MATH 248A. THE CHARACTER GROUP OF Q. 1. Introduction

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1 MATH 248A. THE CHARACTER GROUP OF Q KEITH CONRAD 1. Introduction The characters of a finite abelian grou G are the homomorhisms from G to the unit circle S 1 = {z C : z = 1}. Two characters can be multilied ointwise to define a new character, and under this oeration the set of characters of G forms an abelian grou, with identity element the trivial character, which sends each g G to 1. Characters of finite abelian grous are imortant, for examle, as a tool in estimating the number of solutions to equations over finite fields. [3, Chaters 8, 10]. The extension of the notion of a character to nonabelian or infinite grous is essential to many areas of mathematics, in the context of harmonic analysis or reresentation theory, but here we will focus on discussing characters on one of the simlest infinite abelian grous, the rational numbers Q. This is a secial case of a situation that is well-known in algebraic number theory, but all references I could find in the literature are based on [4], where one can t readily isolate the examination of the character grou of Q without assuming algebraic number theory and Fourier analysis on locally comact abelian grous. (In [2, Chater 3, 1], the determination of the characters of Q is made without algebraic number theory, but the Pontryagin duality theorem from Fourier analysis is used at the end.) The rerequisites for the discussion here are more elementary: familiarity with the comlex exonential function, the -adic numbers Q, and a few facts about abelian grous. In articular, this discussion should be suitable for someone who has just learned about the -adic numbers and wants to see how they can arise in answering a basic tye of question about the rational numbers. Concerning notation, r and s will denote rational numbers, and q will denote rime numbers, and x and y will denote real or -adic numbers (deending on the context). The word homomorhism will always mean grou homomorhism, although sometimes we will add the word grou for emhasis. The sets Q, R, Q, and the -adic integers Z, will be regarded 1

2 2 KEITH CONRAD rimarily as additive grous, with multilication on these sets being used as a tool in the study of the additive structure. 2. Definition and Examles The definition of characters for finite abelian grous makes sense for any grou. Definition 2.1. A character of Q is a grou homomorhism Q S 1. As in the case of characters of finite abelian grous, under ointwise multilication the characters of Q form an abelian grou, which we denote by Q. Examles of nontrivial characters of Q are the homomorhisms r e ir and r e 2πir. Our goal is to write down all elements of Q using exlicit functions. The usefulness of Q can t be discussed here, but hoefully the reader can regard its determination as an interesting roblem, esecially since the end result will be more comlicated than it may seem at first glance. The rototye for our task is the classification of continuous grou homomorhisms from R to S 1. The examle x e ix is tyical. In general, for any y R we have the homomorhism x e 2πixy. The use of the scaling factor 2π here is rominent in number theory, since it makes certain formulas much cleaner. For the reader who is interested in a roof that these are all the continuous homomorhisms R S 1, see the aendix. We will not need to know that every continuous homomorhism R S 1 has this form, but we want to analyze Q with a similar goal in mind, namely to find a relatively concrete method of writing down the homomorhisms Q S 1. Kee in mind that we are imosing no continuity conditions on the elements of Q; as functions from Q to S 1, they are merely grou homomorhisms. (Or think of Q as a discrete grou, so grou homomorhisms are always continuous.) Let s now give examles of nontrivial characters of Q. First we will think of Q as lying in the real numbers. For nonzero y R, the ma x e 2πixy is continuous on R, hits all of S 1, and Q is dense in R, so the restriction of this function to Q, i.e., the function r e 2πiry, is a nontrivial character of Q. When y = 0, this is the trivial character.

