# 1 Adeles over Q. 1.1 Absolute values

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1 1 Adeles over Q 1.1 Absolute values Definition (Absolute value) An absolute value on a field F is a nonnegative real valued function on F which satisfies the conditions: (i) x = 0 if and only if x =0, (ii) xy = x y, (iii) x + y x + y, (triangle inequality) for all x, y F. If an absolute value on a field F satisfies the stronger condition x + y max ( x, y ), (1.1.2) then it is called a non-archimedean absolute value. If condition (1.1.2) fails for some x, y F, then is called an archimedean absolute value. It is always possible to define a trivial absolute value trivial on any field F where { 1, if x 0, x trivial = 0, otherwise. Since trivial is not very interesting, we shall usually exclude it in our discussions. Definition (Equivalence of absolute values) Two absolute values 1 and 2, defined on the same field F, are termed equivalent if there exists c > 0 such that x 1 = x c 2 for all x F. Example The field Q of rational numbers has the classical (and very ancient) archimedean absolute value which we denote by which is defined by { x, if x 0, x = x, if x < 0, (1.1.5) 1 in this web service

2 2 Adeles over Q for all x Q. For each prime p one may define the non-archimedean absolute value p as follows. Given x Q with x = p k m with p mn, and k Z, n we define x p = p k m = p k. (1.1.6) n The definition of p has the effect that the non-archimedean absolute values of numbers divisible by high powers of p become small. Theorem (Ostrowski) The only non-trivial absolute values on Q are those equivalent to the p or the ordinary absolute value. Proof See [Cassels, 1986], [Murty, 2002]. Theorem (Product formula) Let α Q with α 0. The absolute values v, given by (1.1.5), (1.1.6), satisfy the product formula α v =1 v where the product is taken over all v {, 2, 3, 5, 7, 11, 13,..., i.e., v = or v is a prime. Proof The proof is elementary and left to the reader. Definition (Finite and infinite primes) Following the modern tradition we shall call v =2, 3, 5, 7, 11, the finite primes and v = the infinite or archimedean prime. Henceforth, we shall adhere to the convention that v refers to an arbitrary prime v (with v finite or infinite), while p refers specifically to a finite prime. 1.2 The field Q p of p-adic numbers An absolute value on a field F allows us to define the notion of distance between two elements x, y F as x y. We may also introduce a topology on F where the basis of open sets consists of the open balls B r (a) with center a F and radius r > 0: B r (a) = { x x a < r. A sequence of elements x 1, x 2, x 3,... F is termed Cauchy provided x m x n 0, (m, n ). (1.2.1) A field F with a non-trivial absolute value is said to be complete if all Cauchy sequences of elements x 1, x 2, x 3,... F have the property that there in this web service

3 1.2 The field Q p of p-adic numbers 3 exists an element x F such that x n x 0asn, i.e., all Cauchy sequences converge. If a field F is not complete, it is possible to complete it by standard methods of analysis. In brief, one adjoins to the incomplete field F all the elements arising from equivalence classes of Cauchy sequences, where two Cauchy sequences {x 1, x 2,..., {y 1, y 2,... are equivalent if lim i x i y i =0. The original elements α F are then realized as the equivalence class of the constant Cauchy sequence {α, α, α,.... Addition, subtraction, and multiplication of the representatives {x i = {x 1, x 2,..., {y i = {y 1, y 2,... of two equivalence classes of Cauchy sequences are defined by {x i ±{y i = {x i ± y i, {x i {y i = {x i y i. The definition of division is the same, except one has to be careful to not divide by zero because in a Cauchy sequence {x 1, x 2, x 3,..., some of the x i may be 0. Happily, this is not a problem, because every Cauchy sequence is equivalent to a Cauchy sequence without any zero terms and we always choose such a representative for performing division. The sequence of quotients will be Cauchy, provided the Cauchy sequence by which we divide does not converge to zero. Definition (p-adic fields) Let p be a prime number. The completion of Q with respect to the p-adic absolute value p, defined by (1.1.6), is denoted as Q p and called the p-adic field. We now present two explicit constructions of Q p. Analytic construction of Q p : The first construction we present is based on the notion of Cauchy sequences. Let k < n be any two integers (positive or negative) and for each i satisfying k i n let 0 a i < p also be an integer. If we assume a k 0, then it easily follows from (1.1.6) that n a i p i = p k. (1.2.3) i=k p Fix k Z. An infinite sequence {a k, a k+1, a k+2,..., where a i {0, 1,..., p 1 for each i k, and a k 0, determines an infinite sequence x 1 = a k p k x 2 = a k p k + a k+1 p k+1 x 3 = a k p k + a k+1 p k+1 + a k+2 p k+2. in this web service

