 ´ 2 Then noting that 푝 “ p´1q , we obtain the result.

´ ¯ 푞 푞 Step 2. Let 휁푞 P F푝 be a solution to 푥 “ 1 other than 1; such a 휁푞 exists because 푥 ´ 1 is a separable polynomial over F푝. We deﬁne the Gauss sum G P F푝 as:

푥 푥 G “ ¨ 휁푞 . ˆ 푞 푥ÿPF푞 ˆ ˙ 2 푞´1 Step 3. We claim that G “ p´1q 2 푞. Indeed, we compute:

2 푥 푦 푥`푦 푥푦 푥`푦 푧 푥`푥푧 G “ 휁푞 “ 휁푞 “ 휁푞 . ˆ 푞 푞 ˆ 푞 ˆ 푞 ˆ 푥,푦ÿPF푞 ˆ ˙ ˆ ˙ 푥,푦ÿPF푞 ˆ ˙ 푧ÿPF푞 ˆ ˙ 푥ÿPF푞 푦 Here we have applied the change of variables 푧 “ 푥 , noting that:

푦 푥 푦 푥푦 푥 “ “ . 푝 푝 푝 ˆ ˙ ˜ ¸ ˆ ˙

13 ˆ 푝´1 Proof: Under an isomorphism F푞 » Z{p푝 ´ 1qZ, clearly ´1 corresponds to 2 , and this element is a multiple of 2 if and only if 4 | 푝 ´ 1. INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 51

푥`푥푧 푧`1 푥 Now observe that for 푧 ‰ ´1, 휁푞 “ p휁푞 q ranges exactly over the primitive 푞th roots of unity in F푝. Because the sum of all the 푞th roots of unity is 0, this means that for 푧 ‰ 1 the corresponding summand is ´1. Therefore, the above sum is:

´1 푧 1 ` ´ . 푞 푞 ˆ 푧 0, 1 ˆ ˙ 푥ÿPF푞 PF푞ÿzt ´ u ˆ ˙ where the first term corresponds to 푧 “ ´1 and the rest corresponds to 푧 ‰ ´1. We note that 푧 ˆ “ 0, since there are equal numbers of quadratic residues and non-residues, so the 푧PF푞 푞 ´ ¯ ´1 ´1 řsecond term above is 푞 . Clearly the first term sums to p푞 ´ 1q 푞 , giving the result. ´ ¯ ´ ¯ Step 4. Next, we recall a general technique for checking if an element of F푝 lies in F푝. 푝 For 푘 a field of characteristic 푝, note that by the binomial formula that Fr푝p푥q :“ 푥 is a homomorphism. 푝 Then we claim that 푥 P F푝 lies in F푝 if and only if 푥 “ 푥. Indeed, this is obviously satisfied for elements of F푝, and for degree reasons there are at most 푝 roots of this polynomial.

푝 Step 5. It remains to show that G lies in F푝 exactly when 푞 “ 1. It suﬃces to check when G is ﬁxed by the Frobenius in F푝. ´ ¯ We have:

푝 ´1 푥 푝푥 푥 푝푥 푝 푥 푥 푝 Fr푝pGq “ ¨ 휁푞 “ ¨ 휁푞 “ ¨ 휁푞 “ G ˆ 푞 ˆ 푞 ˆ 푞 푞 푥ÿPF푞 ˆ ˙ 푥ÿPF푞 ˆ ˙ 푥ÿPF푞 ˆ ˙ ˆ ˙ as desired.

Proof of Theorem 4.7.1? (2). The strategy is similar to that of Theorem 4.7.1 (1): we ﬁnd a convenient expression for 2 P F푝 and then test by the Frobenius when it lies in F푝. ´1 14 Let 휁8 P F푝 be a primitive 8th root of unity. We claim that 휁8 ` 휁8 is a square root of 2. Indeed, we compute:

´1 2 2 ´2 p휁8 ` 휁8 q “ p휁8 ` 2 ` 휁8 q. 2 ´2 Observing that 휁8 and 휁8 are distinct square roots of ´1, their sum must be zero, giving the claim. Now we observe that:

´1 푝 ´푝 Fr푝p휁8 ` 휁8 q “ 휁8 ` 휁8 2 ´1 푝 ´1 which equals 휁8 `휁8 if and only if 푝 “ 1, ´1 mod 8. This is equivalent to requiring that p´1q 8 “ 1.

