MAZUR S CONSTRUCTION OF THE KUBOTA LEPOLDT -ADIC L-FUNCTION XEVI GUITART Abstract. We give an overview of Mazur s construction of the Kubota Leooldt -adic L- function as the -adic Mellin transform of a Bernoulli measure. We follow (or rather just coy in many occasions) Lang [Lan78] ( 2 of Chater 2 and 3 of Chater 4) and Koblitz [Kob84, Chater II].. Introduction Let ζ(s) be the zeta function, defined for Re(s) > as (.) ζ(s) ( s ). n n s We aim to give a construction of the Kubota Leooldt -adic zeta function ζ, which is a function of a -adic variable that interolates values of ζ at negative integers. One might thin, on first thought, that the interolation roerty of ζ should be (.2) ζ ( ) ζ( ), for all Z >0. However, as we will see, this is not exactly the interolation roerty that ζ satisfies. The actual one differs from (.2) in two asects: () it interolates ζ with the -th Euler factor removed, and (2) only at those integers 0 (mod ). That is to say, ζ satisfies the interolation roerty (.3) Recall the following roerties of ζ(s): () Integral reresentation: (.4) ζ ( ) ( )ζ( ) for all 0 (mod ). ζ(s) Γ(s) x s 0 dx e x. In other words, ζ(s) is essentially (i.e., excet for the term Γ(s)) the Mellin transform of the measure dx/(e x ). (2) Secial values: They can be given in terms of Bernoulli numbers. For any Z >0 we have that ζ( ), Date: November 7, 203. 200 Mathematics Subject Classification. G40 (F4, Y99).
2 XEVI GUITART where the Bernoulli numbers are defined by t e t t!. 0 The strategy for roving (.4) is to exlicitly comute by a direct calculation using (.) ζ() for ositive and even. Then one uses the functional equation relating ζ(s) and ζ( s) to derive (.4) for all Z >0. We will resent Mazur s construction of ζ, in which ζ is defined as a -adic Mellin transform of a so-called Bernoulli distribution. Actually, we will treat a more general case. Let χ: (Z/ m Z) Q be a Dirichlet character, which we lift to a function χ: Z Q extending by 0 to the non rime to integers. We fix an embedding Q C Q, and we regard the values of χ either as algebraic numbers or as elements of C via this fixed embedding. The L-function of χ is given for Re(s) > by L(s, χ) n χ(n) n s l ( χ(l) ) l s. It can be extended by analytic continuation to C and it turns out that for ositive integers its secial values are L(, χ),χ, where the,χ are the so-called generalized Bernoulli numbers (see Definition 3.5 below for the definition). The goal of this notes is to give a roof of the following result. Theorem. (Kubota Leooldt, Iwasawa). There exists a unique -adic meromorhic (analytic if χ χ triv ) function L (s, χ), s Z, such that for Z >0 L (, χ) L(, χω ), where ω : (Z/Z) Q denotes the Teichmüller character. Remar.2. If we tae χ χ triv : (Z/Z) Q the trivial character modulo and 0 (mod ) we see that L (, χ) L(, χ) ( )ζ( ), so that we obtain the -adic zeta function ζ (s) as a articular case. Let {X n } be a collection of finite sets and 2. -adic distributions π n+ : X n+ X n a collection of surjective mas. Thus we can consider the rojective limit X lim X n and the rojections r n : X X n. We endow X with the limit toology, meaning that a basis of oens is