4 4 Adeles over Q of elements in Q. By (1.2.3) it is easy to see that the sequence x 1, x 2, x 3... is a Cauchy sequence. Formally, we may define lim x i = i a i p i, (with x i p = p k for all i =1, 2,...). i=k Let Z p denote the set of all elements x of the completed field Q p which satisfy x p 1. By (1.1.2) it easily follows that Z p must be a ring with maximal ideal { π = x Z p x p < 1. It is easy to check that π = p Z p. Every x Z p can be uniquely realized as the equivalence class of a Cauchy sequence of the form { a 0, a 0 + a 1 p, a 0 + a 1 p + a 2 p 2, a 0 + a 1 p + a 2 p 2 + a 3 p 3,... where 0 a i < p for i =0, 1, 2,... One may check this by first showing that every element of Z p contains a sequence consisting entirely of integers. Every integer may be expressed as a finite sum a a N p N. One then shows that for the sequence to be Cauchy, the digit a i must be eventually constant for each i. The ring Z p can thus be realized as the set of all sums of the type: a i p i (1.2.4) i=0 where 0 a i < p for each i 0. Suppose x Q p does not satisfy x p 1. Then we can multiply x by a suitable power p n with n > 0 so that p n x p 1. It immediately follows that the field Q p, can thus be realized as the set of all sums of the type: a i p i (1.2.5) i=k where 0 a i < p for each i k and k Z arbitrary. The actual mechanics of performing addition, subtraction, multiplication, and division in the field Q p is very similar to what we do in the field R where every element is of the form a k 10 k + a k 1 10 k 1 + (1.2.6) with 0 a i 9 for all i k. The main difference in Q p is that the expansion goes up instead of down as in (1.2.6). a k p k + a k+1 p k+1 + a k+2 p k+2 in this web service

5 1.2 The field Q p of p-adic numbers 5 Here is an example of multiplication in Q 5. Note that the multiplication and carrying procedures mimic the case of multiplication in R except that we move from left to right instead of right to left We give one more example of the type of infinite expansion that occurs in Q p which is analogous to the expansion 1 = that occurs in R. 3 Example Let a be an integer coprime to the prime p. Let f 1bea fixed integer. Then there exist integers ā, a 1, a 2,... such that 1 a = ā + a f p f + a f +1 p f +1 + a f +2 p f +2 + Q p where a a 1(modp f ) with 0 < ā < p f and 0 a i < p for i = f, f +1, f +2,... Since a 1 p = 1 it follows that a 1 must be in Z p and, thus, have an expansion of type (1.2.4). We require a (ā + a f p f + a f +1 p f +1 + ) =1 from which it easily follows that aā 1(modp f ). Note that p-adic expansions of p-adic numbers are always unique. This is not the case for decimal expansions of real numbers. For example: = Algebraic construction of Q p : Let A 1, A 2, A 3,... be an infinite set of groups, rings, or fields. We assume that for every pair of positive integers i, j with i > j there exists a homomorphism f i, j : A i A j. (1.2.8) in this web service