? ? 14 2 2 As a heuristic for this expression, note that over the complex numbers 2 ` 푖 ¨ 2 is visibly an 8th root of unity ´1 ? 휁8, and we have 휁8 ` 휁8 “ 휁8 ` 휁8 “ 2 Rep휁8q “ 2. 52 SAM RASKIN

4.8. Global properties of the Hasse-Minkowski invariant. We have the following result, which we will see is essentially a repackaging of the quadratic reciprocity law.

Proposition 4.8.1 (Hilbert reciprocity). Let 푎, 푏 P Qˆ. For almost every15 prime 푝, we have:

p푎, 푏q푝 “ 1. Moreover, we have:

p푎, 푏q8 ¨ p푎, 푏q푝 “ 1. 푝 prime ź Proof. For the ﬁrst claim, note that if 푝 is an odd prime with 푣푝p푎q “ 푣푝p푏q “ 0, then Proposition 3.16.2 implies that p푎, 푏q푝 “ 1. For the second part, we proceed by cases, applying Propositions 3.16.2 and 3.16.3 freely. Case 1. 푎 “ 푝, 푏 “ 푞 are distinct odd prime numbers. Then we have:

푝´1 푞´1 p푝, 푞q2 “ p´1q 2 2 푞 $p푝, 푞q푝 “ 푝 ’ ’ 푝, 푞 ´푝 ¯ ’p q푞 “ 푞 ’ &p푝, 푞qℓ “ 1´ ¯ if ℓ ‰ 2, 푝, 푞 ’p푝, 푞q8 “ 1. ’ Then Theorem 4.7.1 (1) immediately’ gives the claim. %’ Case 2. 푎 “ 푏 “ 푝 an odd prime number. Then we have:

p 푝´1 q2 푝´1 p푝, 푝q2 “ p´1q 2 “ p´1q 2 ´1 $p푝, 푝q푝 “ 푝 ’ ’p푝, 푝q “ 1´ ¯ ifℓ ‰ 푝 &’ ℓ p푝, 푝q8 “ 1 ’ giving the claim. ’ %’ Case 3. 푎 “ 푝 is an odd prime and 푏 “ 2. Then we have:

푝2´1 p푝, 2q2 “ p´1q 8 2 $p푝, 2q푝 “ 푝 ’ &p푝, 2qℓ “ 1´ ¯ if ℓ ‰ 2, 푝.p푝, 2q8 “ 1. Then Theorem 4.7.1 (2) gives’ the claim. %’

15This phrase means “all but finitely many.” INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 53

Case 4. 푎 “ 푏 “ 2. Then we have p2, 2q푝 “ 1 for 푝 prime and for 푝 “ 8 because: 1 1 2p q2 ` 2p q2 “ 1. 2 2 Case 5. 푎 “ 푝 is an odd prime and 푏 “ ´1. Then we have:

푝´1 p푝, ´1q2 “ p´1q 2 ´1 $p푝, ´1q푝 “ 푝 ’ ’p푝, ´1q “ 1´ ¯ if ℓ ‰ 2, 푝 &’ ℓ p푝, ´1q8 “ 1 ’ giving the result (one can alternatively’ reduce this case to Case 2). %’ Case 6. 푎 “ 2 and 푏 “ ´1. Then we have p2, ´1q푝 “ 1 for all 푝 prime and 푝 “ 8 because:

2 ¨ p1q2 ´ p12q “ 1. Case 7. 푎 “ ´1 and 푏 “ ´1. Then we have:

p´1, ´1q2 “ ´1 $p´1, ´1qℓ “ 1 if ℓ ‰ 2 &’p´1, ´1q8 “ ´1 giving the result. %’ Case 8. General case. By the bimultiplicativity of the Hilbert symbol (Theorem 3.14.5 (1)), we reduce to the above cases. Remark 4.8.2. Hilbert’s form of the quadratic reciprocity law should be viewed as analogous to the conclusion of Exercise 3.3.

Corollary 4.8.3. For any quadratic form 푞 over Q, we have:

휀8p푞q ¨ 휀푝p푞q “ 1. 푝 prime ź Corollary 4.8.4. If 푞 is a non-degenerate binary quadratic form over Q, 푞 represents 휆 P Q if and only if it represents it in each Q푝 and in R with possibly one exception.