3 MATH 248A. THE CHARACTER GROUP OF Q 3 It is easy to believe that these may be all of the characters of Q, since the usual icture of Q inside R makes it hard to think of any way to write down characters of Q besides those of the form r e 2πiry (y R). However, such characters of Q are only the ti of the iceberg. We will now show how to make sense of e 2πiy for -adic y, and the new characters of Q which follow from this will allow us to easily write down all characters of Q. That is, in a loose sense, every character of Q is a mixture of functions that look like r e 2πiry for y in R or some Q. Technically, e 2πiry is meaningless if y is a general -adic number. We introduce a formalism that should be thought of as allowing us to make sense of this exression anyway. For x Q, define the -adic fractional art of x, denoted {x}, to be the sum of the negative-ower-of- terms in the usual -adic exansion of x. Let s comute the -adic fractional art of 21/50 for several. In Q 2, Q 3, and Q 5, = , Therefore = , { } 21 = 1 { } , = 0, = { } 21 = = For any 2, 5, { } = 0. More generally, any r Q is in Z for all but finitely many, so {r} = 0 for all but finitely many. Note that for 0 m n 1, { m n } = m n. In thinking of -adic exansions as analogous to Laurent series in comlex analysis, the -adic fractional art is analogous to the olar art of a -adic number. (One difference: the olar art of a sum of two meromorhic functions at a oint is the sum of the olar arts, but the -adic fractional art of a sum of two -adic numbers is not usually the sum of their individual -adic fractional arts. Carrying in -adic addition can allow the sum of -adic fractional arts to leak into a -adic integral art; consider 3/5 + 4/5 = 2/5 + 1.) There is an analogue in Q of the artial fraction decomosition of rational functions: (2.1) r = {r} + integer. This sum over all makes sense, since most terms are 0. It exresses r as a sum of rational numbers with rime ower denominator, u to addition by

4 4 KEITH CONRAD an integer. For examle, { } = { } = = = { } To rove (2.1), we show the difference r {r}, which is rational, has no rime in its denominator. For q, {r} Z q, while r {r} q Z q by the definition of {r} q. Thus r {r} = r {r} q q {r} is in Z q. Alying this to all q, r {r} is in Z. Remark 2.2. If we view Equation 2.1 not in Q but in Q/Z, it exresses an element of Q/Z as a sum of elements of -ower order for different rimes. This is the exlicit realization of the torsion abelian grou Q/Z as the direct sum of its -ower torsion subgrous. The -adic fractional art should be thought of as a -adic analogue of the usual fractional art function { } on R, where {x} [0, 1) and x {x} Z. In articular, for real x we have {x} = 0 recisely when x Z. The -adic fractional art has similar features. For x Q, {x} = m n [0, 1) (0 m n 1), x {x} Z, {x} = 0 x Z. While the ordinary fractional art on R is not additive, the deviation from additivity is given by an integer: {x + y} = {x} + {y} + integer. This deviation from additivity gets wied out when we take the comlex exonential of both sides: e 2πi{x+y} = e 2πi{x} e 2πi{y}. Of course e 2πi{x} = e 2πix for x R, so there is no need to use the fractional art. But in the -adic case, the fractional-art viewoint gives us the following basic definition: For x Q, set ψ (x) = e 2πi{x}. This is not the exonential of a -adic number: {x} is a rational number with -ower denominator, so e 2πi{x} is some -th ower root of unity.

5 MATH 248A. THE CHARACTER GROUP OF Q 5 Examle 2.3. ψ 2 (21/50) = e 2πi(1/2) = 1, ψ 3 (21/50) = 1, ψ 5 (21/50) = e 2πi(23/25). Examle 2.4. Let a Q 7 be the solution to x 2 = 2 such that a 3 mod 7Z 7. Then a mod 7 3 Z 7, so ψ 7 (a/7 3 ) = e 2πi{a/73 } 7 = e 2πi( )/7 3 = e 2πi 59/343. The function ψ is a grou homomorhism: e 2πi{x+y} = e 2πi{x} e 2πi{y}. To see this, we have to understand the extent to which the -adic fractional art fails to be additive. For x, y Q, x {x}, y {y}, x + y {x + y} Z. Adding the first two terms and subtracting the third, we see that the rational number {x} + {y} {x + y} is a -adic integer. As a sum/difference of -adic fractional arts, it has a ower of as denominator. Since it is also a -adic integer, there is no ower of in the denominator, so the denominator must be 1. Therefore {x} + {y} {x + y} Z. Thus ψ is a grou homomorhism. This is the technical way we make sense of the meaningless exression e 2πix for x Q ; e 2πix can be thought of recisely as e 2πi{x}. (Warning: when x is rational, so e 2πix makes sense in the usual way, but this is generally not the same as e 2πi{x}. Referring to e 2πi{x} as just a sensible interretation of e 2πix for -adic x should only be considered as a loose manner of seaking.) Remark 2.5. The grou Q /Z is isomorhic to the -ower torsion subgrou of Q/Z: Q/Z = Q /Z. The character ψ fits into the commutative diagram Q /Z ψ inclusion Q/Z r e 2πir S 1 Unlike the function x e 2πix for real x, the function ψ : x e 2πi{x} does not take on all values in S 1. Its image is exactly the th ower roots

