The Character Group of Q

MATH 676. THE CHARACTER GROUP OF Q KEITH CONRAD 1. Introduction The characters of a nite abelian group G are the homomorphisms from G to the unit circle S1 =fz2C :jzj= 1g. Two characters can be multiplied pointwise to de ne a new character, and under this operation the set of characters of G forms an abelian group, with identity element the trivial character, which sends each g2G to 1. Characters of nite abelian groups are important, for example, as a tool in estimating the number of solutions to equations over nite elds. [3, Chapters 8, 10]. The extension of the notion of a character to nonabelian or in nite groups is essential to many areas of mathematics, in the context of harmonic analy- sis or representation theory, but here we will focus on discussing characters on one of the simplest in nite abelian groups, the rational numbers Q. This is a special case of a situation that is well-known in algebraic number theory, but all references I could nd in the literature are based on [4], where one can?t readily isolate the examination of the character group of Q without assuming algebraic number theory and Fourier analysis on locally compact abelian groups. (In [2, Chapter 3, x1], 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 prerequisites for the discussion here are more elementary: familiarity with the complex exponen- tial function, the p-adic numbers Qp, and a few facts about abelian groups. In particular, this discussion should be suitable for someone who has just learned about the p-adic numbers and wants to see how they can arise in answering a basic type of question about the rational numbers. Concerning notation, r and s will denote rational numbers, p and q will denote prime numbers, and x and y will denote real or p-adic numbers (depending on the context). The word \homomorphism" will always mean \group homomorphism", although sometimes we will add the word \group" for emphasis. The sets Q, R, Qp, and thep-adic integers Zp, will be regarded primarily as additive groups, with multiplication on these sets being used as a tool in the study of the additive structure. 2. De nition and Examples The de nition of characters for nite abelian groups makes sense for any group. 1 2 KEITH CONRAD De nition. A character of Q is a group homomorphism Q!S1. As in the case of characters of nite abelian groups, under pointwise multiplication the characters of Q form an abelian group, which we denote by bQ. Examples of nontrivial characters of Q are the homomorphisms r7!eir and r7!e2 ir. Our goal is to write down all elements of bQ using explicit functions. The usefulness of bQ can?t be discussed here, but hopefully the reader can regard its determination as an interesting problem, especially since the end result will be more complicated than it may seem at rst glance. The prototype for our task is the classi cation of continuous group ho- momorphisms from R to S1. The example x7!eix is typical. In general, for any y2R we have the homomorphism x7!e2 ixy: The use of the scaling factor 2 here is prominent in number theory, since it makes certain formulas much cleaner. For the reader who is interested in a proof that these are all the continuous homomorphisms R !S1, see the appendix. We will not need to know that every continuous homomor- phism R!S1 has this form, but we want to analyze bQ with a similar goal in mind, namely to nd a relatively concrete method of writing down the homomorphisms Q!S1. Keep in mind that we are imposing no continu- ity conditions on the elements of bQ; as functions from Q to S1, they are merely group homomorphisms. (Or think of Q as a discrete group, so group homomorphisms are always continuous.) Let?s now give examples of nontrivial characters of Q. First we will think of Q as lying in the real numbers. For nonzero y2R, the map x7!e2 ixy is continuous on R, hits all of S1, and Q is dense in R, so the restriction of this function to Q, i.e., the function r7!e2 iry; is a nontrivial character of Q. When y = 0, this is the trivial character. It is easy to believe that