Proof. Indeed, we have seen in Proposition 3.15.1 that 푞 represents 휆 in Q푝 (allowing 푝 “ 8 to correspond to R) if and only if p´ discp푞q, 휆q푝 “ 휀p푞q. Therefore, Hilbert reciprocity implies that if 푞 represents 휆 locally with one exception, then 푞 represents 휆 locally with no exceptions, and therefore, the Hasse principle for 푛 “ 2 implies that 푞 represents 휆 in Q. 54 SAM RASKIN

4.9. We now conclude the proof of the Hasse principle. Lemma 4.9.1. Let 푆 be a finite set consisting of primes and possibly the symbol 8. ˆ Suppose that we are given 푥푝 P Q푝 for every 푝 P 푆 (where Q8 is understood as R). Then there exists a prime ℓ and 푥 P Q such that:

ˆ 2 푥 P 푥푝pQ푝 q for all 푝 P 푆 (4.6) ˆ 푥 P Z푝 for all primes 푝 R 푆 Y tℓu. Proof. We suppose 8 P 푆 for convenience (certainly it does not harm either our hypothesis or our conclusion!). Let 휀 P t1, ´1u be the sign of 푥8. By Dirichlet’s theorem on primes in arithmetic progressions, there exists a prime ℓ such that:

´푣푝p푥q ´푣푞p푥푞q ℓ “ 휀 ¨ p푝 푥푝q ¨ 푞 mod 푝 푞P푆 prime 푞ź‰푝 for every odd prime 푝 in S, and for 푝 “ 2, we impose the same congruence but require it to hold mod 푝3 “ 8. We now deﬁne:

푣 p푥 q 푥 “ 휀 ¨ ℓ ¨ 푞 푞 푞 P Q. 푞P푆 prime ź Clearly 푥 has no primes other than those in 푆 and ℓ divides the numerator or denominator of 푥, so the second requirement of (4.6) holds for it. We see that 푥 has the same sign as 푥8. Moreover, for 푝 P 푆 an odd prime, we have:

´1 푥푥푝 P 1 ` 푝Z푝 ´1 and therefore is a square. For 푝 “ 2, we similarly find 푥푥2 P 1 ` 8Z2, and again, therefore is a square. Completion of the proof of Theorem 4.1.1. Case 1. To prove the 푛 “ 3 case of the Hasse principle, it suffices to show that any non-degenerate form 푞 of rank 4 that is locally isotropic (i.e., isotropic over each Q푝 and R) is globally isotropic (i.e., isotropic over R). Let 푞 be such a form. Diagonalizing 푞 we see that we can write 푞 as the difference of two binary quadratic forms 푞1 ´ 푞2. Let 푆 be the set consisting of 2, 8, and the (finite) set of primes 푝 with at least one of 푣푝pdiscp푞1qq or 푣푝pdiscp푞2qq odd. Because 푞 is locally isotropic, for every 푝 P 푆, we can find 훼푝, 훽푝 P Q푝 ˆ Q푝 (with Q푝 meaning R for 푝 “ 8) not both the zero vector such that 푞1p훼푝q “ 푞2p훽푝q. We define 푥푝 to be the common value 푞1p훼푝q “ 푞2p훽푝q. We can take 푥푝 to be non-zero: indeed, e.g., if 훼푝 ‰ 0 and 푞1p훼푝q “ 0, then 푞1 is the hyperbolic plane and therefore represents any non-zero value represented by 푞2p훽푝q. ˆ Therefore, we can find 푥 P Q associated with these 푥푝 as in Lemma 4.9.1. Let ℓ be defined as in loc. cit. Observe that 푞1 represents 푥 in R and in each Q푝, 푝 ‰ ℓ. Indeed, for 푝 P 푆 this follows because 푞1 ˆ 2 represents 푥푝 and therefore every element of 푥푝pQ푝 q . Then for 푝 R 푆, 푝 ‰ ℓ: by Proposition 3.15.1, INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 55 we needs to show that p´ discp푞1q, 푥푝q “ 1. Because 푝 R 푆, 푣푝pdiscp푞1qq is even, and therefore the ˆ ˆ 2 ˆ coset discp푞1q P Q푝 {pQ푝 q is represented by an element of Z푝 . Then because 푝 is odd (by virtue of ˆ not being in 푆) and 푥푝 P Z푝 , we obtain the required identity of Hilbert symbols from Proposition 3.16.2. Now by Corollary 4.8.4, 푞1 represents 푥 over Q, say, 푞1p훼q “ 푥. By the same logic, we can find 훽 P Q ˆ Q with 푞2p훽q “ 푥. Then 푞1p훼q ´ 푞2p훽q “ 0, proving that 푞 was isotropic. Case 2. We now treat the 푛 ě 4 case of the Hasse principle. Again, it suffices to show a non-degenerate rational form 푞 of rank ě 5 that is locally isotropic is globally isotropic. We again write 푞 “ 푞1 ´ 푞2, where 푞1 is a non-degenerate binary form and 푞2 is a form of rank ě 3. Let 푆 be set consisting of 8, 2, and that finite set of primes for which 푣푝pdiscp푞2qq is odd or 휀푝p푞2q “ ´1. Because 푞 is locally isotropic, we can find 푥푝 and 푦푝 vectors over Q푝 with 푞p푥푝q “ 푞p푦푝q ‰ 0 for every 푝 P 푆. ´1 ˆ 2 By the Chinese remainder theorem, we can find 푥 P Q such that 푞1p푥q푞1p푥푝q P pQ푝 q for every 푝 P 푆.16 As before, it suffices to show that 푞2 represents 푞1p푥q over Q. By induction, it suffices to show that 푞2 represents 푞1p푥q locally. This is clear for 푝 P 푆, in particular, for 푝 “ 2 or 푝 “ 8. For all other primes 푝, we have:

ˆ ˆ 2 ˆ ˆ 2 discp푞2q P Z푝 ¨ pQ푝 q Ď Q푝 {pQ푝 q so that p´ discp푞2q, ´1q푝 “ 1 (by Proposition 3.16.2) and 휀푝p푞2q “ 1, and therefore we obtain that 푞2 represents every value over Q푝 by Proposition 3.18.1. 4.10. Three squares theorem. As an application, we now deduce the following result of Gauss. ě0 2 2 2 Theorem 4.10.1. Any integer 푛 P Z can be written as the sum 푥1 ` 푥2 ` 푥3, 푥푖 P Z, if and only if 푥 is of the form 4푛p8푘 ` 7q. 2 2 2 Exercise 4.3. (1) Show that 푞p푥, 푦, 푧q “ 푥 ` 푦 ` 푧 represents every 휆 P Q푝 for 푝 odd, and 2 represents 휆 P Q2 if and only if ´휆 P 2 Z ¨ p1 ` 8Z2q. (2) Deduce the rational version of Theorem 4.10.1 from the Hasse principle, i.e., the version in which 푥1, 푥2, 푥3 are taken to be rational numbers instead of integers. By Exercise 4.3, it suﬃces to pass from rational solutions to integral solutions. The following result gives a way to do this in some cases. Proposition 4.10.2 (Hasse-Davenport). Let Λ be a finite rank free abelian group equipped with an 17 integral quadratic form 푞 :Λ Ñ Z. Let ΛQ “ ΛbQ, and we abuse notation in letting 푞 :ΛQ Ñ Q denote the induced quadratic form. Suppose that the quadratic form 푞 is anisotropic as a quadratic form over Q. Suppose moreover that for every 푣 P ΛQ :“ Λ b Q, there exists 푤 P Λ with:

|푞p푣 ´ 푤q| ă 1.

16We emphasize that we do not need Lemma 4.9.1 here: we are not imposing any conditions away from the primes in 푆. 17Since we’ve only spoken about quadratic forms on vector spaces before, this warrants an explanation: all this 2 means is that 푞p푛푣q “ 푛 푞p푣q for every 푛 P Z and 푣 P Λ. 56 SAM RASKIN

Then 푞 represents 휆 P Z by a vector in Λ if and only if 푞 represents 휆 with a vector in ΛQ. Proof of Theorem 4.10.1. By Exercise 4.3, it suffices to see that 푞p푥, 푦, 푧q “ 푥2 ` 푦2 ` 푧2 satisfies the hypotheses of Proposition 4.10.2. Observe that 푞 is anisotropic over Q because it is so over R (by positive-definiteness). 3 1 1 1 3 1 1 Then for every p푥, 푦, 푧q P Q , we can find p푥 , 푦 , 푧 q P Z with |푥 ´ 푥 | ď 2 , and similarly for 푦 and 푧. Then:

1 1 1 푞p푥 ´ 푥1, 푦 ´ 푦1, 푧 ´ 푧1q ď p푥 ´ 푥1q2 ` p푦 ´ 푦1q2 ` p푧 ´ 푧1q2 ď ` ` ă 1 4 4 4 as desired.