6 6 KEITH CONRAD of unity. It is also locally constant (so continuous in an elementary way), since ψ (x + y) = ψ (x) for y Z, and Z is a neighborhood of 0. By the same tye of interior scaling argument as for the basic character e 2πix on the reals, we can use ψ to construct many continuous characters of Q. Choosing any y Q, we have a grou homomorhism Q S 1 by x ψ (xy) = e 2πi{xy}. By varying y in Q, we get lots of examles of nontrivial continuous homomorhisms Q S 1. It is a fact that all continuous homomorhisms Q S 1 are of the above tye, but we will not need this fact in our study of the character grou Q of Q. However, our analysis of Q contains the essential ingredients for a roof, so we will discuss this again in the aendix. By restricting a character of Q to the subset Q, we get new characters of Q. For fixed y Q, let r e 2πi{ry}. This character of Q takes as values only the th ower roots of unity, so it is quite different (for y 0) from the functions r e 2πiry for real y. Examle 2.6. For a 5-adic number y, let χ: Q S 1 be defined by χ(r) = e 2πi{ry} 5. We want to calculate χ(21/50). Since = , we need to know the 5-adic exansion of y out to the multile of 5. If, for instance, y = , then (21/50)y = 2/25 + 1/5 +..., so χ(21/50) = e 14πi/25. To comute χ(r) for a secific rational number r, we only need to know the 5-adic exansion of y to an aroriate finite number of laces, deending on the 5-adic exansion of r. 3. Q and the Adeles We shall now use the above homomorhisms from R and the various Q s to S 1 to construct all homomorhisms from Q to S 1, i.e., all elements of Q. This is a simle examle of the local-global hilosohy in number theory, which says that one should try to analyze a roblem over Q (the global field) by first analyzing it over each of the comletions R, Q 2, Q 3,... of Q (the local fields), and then use this information over the comletions to solve the roblem over Q. The roblem we are concerned with is the

7 MATH 248A. THE CHARACTER GROUP OF Q 7 construction of homomorhisms to S 1, and we have already taken care of the local roblem, at least for our uroses. Define ψ : R S 1 by x e 2πix (the reason for the minus sign will be aarent later). Define, as before, ψ : Q S 1 by x e 2πi{x}. Now choose any elements a R and a Q for all rimes, with the roviso that a Z for all but finitely many. We define a function Q S 1 by r ψ (ra ) ψ (ra ) = e 2πira e 2πi{ra}. To show this ma makes sense and is a character, note that for any rational r, r Z for all but finitely many rimes, so by our convention on the a s, ra Z for all but finitely many rimes. (The finitely many such that ra Z will of course vary with r. It is not the case that factors where a Z lay no role, since we may have ra Z when a Z, if the denominator of r has a large ower of.) Thus ψ (ra ) = 1 for all but finitely many, so for each r Q the infinite roduct defining the above function at r is really a finite roduct. Each local function ψ, ψ 2, ψ 3,... is a homomorhism, so our ma above is a homomorhism, hence is an element of Q. The homomorhisms ψ, ψ 2, ψ 3,... are our basic mas, and the numbers a, a 2, a 3,... should be thought of as interior scaling factors that allow us to define many characters of Q in terms of the one basic character r e 2πir e 2πir. To understand this construction better, we want to look at the sequences of elements (a, a 2, a 3,... ) that have just been used to define characters of Q. This leads us to introduce a ring which lays a rominent role in number theory. Definition 3.1. The adeles, A Q, are the elements (a, a 2, a 3,... ) in the roduct set R Q such that a lies in Z for all but finitely many. A random adele will not have any rational coordinates. As an examle of something which is not an adele, consider any element of the roduct set R Q whose -adic coordinate is 1/ for infinitely many rimes. The adele ring of Q lies between the direct sum v Q v and the direct