these may be all of the characters of Q, since the usual picture of Q inside R makes it hard to think of any way to write down characters of Q besides those of the form r7!e2 iry (y2R). However, such characters of Q are only the tip of the iceberg. We will now show how to make sense of e2 iy for p-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 r7!e2 iry for y in R or some Qp. Technically, e2 iry is meaningless if y is a general p-adic number. We introduce a formalism that should be thought of as allowing us to make sense of this expression anyway. MATH 676. THE CHARACTER GROUP OF Q 3 For x2Qp, de ne the p-adic fractional part of x, denotedfxgp, to be the sum of the negative-power-of-p terms in the usual p-adic expansion of x. Let?s compute the p-adic fractional part of 21=50 for several p. In Q2;Q3, and Q5, 21 50 = 1 2+2+2 2+23+:::; 21 50 = 2 3+3 2+36+:::; 21 50 = 3 25+ 4 5+2+2 5+:::: Therefore 21 50 2 = 12; 21 50 3 = 0; 21 50 5 = 325 + 45 = 2325: For any p6= 2;5, f2150gp = 0. More generally, any r2Q is in Zp for all but nitely many p, so frgp = 0 for all but nitely many p. Note that for 0 m pn 1, fmpngp = mpn. In thinking of p-adic expansions as analogous to Laurent series in com- plex analysis, the p-adic fractional part is analogous to the polar part of a p-adic number. (One di erence: the polar part of a sum of two mero- morphic functions at a point is the sum of the polar parts, but the p-adic fractional part of a sum of two p-adic numbers is not usually the sum of their individual p-adic fractional parts. Carrying in p-adic addition can allow the sum of p-adic fractional parts to \leak" into a p-adic integral part; consider 3=5 + 4=5 = 2=5 + 1.) There is an analogue in Q of the partial fraction decomposition of rational functions: (1) r = X p frgp + integer: This sum over all p makes sense, since most terms are 0. It expresses r as a sum of rational numbers with prime power denominator, up to addition by an integer. For example, X p 21 50 p = 21 50 2 + 21 50 5 = 12 + 2325 = 7150 = 2150 + 1: To prove (1), we show the di erence r Ppfrgp, which is rational, has no prime in its denominator. For p6= q, frgp2Zq, while r frgq2Zq by the de nition of frgq. Thus r Ppfrgp = r frgq Pp6=qfrgp is in Zq. Applying this to all q, r Ppfrgp is in Z. The p-adic fractional part should be thought of as a p-adic analogue of the usual fractional part functionf gon R, wherefxg2[0;1) and x fxg2Z. 4 KEITH CONRAD In particular, for real x we have fxg= 0 precisely when x2Z. The p-adic fractional part has similar features. For x2Qp, fxgp = mpn 2[0;1) (0 m pn 1); x fxgp2Zp; fxgp = 0,x2Zp: While the ordinary fractional part on R is not additive, the deviation from additivity is given by an integer: fx+yg=fxg+fyg+ integer: This deviation from additivity gets wiped out when we take the complex exponential of both sides: e2 ifx+yg = e2 ifxge2 ifyg. Of course e2 ifxg = e2 ix for x 2 R, so there is no need to use the fractional part. But in the p-adic case, the fractional-part viewpoint gives us the following basic de nition: For x2Qp, set p(x) = e2 ifxgp. For example, 2(21=50) = e2 i(1=2) = 1; 3(21=50) = 1; 5(21=50) = e2 i(23=25): The function p is a group homomorphism: e2 ifx+ygp = e2 ifxgpe2 ifygp. To see this, we have to understand the extent to which the p-adic fractional part fails to be additive. For x;y2Qp, x fxgp; y fygp; x+y fx+ygp2Zp: Adding the rst two terms and subtracting the third, we see that the rational number fxgp +fygp fx + ygp is a p-adic integer. As a sum/di erence of p-adic fractional parts, it has a power of p as denominator. Since it is also a p-adic integer, there is no power of p in the denominator, so the denominator must be 1. Therefore fxgp +fygp fx+ygp2Z: Thus p is a group homomorphism. This is the technical way we make sense of the meaningless expression e2 ix for x2Qp; e2 ix can be thought of precisely as e2 ifxgp. (Warning: when x is rational, so e2 ix makes sense in the usual way, but this is generally not the same as e2 ifxgp. Referring to e2 ifxgp as just a \sensible interpretation" of e2 ix for p-adic x