6 6 Adeles over Q Assume also that whenever i, j, k are positive integers satisfying i > j > k, that f i,k = f j,k f i, j. (1.2.9) Definition (Inverse limit) Let A 1, A 2, A 3,... be an infinite set of groups, rings, or fields. Assume that for all positive integers i > j that homomorphisms f i, j exist satisfying (1.2.8), (1.2.9). Then the inverse limit of the A i, denoted lim A i is defined to be the set of all infinite sequences (a 1, a 2, a 3,...) where a i A i for all i 1 and f i, j (a i )=a j for all i > j 1. The inverse limit inherits the algebraic structure of the sets A i. It will be either a group, ring or field. In the algebraic approach to the construction of Q p we first construct (using the inverse limit) the ring of p-adic integers, denoted Z p. The field Q p is then constructed as the field of quotients of Z p, consisting of all elements of the form a/b with a, b Z p and b 0. Note that Z p is an integral domain. Let p be a prime and let i be a positive integer. Then the set { A i := a 0 + a 1 p + a i 1 p i 1 0 al < p for all 0 l<i (1.2.11) determines a finite ring with p i elements which is canonically identified with the quotient ring (Z/p i Z). The algebraic operations are addition and multiplication modulo p i. For every i > j, we have the canonical homomorphism f i, j : A i A j defined by f i, j ( a0 + a 1 p + a i 1 p i 1) = a 0 + a 1 p + a j 1 p j 1, which simply drops off the tail end terms in the sum. It easily follows from Definition that an element of the inverse limit is a sequence of the form (a 0, a 0 + a 1 p, a 0 + a 1 p + a 2 p 2, a 0 + a 1 p + a 2 p 2 + a 3 p 3,...). Formally, we define the infinite sum i=0 a i p i to be the sequence above. Then ( / lim Z p i Z ) { = a i p i 0 a i < p for all i 0. (1.2.12) i=0 Definition (Ring of p-adic integers Z p ) Let p be a prime number. The ring of p-adic integers Z p is defined to be the inverse limit of finite rings given by (1.2.11). in this web service

7 1.3 Adeles and ideles over Q Adeles and ideles over Q The completion of Q with respect to the archimedean absolute value is just R which we also denote as Q. Formally, the ring of adeles over Q, denoted A Q, is a ring determined by the restricted product (relative to the subgroups Z p ) A Q = R p Q p, where restricted product (relative to the subgroups Z p ) means that all but finitely many of the components in the product are in Z p. Definition (Adeles) The ring of adeles over Q, denoted A Q, is defined by { A Q := {x, x 2, x 3,... x v Q v ( v ), x p Z p, ( but finitely many p). Given two adeles x = {x, x 2, x 3,..., x = {x, x 2, x 3,..., we define addition and multiplication (the ring operations) as follows x + x := {x + x, x 2 + x 2, x 3 + x 3,... x x := {x x, x 2 x 2, x 3 x 3,... Recall that a topological space X is called locally compact if every point of X has a compact neighborhood. For example, Q p is locally compact and Z p is compact. Furthermore, A Q can be made into a locally compact topological ring by taking as a basis for the topology all sets of the form U p S Z p where S is any finite set of primes containing, and U is any open subset in the product topology on the finite product v S Q v. (This follows the Tychonoff theorem, see [Munkres, 1975].) The ideles of Q are defined to be the multiplicative subgroup of A Q, denoted A Q. Definition (Ideles) The multiplicative group of ideles over Q, denoted A Q, is defined by A Q {{x :=, x 2,... A Q xv Q v ( v), x p Z p, ( but finitely many p). in this web service

8 8 Adeles over Q Here Z p denotes the multiplicative group of units of Z p. Clearly, u Z p if and only if u p =1. The ideles over Q also form a locally compact topological group with the basis of the topology consisting of the open sets U p S Z p where U is an open set in v S Q v and S is any finite set of primes containing. Here, the topology on the finite product v S Q v is the product topology. Warning: The topology of the ideles is not the topology induced from the adeles. It is quite different. Definition (Finite adeles) The ring of finite adeles over Q, denoted A finite, is defined by { A finite := {x 2, x 3,... x p Q p ( p < ), x p Z p, ( but finitely many p). There is a natural embedding of A finite into A Q given by {x 2, x 3,... {0, x 2, x 3,... Definition (Finite ideles) The group of finite ideles over Q, denoted A, is defined by finite A {{x := finite 2, x 3,... x p Q p ( p < ), x p Z p, ( but finitely many p). There is a natural embedding of A into A finite Q given by {x 2, x 3,... {1, x 2, x 3, Action of Q on the adeles and ideles The ring Q can be embedded in the adeles as follows. It is clear that for any fixed q Q that q v > 1 for only finitely many v. Thus q lies in Z p for all but finitely many p <. Let q Q. Then {q, q, q,... A Q. This is usually referred to as a diagonal embedding. It follows that Q may be considered as a subring of A Q.ViewingA Q and Q as additive groups, it is in this web service