Proof of Proposition 4.10.2. Here is the strategy of the proof: we start we some 푣 P ΛQ representing 1 휆 ‰ 0, and we need to find some 푣 P Λ Ď ΛQ representing 휆. We know from (the proof of) 1 1 Proposition 2.14.2 that we can obtain any such 푣 P ΛQ with 푞p푣 q “ 푞p푣q by acting on 푣 by reflections. So we expect to find our 푣1 by such a process. On to the actual argument: Choose 푤 P Λ with |푞p푣 ´ 푤q| ă 1. If 푞p푣 ´ 푤q “ 0, then by the anisotropic assumption, we have 푣 “ 푤 and we are done. 1 Otherwise, since 푞p푣 ´ 푤q ‰ 0 we have the reflection 푠푣´푤 as in Example 2.13.1. So we define 푣 as:

1 퐵푞p푣, 푣 ´ 푤q 푣 “ 푠푣 푤p푣q :“ 푣 ´ ¨ p푣 ´ 푤q. ´ 푞p푣 ´ 푤q By Example 2.13.1, we have 푞p푣1q “ 푞p푣q. We want to show that this process produces a vector in Λ after some number of iterations. To this end, let 푛 be a positive integer such that 푛푣 P Λ Ď ΛQ. It suﬃces to show that:

푞p푣 ´ 푤q ¨ 푛 P Z (4.7) and:

p푞p푣 ´ 푤q ¨ 푛q푣1 P Λ (4.8) since by assumption, |푞p푣 ´ 푤q푛| ă 푛. Indeed, we then have a sequence of vectors 푣, 푣1,... with 푞p푣q “ 푞p푣1q “ ..., and a strictly decreasing sequence of positive integers 푛, 푛1 :“ |푞p푣 ´ 푤q푛|,... with 푛푣 P Λ, 푛1푣1 P Λ, etc., and eventually we must obtain an honest element of Λ. For (4.7), we compute:

푞p푣 ´ 푤q푛 “ p푞p푣q ` 푞p푤q ´ 퐵푞p푣, 푤qq푛 “ 푞p푣q푛 ` 푞p푤q푛 ´ 퐵푞p푛푣, 푤q. Then 푞p푣q “ 휆 P Z, 푞p푤q P Z because 푤 P Λ, and 퐵p푛푣, 푤q P Z because 푛푣, 푤 P Λ. Then for (4.8), we expand:

퐵푞p푣, 푣 ´ 푤q p푞p푣 ´ 푤q ¨ 푛q푣1 “ p푞p푣 ´ 푤q ¨ 푛qp푣 ´ ¨ p푣 ´ 푤qq “ 푞p푣 ´ 푤q

푞p푣 ´ 푤q ´ 퐵푞p푣, 푣 ´ 푤q 푛푣 ` 퐵푞p푛푣, 푣 ´ 푤q푤. Because 푛푣 and 푤 lie in Λ,` it suffices to see that each˘ of our coefficients lie in Z. For the first coefficient, we find: INTRODUCTION TO THE ARITHMETIC THEORY OF QUADRATIC FORMS 57

푞p푣 ´ 푤q ` 푞p푣q “ 푞p푤 ´ 푣q ` 푞p푣q “ 푞p푤q ´ 퐵푞p푤 ´ 푣, 푣q “ 푞p푤q ` 퐵푞p푣 ´ 푤, 푣q so that 푞p푣 ´ 푤q ´ 퐵푞p푣 ´ 푤, 푣q “ 푞p푤q ´ 푞p푣q “ 푞p푤q ´ 휆 P Z. For the second coeﬃcient, we see:

퐵푞p푛푣, 푣 ´ 푤q “ 푛퐵푞p푣, 푣q ´ 퐵푞p푛푣, 푤q and the former term is 푛 ¨ 2푞p푣q P Z, and the latter term is integral because 푛푣, 푤 P Λ. References [Cas] J.W.S. Cassels. Rational Quadratic Forms. Dover Books on Mathematics Series. Dover Publications, Incorpo- rated, 2008. [Ser] Jean-Pierre Serre. A Course in Arithmetic. Graduate Texts in Mathematics. Springer, 1973.