should only be considered as a loose manner of speaking.) Unlike the function x 7! e2 ix for real x, the p-adic function p: x 7! e2 ifxgp does not take on all values in S1. Its image is exactly the pth power roots of unity. It is also locally constant (so continuous in an elementary way), since p(x+y) = p(x) for y2Zp, and Zp is a neighborhood of 0. By the same type of \interior scaling" argument as for the basic character e2 ix on the reals, we can use p to construct many continuous characters of Qp. Choosing any y2Qp, we have a group homomorphism Qp!S1 by x7! p(xy) = e2 ifxygp: MATH 676. THE CHARACTER GROUP OF Q 5 By varying y in Qp, we get lots of examples of nontrivial continuous homomorphisms Qp!S1. It is a fact that all continuous homomorphisms Qp ! S1 are of the above type, but we will not need this fact in our study of the character group bQ of Q. However, our analysis of bQ contains the essential ingredients for a proof, so we will discuss this again in the appendix. By restricting a character of Qp to the subset Q, we get new characters of Q. For xed y2Qp, let r7!e2 ifrygp: This character of Q takes as values only the pth power roots of unity, so it is quite di erent (for y6= 0) from the functions r7!e2 iry for real y. Example. For a 5-adic number y, let : Q!S1 be de ned by (r) = e2 ifryg5. We want to calculate (21=50). Since 21 50 = 3 25 + 4 5 +:::; we need to know the 5-adic expansion of y out to the multiple of 5. If, for instance, y = 4 + 1 5 + :::, then (21=50)y = 2=25 + 1=5 + :::, so (21=50) = e14 i=25. To compute (r) for a speci c rational number r, we only need to know the 5-adic expansion of y to an appropriate nite number of places, depending on the 5-adic expansion of r. 3. bQ and the Adeles We shall now use the above homomorphisms from R and the various Qp?s to S1 to construct all homomorphisms from Q to S1, i.e., all elements of bQ. This is a simple example of the \local-global" philosophy in number theory, which says that one should try to analyze a problem over Q (the \global" eld) by rst analyzing it over each of the completions R;Q2;Q3;::: of Q (the \local" elds), and then use this information over the completions to solve the problem over Q. The problem we are concerned with is the construction of homomorphisms to S1, and we have already taken care of the \local" problem, at least for our purposes. De ne 1: R !S1 by x7!e 2 ix (the reason for the minus sign will be apparent later). De ne, as before, p: Qp !S1 by x7!e2 ifxgp. Now choose any elements a12R and ap2Qp for all primes p, with the proviso that ap2Zp for all but nitely many p. We de ne a function Q!S1 by r 7! 1(ra1) Y p p(rap) = e 2 ira1 Y p e2 ifrapgp: 6 KEITH CONRAD To show this map makes sense and is a character, note that for any rational r, r 2 Zp for all but nitely many primes p, so by our convention on the ap?s, rap2Zp for all but nitely many primes p. (The nitely many p such that rap 62 Zp will of course vary with r. It is not the case that factors where ap 2 Zp play no role, since we may have rap 62 Zp when ap 2 Zp, if the denominator of r has a large power of p.) Thus p(rap) = 1 for all but nitely many p, so for each r2 Q the in nite product de ning the above function at r is really a nite product. Each \local function" 1; 2; 3;::: is a homomorphism, so our map above is a homomorphism, hence is an element of bQ. The homomorphisms 1; 2; 3;::: are our basic maps, and the numbers a1;a2;a3;::: should be thought of as interior scaling factors that allow us to de ne many characters of Q in terms of the one basic character r7!e 2 ir Y p e2 ir: To understand this construction better, we want to look at the sequences of elements (a1;a2;a3;:::) that have just been used to de ne characters of Q. This leads us to introduce a ring which plays a prominent role in number theory. De nition. The adeles, AQ, are the elements (a1;a2;a3;:::) in the product set R Qp Qp such that ap lies in Zp for all but nitely many p. A \random" adele will not have any rational coordinates. As an example of something which is not an adele, consider any element