9 1.4 Action of Q on the adeles and ideles 9 then natural to take the quotient Q\A Q. Another way to view this quotient is to define an additive action (denoted +) of Q on A Q by the formula q + x := {q + x, q + x 2, q + x 3,... for all x = {x, x 2, x 3,... A Q and all q Q. Here q + x v denotes addition in Q v. This is a continuous action and Q is a discrete subgroup of A Q in the sense that for each q Q, there is a subset U A Q, which is open in the topology on A Q, such that U Q = {q. We now introduce the notion of a fundamental domain for the action of an arbitrary group on an arbitrary set X. Definition (Fundamental domain) Let a group G act on a set X (on the left). A fundamental domain for this action is a subset D X which satisfies the following two properties: (1) For each x X, there exists d D and g G such that gx = d. (2) The choice of d in (1) is unique. Remarks A fundamental domain is precisely a choice of one point from each orbit of G. IfG\X is the quotient space with the quotient topology and π : X G\X is the quotient map, then the fundamental domain is the image of a section σ : G\X X. (This is a set theoretic section, it need not be continuous.) The construction of an explicit fundamental domain for the action of the additive group Q on the adele group A Q is equivalent to a generalization of the ancient Chinese remainder theorem. Theorem (Chinese Remainder Theorem) Let p 1, p 2,...p n be distinct primes. Let e 1, e 2,...,e n be positive integers and c 1, c 2,...,c n be arbitrary integers. Then the system of linear congruences x c 1 (mod p e 1 1 ) x c 2 (mod p e 2 2 ). x c n (mod p e n n ) has a unique solution x (mod p e 1 1 pe 2 2 pe n n ). Proof A simple proof can be obtained by explicitly constructing a solution to the system of linear congruences. Set N = p e 1 1 pe 2 2 pe n n. For each 1 i n define an integer u i by the condition N p e i i u i 1 (mod p e i i ). in this web service

10 10 Adeles over Q Then one easily checks that the element x c 1 N p e 1 1 u 1 + c 2 N p e 2 2 u c n N p e n n satisfies x c i (mod p e i i ) for all 1 i n. We leave the proof of uniqueness to the reader. Example Consider the system of linear congruences x 2(mod3 2 ) x 1(mod5 3 ) x 3(mod7). Then u 1 is defined by the congruence u 1 1(mod3 2 ), and u 1 =5. Similarly, u 2 1(mod5 3 ) and u 2 = 2, while u 3 1(mod7) and u 3 =3. It follows that x (mod ). A modern version of the Chinese Remainder Theorem (Theorem 1.4.2) can be given in terms of p-adic absolute values. Theorem (Weak approximation) Let p 1, p 2,...,p n be distinct primes. Let c i Q pi for each i =1, 2,...,n. Then for every ɛ>0, there exists an α Q such that α c i pi <ɛ for all 1 i n. Furthermore, α may be chosen so that the denominator, when written in lowest terms, is not divisible by any primes other than p 1,...,p n. Proof The general case follows easily from the case when c i Z pi for all i. As Z is dense in Z p, we may then replace c i by c i Z. At this point the statement reduces to the classical form, given in Theorem Proposition (Strong approximation for adeles) A fundamental domain DforQ\A Q is given by { D = {x, x 2, x 3,... 0 x < 1, x p Z p for all finite primes p =[0, 1) Z p. That is, we have p A Q = β Q {β + D, (disjoint union). u n in this web service

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