8 8 KEITH CONRAD roduct v Q v. It is called the restricted direct roduct of the Q v s, and the restricted direct roduct notation is v Q v If a is a tyical adele, its real coordinate will be written as a and its -adic coordinate will be written as a. While ψ includes an awkwardlooking minus sign (whose rationale will be exlained below), a does not. It is the real coordinate of a, not its negative. Under comonentwise addition and multilication, A Q is a commutative ring (but not an integral domain). For our uroses, the additive grou structure of A Q is its most imortant algebraic feature. Since any rational number r is in Z for all but finitely many rimes, we see that Q naturally fits into A Q by the diagonal ma r (r, r, r,... ), making Q a subring of A Q. We shall call an adele rational if all of its coordinates are the same rational number, so the rational adeles are naturally identified with the rational numbers. We will write the rational adele (r, r, r,... ) just as r. Let s see how the adeles are a useful notation to describe Q. From what has been done so far, for any adele a = (a, a 2, a 3,... ) we have defined a character Ψ a of Q by Ψ a (r) = ψ (ra ) ψ (ra ) = e 2πira e 2πi{ra}. Since addition in A Q is comonentwise, a comutation shows that for adeles a and b Ψ a+b (r) = Ψ a (r)ψ b (r) for all rational numbers r, so in Q we have Ψ a+b = Ψ a Ψ b. Clearly Ψ 0 is the trivial character. For a rational adele s and any r Q, Ψ s (r) = e 2πi( rs+p {rs}) = 1 by (2.1). Thus Ψ s is the trivial character for all rational adeles s. This is why the minus sign was used in the definition of ψ. If a and b are two adeles whose difference is a rational adele, Ψ a = Ψ b. The following theorem, which will be shown in the next section, tells us that we have found all of the characters of Q, and can decide when we have described a character in two different ways.

9 MATH 248A. THE CHARACTER GROUP OF Q 9 Theorem 3.2. Every character of Q has the form Ψ a for some a A Q, and Ψ a = Ψ b if and only if a b is a rational adele. In other words, the ma Ψ: A Q Q given by a Ψ a is a surjective homomorhism with kernel equal to the rational adeles Q, so Q = A Q /Q. 4. The Image and Kernel of Ψ Let χ: Q S 1 be a character. We want to write χ = Ψ a for some adele a. We begin by considering χ(1), which is some number on the unit circle, so χ(1) = e 2πiθ for a (unique) real θ [0, 1). Define χ : Q S 1 by χ (r) = ψ (rθ) = e 2πirθ. Then χ is a character of Q and χ (1) = χ(1). Let χ (r) = χ(r)/χ (r), so χ is a character with χ (1) = 1 and χ(r) = χ (r)χ (r). For any rational r = m/n, χ (r) n = χ (m) = χ (1) m = 1. The image of χ is inside the roots of unity, so dividing χ by χ to give us χ uts us in an algebraic setting. Every root of unity, say e 2πis for s Q, is a unique roduct of rime ower roots of unity. Indeed, by (2.1) e 2πis = e 2πi{s}, where e 2πi{s} is a th ower root of unity, equal to 1 for all but finitely many. Let χ (r) denote the -th ower root of unity that contributes to the root of unity χ (r). For examle, if χ (r) = e 3πi/7, then χ 2 (r) = 1, χ 7 (r) = e 10πi/7, and χ (r) = 1 for all 2, 7. The function χ : Q S 1 is a character of Q. Since χ (1) = 1, χ (1) = 1 for all rimes. Since χ (r) = χ (r), we have χ(r) = χ (r) χ (r). This decomosition of χ can be viewed as the main ste in describing Q in terms of adeles. We have broken u our character χ into local characters χ, χ 2, χ 3,..., and now roceed to analyze each one individually.