of the product set R Qp Qp whose p-adic coordinate is 1=p for in nitely many primes p. If a is a typical adele, its real coordinate will be written as a1 and its p-adic coordinate will be written as ap. While 1 includes an awkward- looking minus sign (whose rationale will be explained below), a1 does not. It is the real coordinate of a, not its negative. Under componentwise addition and multiplication, AQ is a commutative ring (but not an integral domain). For our purposes, the additive group structure of AQ is its most important algebraic feature. Since any rational number r is in Zp for all but nitely many primes p, we see that Q naturally ts into AQ by the diagonal map r7!(r;r;r;:::); making Q a subring of AQ. We shall call an adele rational if all of its coordinates are the same rational number, so the rational adeles are natu- rally identi ed 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 bQ. From what has been done so far, for any adele a = (a1;a2;a3;:::) we have de ned a MATH 676. THE CHARACTER GROUP OF Q 7 character a of Q by a(r) = 1(ra1) Y p p(rap) = e 2 ira1 Y p e2 ifrapgp: Since addition in AQ is componentwise, a computation shows that for adeles a and b a+b(r) = a(r) b(r) for all rational numbers r, so in bQ we have a+b = a b. Clearly 0 is the trivial character. For a rational adele s and any r2Q, s(r) = e2 i( rs+ P pfrsgp) = 1 by (1). Thus s is the trivial character for all rational adeles s. This is why the minus sign was used in the de nition of 1. If a and b are two adeles whose di erence is a rational adele, a = 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 di erent ways. Theorem. Every character of Q has the form a for some a2AQ, and a = b if and only if a b is a rational adele. In other words, the map : AQ ! bQ given by a7! a is a surjective homomorphism with kernel equal to the rational adeles Q, so bQ = AQ=Q. 4. The Image and Kernel of Let : Q!S1 be a character. We want to write = a for some adele a. We begin by considering (1), which is some number on the unit circle, so (1) = e 2 i for a (unique) real 2 [0;1). De ne 1: Q ! S1 by 1(r) = 1(r ) = e 2 ir . Then 1 is a character of Q and 1(1) = (1). Let 0(r) = (r)= 1(r), so 0 is a character with 0(1) = 1 and (r) = 1(r) 0(r). For any rational r = m=n, 0(r)n = 0(m) = 0(1)m = 1: The image of 0 is inside the roots of unity, so dividing by 1 to give us 0 puts us in an algebraic setting. Every root of unity, say e2 is for s 2 Q, is a unique product of prime power roots of unity. Indeed, by (1) e2 is = Y p e2 ifsgp; where e2 ifsgp is a pth power root of unity, equal to 1 for all but nitely many p. 8 KEITH CONRAD Let p(r) denote the p-th power root of unity that contributes to the root of unity 0(r). For example, if 0(r) = e3 i=7, then 2(r) = 1, 7(r) = e10 i=7, and p(r) = 1 for all p 6= 2;7. The function p: Q ! S1 is a character of Q. Since 0(1) = 1, p(1) = 1 for all primes p. Since 0(r) = Qp p(r), we have (r) = 1(r) Y p p(r): This decomposition of can be viewed as the main step in describing bQ in terms of adeles. We have broken up our character into \local" characters 1; 2; 3;:::, and now proceed to analyze each one individually. By construction, 1(r) = e 2 ir for some real 2[0;1). We now want to show that p(r) = e2 ifrcgp for some c2Zp. This will involve giving an explicit method for constructing c. Since p(1) = 1, p(1=pn)pn = 1, so p(1=pn) = e2 icn=pn for some (unique) integer cn with 0 cn pn 1. Since p(1=pn+1)p = p(1=pn), we get cn+1 cn mod pnZ. Thus fc1;c2;c3;:::g is a p-adic Cauchy se- quence in Z, so it has a limit c2Zp, and c cn mod pnZp for all n. Since 0 cn pn 1, c pn p = cnpn: We now show p(r) = e2 ifrcgp = p(rc) for all r 2 Q. Write r = s=t where s;t2Z with t6= 0. Let t = pmt0 for (p;t0) = 1. Then p(r)t0 = p(t0r) = p s pm = ? 