MAZUR S CONSTRUCTION OF THE KUBOTA LEPOLDT -ADIC L-FUNCTION 3 {rn (x): x X n } n. Let also K be a comlete local field with resect to a non-archimedean norm, and let ϕ n : X n K be a collection of mas. Definition 2.. {ϕ n } n is said to be comatible if for each x X n ϕ n+(y) ϕ(x). y π n+ (x) Definition 2.2. A function f : X K is locally constant if it factors through X n for some n. We denote by LC(X, K) the set of locally constant functions. Observe that if f factors through X n it also factors through X m for any m n. comatibility of {ϕ n } imlies that f(x)ϕ n (x) f(x)ϕ m (x), x X n x X m so that there is a well defined K-linear functional dϕ: LC(X, K) K f fdϕ : x X n f(x)ϕ n (x). Then {ϕ n } (or the functional dϕ) is called a distribution on X. Observe that fdϕ (2.) f(x)ϕ n (x) max f(x) ϕ n (x) f ϕ, x X n x X n where denotes the su norm and ϕ su n ϕ n. Then the Proosition 2.3. Every continuous function f C(X, K) is a uniform limit of locally constant functions. That is to say, there exist functions f n LC(X, K) such that f n f 0. Proof. One can define f n : X n K by f n (x) f(s) for any s such that r n (s) x, and then tae f n f n rn. We also have that f n f m 0, and by (2.) we see that (f n f m )dϕ f n f m ϕ, so that the sequence f n dϕ converges if ϕ <. Definition 2.4. A measure is a bounded distribution. By the discussion above, a measure ϕ defines a functional dϕ: C(X, K) K f fdϕ : lim n fn dϕ(x). We end this section with a reformulation of the above formalism, also articularized to the case of interest to us, which is when X Z. In this case a distribution is a collection of mas ϕ n : Z/ n Z K satisfying the comatibility condition ϕ n (a) ϕ n+ (a + b n ). b0
4 XEVI GUITART Let us denote by U the set of oen comacts in Z. Any U U is a disjoint union of balls of the form a + n Z. In articular, we have that Then we can define a ma a + n Z b0 a + b n + n+ Z. and the comatibility condition is equivalent to ϕ: U K a + n Z ϕ(a + n Z ) : ϕ n (a), ϕ(a + n Z ) ϕ(a + b n + n+ Z ). b0 If we let U n {a+ n Z },...,n denote the set of balls of radius n we can rewrite the integral of a continuous function f as fdϕ lim f(t U )ϕ(u), Z n U U n where t U U is any samle oint. Finally, if Y Z we will use the notation fdϕ Y Z Y fdϕ. 3. Bernoulli distributions Recall the definition of the Bernoulli numbers : t e t t!. The Bernoulli olynomials (X) are defined by: 0 te tx e t 0 (X) t!. Observe that (0). The first Bernoulli olynomials are B 0 (X), B (X) X 2, B 2(X) X 2 X 6. Proosition 3.. The Bernoulli olynomials satisfy the relation (3.) ( ) X + a (X).
MAZUR S CONSTRUCTION OF THE KUBOTA LEPOLDT -ADIC L-FUNCTION 5 Proof. We have that te (X+a)t e t 0 te X+a t e t 0 ( X + a ( ) X + a (t)! ) t!. On the other hand, by summing the geometric series we find that: te (X+a)t e t text e t et e t text e t 0 (X) t!, and the result follows by comaring the terms of t! in the two exressions. Definition 3.2. The -th Bernoulli distribution is defined by the mas Z/ n Z Q Q( x n( ) ) x n where in this exression we tae x to be an integer between 0 and n. If t is a rational number let t denote the least nonnegative in the same class of t modulo Z. Sometimes we will use also a notation lie ( x ) if we don t want to secify that x has to be n taen between 0 and n. Remar 3.3. The mas of Definition 3.2 do secify a distribution. Indeed, the comatibility roerty follows from (3.). It is convenient to wor with a normalized version of the Bernoulli distribution. Definition 3.4. We denote by E the distribution given by the mas (x) n( ) ( x n ) for x Z/n Z. Observe that since the function factorizes through Z/Z we have de (3.2) Z ( a ) (0). Generalized Bernoulli numbers. Let f : Z/ n Z K be a function. Definition 3.5. The generalized Bernoulli olynomials,f are defined by n f(a) te(x+a) t e n t 0,,f (X) t!. The constant terms,f :,f (0) are the generalized Bernoulli numbers (relative to f). The following identity is an immediate consequence of the definitions: n,f n( ) f(a) ( a n ).