10 10 KEITH CONRAD By construction, χ (r) = e 2πirθ for some real θ [0, 1). We now want to show that χ (r) = e 2πi{rc} for some c Z. This will involve giving an exlicit method for constructing c. Since χ (1) = 1, χ (1/ n ) n = 1, so χ (1/ n ) = e 2πicn/n for some (unique) integer c n with 0 c n n 1. Since χ (1/ n+1 ) = χ (1/ n ), we get c n+1 c n mod n Z. Thus {c 1, c 2, c 3,... } is a -adic Cauchy sequence in Z, so it has a limit c Z, and c c n mod n Z for all n. Since 0 c n n 1, { } c n = c n n. We now show χ (r) = e 2πi{rc} = ψ (rc) for all r Q. Write r = s/t where s, t Z with t 0. Let t = m t for (, t ) = 1. Then ( ) ( ) s 1 s χ (r) t = χ (t r) = χ m = ϕ m = e 2πicms/m. In Q/Z, { } { } c m s c cs m == m s m {crt } {cr} t, so χ (r) t = e 2πi{cr}t. Thus the -th ower roots of unity χ (r) and e 2πi{cr} have a ratio that is a t -th root of unity. Since t is rime to, this ratio must be 1, so ( s ) χ = e 2πi{cr} = ψ (rc). t Write c Z as a, so χ (r) = ψ (ra ) for all r Q. Therefore χ = χ χ = Ψ a for the adele a = (θ, a 2, a 3,... ) [0, 1) Z. We now know every character of Q has the form Ψ a for an adele in the secial set [0, 1) Z. When the adele a is in this set, we can determine it from the character. Indeed, in this case all a are in Z, so Ψ a (1) = e 2πia. Since a [0, 1), it is comletely determined from knowing Ψ a (1). We can then multily Ψ a by the character e 2πira to assume a = 0. Then Ψ a has only roots of unity as its values. The th ower comonent of Ψ a (r) is e 2πi{ar}, which uon taking r = 1/, 1/ 2, etc. allows us to successively determine each digit of a, so we can determine a. Every character of Q has the form Ψ a for a unique adele a in [0, 1) Z. It turns out that every adele can be ut into this set uon addition by a suitable rational adele:

11 MATH 248A. THE CHARACTER GROUP OF Q 11 (4.1) A Q = Q + [0, 1) Z. To rove (4.1), fix an adele a. We know a Z as long as is outside of a finite set (say) F. Let r = F {a }, the sum of the various ole arts of a. So a r has no ole arts, hence a r R Z. Let N be the integer such that N a r < N + 1. Since N Z for all rimes, we see that a (r + N) [0, 1) Z, so (4.1) is established. The decomosition (4.1) is direct, since Q ([0, 1) Z ) = {0} (analogous to the relation Z [0, 1) = {0}). We ut direct in quotes since [0, 1) is not a grou, so (4.1) is not quite a direct sum of subgrous. Equation (4.1) is analogous to the decomosition R = Z + [0, 1). This analogy between the air (A Q, Q) and the air (R, Z) can be carried further. For examle, if we define Ẑ to be the character grou of Z, then Ẑ = S 1 (associate to z S 1 the element of Ẑ given by n zn ). The isomorhism Q = A Q /Q should be thought of as analogous to the isomorhism Ẑ = R/Z. Since Ψ a+s = Ψ a if s is a rational adele, and for a [0, 1) Z the character Ψ a is trivial recisely when a = 0, we have roven that the grou homomorhism Ψ: A Q Q is surjective with kernel Q, so every element of Q has the form Ψ a for some a A Q, and two characters Ψ a and Ψ b are equal if and only if a b is a rational adele. This is our desired concrete desrition of Q. Using the comletions of Q to define various exlicit homomorhisms ψ, ψ 2, ψ 3,..., we have shown how to glue them together using the adeles to get all the characters of Q. To areciate the choice of e 2πix over e 2πix to lay the role of the basic real character, we can work through the above argument with e 2πix. This leads to the homomorhism Φ: A Q Q given by Φ a (r) = e 2πia r e 2πi{ar}. It is surjective, like Ψ, but the kernel of Φ is the set of adeles of the form ( r, r, r,... ) for rational r, which is not as elegant. It is left as an exercise for anyone who is familiar with the adeles A k of any finite extension field k of Q to extend our arguments and give an elementary roof that k = A k /k.