1 pm s = e2 icms=pm: In Q=Z, cms pm == c pm p s cs pm p fcrt0gp fcrgpt0; so p(r)t0 = e2 ifcrgpt0. Thus the p-th power roots of unity p(r) and e2 ifcrgp have a ratio that is a t0-th root of unity. Since t0 is prime to p, this ratio must be 1, so p s t = e2 ifcrgp = p(rc): Write c 2 Zp as ap, so p(r) = p(rap) for all r 2 Q. Therefore = 1 Qp p = a for the adele a = ( ;a2;a3;:::)2[0;1) Qp Zp. We now know every character of Q has the form a for an adele in the special set [0;1) Qp Zp. When the adele a is in this set, we can determine it from the character. Indeed, in this case all ap are in Zp, so a(1) = e 2 ia1. Since a12[0;1), it is completely determined from knowing a(1). We can then multiply a by the character e2 ira1 to assume a1 = 0. Then a has only roots of unity as its values. The pth power component of a(r) is e2 ifaprgp, which upon taking r = 1=p;1=p2, etc. allows us to successively MATH 676. THE CHARACTER GROUP OF Q 9 determine each digit of ap, so we can determine ap. Every character of Q has the form a for a unique adele a in [0;1) Qp Zp. It turns out that every adele can be put into this set upon addition by a suitable rational adele: (2) AQ = Q + [0;1) Y p Zp: To prove (2), x an adele a. We know ap2Zp as long as p is outside of a nite set (say) F. Let r = Pp2Ffapgp, the sum of the various pole parts of a. So a r has no pole parts, hence a r2 R Qp Zp. Let N be the integer such that N a1 r 0, (x) has constant sign on its imaginary part. Replacing with 1 if necessary, we may assume that has positive imaginary part on small positive numbers. Since (t)6= 1 for some t> 0, by connectedness ((0;t)) is an arc in S1, so contains a root of unity, say (u) for 0 0 is in the kernel of . Replacing by the continuous homomorphism x 7! (Nux), we may assume that (1) = 1. To summarize, we may assume is a continuous homomorphism from R to S1 which contains 1 in its kernel and which has positive imaginary part for small positive numbers. According to what we are trying to prove, we now expect that (x) = e2 ixy for some (positive) integer y, and this is what we shall show. Since the reciprocals of the prime powers generate the dense subgroup Q of R, by continuity it su ces to nd an integer y such that (1=pn) = e2 iy=pn for all primes p and integers n 1. Actually, we?ll show for any integer m > 1 that there is an integer ym such that (1=mn) = e2 iym=mn for all n 1. Since (1=(pq)n)pn = (1=qn) and (1=(pq)n)qn = (1=pn), it follows that ypq yq 2\nqnZ = f0g, so ypq = yq, and similarly that ypq = yp, so yp = yq. Thus for all primes p, the integers yp are the same, and this common integer y solves our problem. Since (1=mn)mn = (1) = 1, (1=mn) = e2 icn=mn for some integer cn such that 0 cn 0 and sin(2 cn=mn) is positive and arbitrarily small. Since 0 cn < mn, the cosine condition implies cn=mn2(0;1=4)[(3=4;1) for large n. We can?t have cn=mn in (3=4;1) for large n, since then sin(2 cn=mn) is negative (here is where we use the assumption that (x) is in the rst quadrant for small positive x). Thus for all large n, cn=mn 2 (0;1=4). Since sin(2 cn=mn) is arbitrarily small for all large n, it follows by the nature of the sine function and the location of cn=mn that cn=mn is arbitrarily small for n large. To x ideas, 0 < cn=mn < 1=(m + 1) for all large n. Then 0 < cn+1=mn < 12 KEITH CONRAD m=(m + 1) for all large n, so jcn=mn cn+1=mnj< 1 for all large n. Since the left hand side of this last inequality is an integer, it must be zero, so all cn?s are equal for n su ciently large. Call this common value ym. Thus (1=mn) = e2 iym=mn for all large n, hence for all n 1; for example, if (1=m100) = e2 iym=m100, then raising both sides to the m98-th power we see that (1=m2) = e2 iym=m2. Acknowledgments. I thank Randy Scott, Eric Sommers and Ravi Vakil for looking over a preliminary version of this manuscript. References [1] J.B. Conway, A Course in Functional Analysis, 2nd ed., Springer-Verlag, New York, 1990. [2] I. M. Gel?fand, M. I. Graev, I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions, Academic Press, 1990. [3] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Springer-Verlag, New York, 1986. [4] J. Tate, Fourier Analysis in Number Fields and Hecke?s Zeta-Functions, in: Algebraic Number Theory, Academic Press, New York, 1967, 305-347. [5] L. Washington, On the Self-Duality of Qp, American Mathematical Monthly 81 (4) 1974, 369-370.