6 XEVI GUITART Interreting f as a locally constant function on Z we can rewrite this as (3.3),f fde. Z From distributions to measures. Returning to the Bernoulli distributions, we remar the fact that they are not measures. For instance (x) B ( x n ) x n 2x n 2 n, which is not -adically bounded. They can be turned into measures by a standard rocess, called regularization. For this let c be a rational integer such that c. Then we define distributions E,c by means of the mas,c (x) E(n) (x) c (c x), for x Z/ n Z. Here c x denotes multilication in Z/ n Z. It is easy to see that E,c is a distribution (for a linear combination of distributions is a distribution). Passing to the limit the above mas we can write (3.4) E,c (x) E (x) c E (c x) The following two roerties are really ey in the construction of the -adic L-function. Proosition 3.6. () The values of,c are -integral (i.e., they belong to Z ). (2),c (x) x,c (x) (mod n d Z ), where d is the -adic valuation of the least common multile of the denominators of the coefficients of (X). Proof. We will first rove the second assertion. To begin with, let s assume that d 0, so that the coefficients of lie in Z. It is easily checed that (X) is of the form X + X + higher order terms. 2 Now let x Z/ n Z, which we also view as an integer 0 x n. Denote by {c x} the reresentative of c x in Z/ n Z that lies in the range 0,..., n. Then we can write c{c x} x + a n for some a Z. The following congruences are modulo n Z :,c (x) [ ( ) ( x {c n( ) n c )] x} n [ ( ) x n( ) n ( ) ] [ x 2 n ( {c n( ) c ) x} n c ( {c ) ] x} 2 n [ ( ) x n( ) n ( ) ] [ ( x 2 n ) n( ) c {c x} n c ( ) ] c {c x} 2 n [ ( ) x n( ) n ( ) ] [ ( x 2 n ) x n( ) n + a c ( ) ] x 2 n + a ( ) c x a. 2 actually this d is missing in Lang s statement, but I believe it should be there (cf. [Kob84], Chater 4, 5, Theorem 5). In any case, it does not mae any difference for large n, since d is indeendent of n.
MAZUR S CONSTRUCTION OF THE KUBOTA LEPOLDT -ADIC L-FUNCTION 7 On the other hand: (3.5),c (x) B ( ) ( x {c ) x} n cb n x n [ {c 2 c x} n ] 2 x n 2 x + an n c 2 c a. 2 If d > 0, one considers D where D is the least common multile of the denominators of (X). Since D has coefficients in Z, the above calculations go through and give the result in general. Now we rove the first statement. First of all observe that,c taes values in Z because of (3.5) (recall that a Z). Now for n big enough n d 0, and that,c (x) belongs to Z follows from art (2) of the roosition. If n is such that n d < 0, we just tae m such that m d > 0 and use the distribution relation:,c (x) E (m),c (y) Z. y x (mod n ) Remar 3.7. Passing to the limit the second statement in the roosition it can be rewritten as (3.6) E,c (x) x E,c (x). Combining this with the formula (3.2) for we obtain the following imortant exression of in terms of E,c. Proosition 3.8. For any Z >0 (3.7) c x de,c (x) Z Proof. It follows from the following comutation: (3.2) (3.4) de de,c + Z Z ( ) de,c + c de (x) (3.6) Z Z c de (c x) Z Z x de,c + c, where in ( ) we used the change of variables x cx (observe that this transforms Z in Z because by assumtion c ). Observe that the left term in (3.7) is (the negative of) ζ( ). Therefore, if the right hand side of (3.7) was still meaningful when we relace the integer by an arbitrary s Z, we could use this exression on the right to define a -adic function that interolates ζ at the integers. However, in the right hand side of (3.7) we have the function x for x ranging over the domain of integration, which is Z. The roblem is that if x Z and s Z, we can not always consider something lie x s. For instance, if x Z then it is clear that we can not give a meaning to x s for arbitrary s Z. A first ste in order to fix this roblem would be to exress the Bernoulli numbers as integrals over, rather than over Z. The following roosition tells us that this is also ossible, and the cost is the aearance of an Euler factor term.