12 12 KEITH CONRAD Aendix A. Characters of R and Q Here we shall rove that all continuous homomorhisms R S 1 have the form x e 2πixy for a unique real number y, and all continuous homomorhisms Q S 1 have the form x e 2πi{xy} for a unique -adic number y. The similarity in these descritions suggests the ossibility of having a common roof for these results. Using some general theorems about locally comact abelian grous, it is ossible to rove these results in a unified (and rather brief) manner where the only roerty one uses about R and Q is that they are locally comact fields with resect to nontrivial absolute values (see [4, Lemma 2.2.1]). However, we take a more elementary ath. While we handle the real and -adic cases searately, there are some basic similarities in the two roofs. First let s make some general remarks. The above mas certainly are examles of continuous homomorhisms. If y 0 in the real case, then evaluating the ma at x = 1/2y shows the homomorhism is not trivial, while an evaluation at x = 1/y for nonzero y in the -adic case shows the homomorhism is not trivial. It follows easily from this that such homomorhisms are uniquely determined by the scaling factor y, so we only need to show every continuous homomorhism has the indicated form. A common ste in both cases will be to show that any homomorhism has nontrivial kernel, and then reduce to the case where 1 lies in the kernel. The roof that the kernel is nontrivial will oerate on different rinciles in the two cases (real vs. -adic), taking into account secial features of the toology of R and Q. Since the -adic case is easier to handle and may be less familiar to the reader, we resent it first. If χ: Q S 1 is a continuous homomorhism, then for all small x in Q, χ(x) 1 < 1. In articular, χ sends N Z into {z C : z 1 < 1} for large enough N. Since N Z is a grou, χ( N Z ) is a subgrou of C. Clearly there are no nontrivial subgrous of C entirely within the oen ball of radius 1 around 1 in C. Thus χ( N Z ) = {1}, so χ is locally constant and χ( N ) = 1. Let ψ(x) = χ( N x) for all x Q, so ψ is a continuous homomorhism Q S 1 which is trivial on Z, in articular ψ(1) = 1. Thus ψ(1/ n ) is a n -th root of unity, so ψ(1/ n ) = e 2πicn/n

13 MATH 248A. THE CHARACTER GROUP OF Q 13 for some integer c n with 0 c n n 1. Proceeding exactly as in 4, we find that the integers c n form a Cauchy sequence in Z, and their limit c satisfies ψ(r) = e 2πi{rc} for all r Q, so by continuity ψ(x) = e 2πi{xc} for all x Q. Thus χ(x) = ψ(x/ N ) = e 2πi{xy} for y = c/ N. For a slightly different argument, see [5]. We never used the hyothesis that the image of χ is a subset of S 1, only that it is a subset of C ; we roved its image has to lie in S 1. The essential oint is that subgrous of Q and C are quite different. In Q, the identity 0 has a neighborhood basis of subgrous ( n Z for n 0), while in C, the identity 1 has a neighborhood containing no nontrivial grou. It is ossible to establish the real case rather quickly, using integrals (see [1, Theorem 9.11]). However, the basic structure of the -adic case can be carried over to the real case, and we now resent this alternate (but longer) argument. Let χ: R S 1 be a continuous homomorhism. We want to show χ(x) = e 2πixy for some y R. We may assume χ is nontrivial. If the kernel of χ contains nonzero numbers arbitrarily close to 0, then by translations the kernel of χ is dense in R, so by continuity χ is trivial. Thus, for nontrivial χ and all small x > 0, χ(x) has constant sign on its imaginary art. Relacing χ with χ 1 if necessary, we may assume that χ has ositive imaginary art on small ositive numbers. Since χ(t) 1 for some t > 0, by connectedness χ((0, t)) is an arc in S 1, so contains a root of unity, say χ(u) for 0 < u < t. If χ(u) has order N, then Nu > 0 is in the kernel of χ. Relacing χ by the continuous homomorhism x χ(n ux), we may assume that χ(1) = 1. To summarize, we may assume χ is a continuous homomorhism from R to S 1 which contains 1 in its kernel and which has ositive imaginary art for small ositive numbers. According to what we are trying to rove, we now exect that χ(x) = e 2πixy for some (ositive) integer y, and this is what we shall show. Since the recirocals of the rime owers generate the dense subgrou Q of R, by continuity it suffices to find an integer y such that χ(1/ n ) = e 2πiy/n for all rimes and integers n 1. Actually, we ll show for any integer m > 1 that there is an integer y m such that χ(1/m n ) = e 2πiym/mn for all n 1. Since χ(1/(q) n ) n = χ(1/q n ) and χ(1/(q) n ) qn = χ(1/ n ), it follows that y q y q n q n Z = {0}, so y q = y q, and similarly that y q = y, so y = y q. Thus for all rimes, the integers y are the same, and this common integer y solves our roblem.

14 14 KEITH CONRAD Since χ(1/m n ) mn = χ(1) = 1, χ(1/m n ) = e 2πicn/mn for some integer c n such that 0 c n < m n. Note c n deends on m (well, a riori at least). Since χ(1/m n+1 ) m = χ(1/m n ), c n+1 /m n c n /m n Z. For large n, contintuity of χ imlies cos(2πc n /m n ) > 0 and sin(2πc n /m n ) is ositive and arbitrarily small. Since 0 c n < m n, the cosine condition imlies c n /m n (0, 1/4) (3/4, 1) for large n. We can t have c n /m n in (3/4, 1) for large n, since then sin(2πc n /m n ) is negative (here is where we use the assumtion that χ(x) is in the first quadrant for small ositive x). Thus for all large n, c n /m n (0, 1/4). Since sin(2πc n /m n ) is arbitrarily small for all large n, it follows by the nature of the sine function and the location of c n /m n that c n /m n is arbitrarily small for n large. To fix ideas, 0 < c n /m n < 1/(m + 1) for all large n. Then 0 < c n+1 /m n < m/(m + 1) for all large n, so c n /m n c n+1 /m n < 1 for all large n. Since the left hand side of this last inequality is an integer, it must be zero, so all c n s are equal for n sufficiently large. Call this common value y m. Thus χ(1/m n ) = e 2πiym/mn for all large n, hence for all n 1; for examle, if χ(1/m 100 ) = e 2πiym/m100, then raising both sides to the m 98 -th ower we see that χ(1/m 2 ) = e 2πiym/m2. Acknowledgments. I thank Randy Scott, Eric Sommers and Ravi Vakil for looking over a reliminary version of this manuscrit. References [1] J.B. Conway, A Course in Functional Analysis, 2nd ed., Sringer-Verlag, New York, [2] I. M. Gel fand, M. I. Graev, I. I. Pyatetskii-Shairo, Reresentation Theory and Automorhic Functions, Academic Press, [3] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Sringer-Verlag, New York, [4] J. Tate, Fourier Analysis in Number Fields and Hecke s Zeta-Functions, in: Algebraic Number Theory, Academic Press, New York, 1967, [5] L. Washington, On the Self-Duality of Q, American Mathematical Monthly 81 (4) 1974,