8 XEVI GUITART Proosition 3.9. For any Z >0 (3.8) ( ) B c Proof. Since Z Z we have: x de,c (x) Z x de,c (x). x de,c (x) + x de,c (x). Z Now the result follows immediately from the following claim: Claim. x de,c x de,c. Z Z In order to rove the claim, we consider the decomosition of Z Z n x0 which induces the following decomosition of Z : Z Then, by definition of the integral we have n x0 Z x de,c lim n x + n Z, x + n+ Z. n x0 (x) E n+,c (x) ( ) n lim x E,c(x) n n x0 Z x de,c (x), where in ( ) we have used that E (n+),c (x),c (x). This follows directly from the shae of the Bernoulli olynomial B (X) X 2. Indeed, B (n) (x) x n x n+ 2 B(n+) (x). Remar 3.0. The exression on the right hand side of (3.8) still does not mae sense when relacing by an arbitrary s Z. Indeed, the exression x s is well defined in general only when x belongs + Z. But at least now x in (3.8), so that x : xω(x) lies in + Z and x s maes sense for all s Z. This is is how we will obtain the desired -adic L-function, but we ostone the details until the next section. From the exressions of the Bernoulli numbers in terms of -adic integrals of Proosition 3.9 one recovers, as a consequence of rather elementary roerties of the integrals, the following classical congruences. Corollary 3. (Congruences of Kummer and von Staudt). Let be a ositive integer.
MAZUR S CONSTRUCTION OF THE KUBOTA LEPOLDT -ADIC L-FUNCTION 9 () If then. (2) If and (mod ( ) n ) then ( ) ( ) (mod n+ ). (3) If 0 (mod ( )) and is even then (mod Z ). Proof. We just rove the second statement, the others are roved similarly. First of all, observe that the congruence is equivalent to c x (3.9) de,c (x) x de,c (x) (mod n+ ). c It is enough to rove that: c (mod n+ Z ), and c (3.0) x de,c (x) (mod n+ ) x de,c (x) (mod n+ Z ). For this choose c a rimitive root modulo, so that c (mod ). This imlies that c and c are -adic units. In addition, if we write + a n ( ) for some a we find that c c c ( c ( )na ). But c (mod ), and an easy exercise using the binomial exansion shows that c ( )n (mod n+ ). Equivalently, c c (n+), and from this it also follows that c c (n+). Alying the same argument to x and x (which we can since x ) we also find that x x (n+). Now (3.0) follows from the basic roerty of the integrals (2.), together with the fact that E,c taes values in Z. Now let χ: Z/ n Z Q C be an Dirichlet character (we assume n > 0, and we extend by 0 on the non-invertible elements). By (3.3) and the fact that χ is zero on Z we see that χde. Proosition 3.2. For Z >0 we have that (3.),χ χ(c)c χ(x)x de,c (x)
0 XEVI GUITART Proof. It is comutation, similar to that of Proosition 3.7:,χ χde χde,c + χ(x)c de (c x) x χ(x)de,c (x) + χ(c)χ(x)c de (x) x χ(x)de,c (x) + c χ(c),χ. 4. Definition of the -adic L-function Let ω : µ denote the Teichmüller character (we assume 2 for simlicity here), characterized by the roerty that ω(a) a (mod ). Then any a can be written uniquely as a a ω(a), with a (mod ). Definition 4.. The Mellin transform of a measure µ is the function on the -adic variable s defined by M µ(s) a s a dµ(a). The Mellin transform is analytic on Z. This follows from the following lemma. Lemma 4.2. Let µ be a measure on. Then there exist b n Z with b n 0 such that a s dµ b n s n, for s Z. n Proof. We can decomose the integral as a s dµ b ω(b) (+Z ) a s dµ(a). In each integral we can mae the change of variables a ω(b)x (and relace µ by another measure), so that we are reduced to rove the roosition for integrals of the form +Z x s dµ(x). Now we have that x s dµ +Z x s dµ +Z +Z n0 +Z n0 s(s ) (s n + ) ( ) n (x ) n dµ(x) s (x )n dµ(x). n!
MAZUR S CONSTRUCTION OF THE KUBOTA LEPOLDT -ADIC L-FUNCTION Now x runs over +Z, so that x (mod Z ). This imlies that (x )n n! is -integral and tends to 0 -adically. Thus we can interchange the sum and the integral: (4.) x s dµ s(s ) (s n + )c n, +Z (x ) n n0 where the c n +Z n! dµ(x) are -integral and tend to 0 as n. Now it is clear that we can reorder (4.) and write it as a ower series of the shae stated in the roosition. Fix a c such that χ(c) c s is not identically. Definition 4.3. The -adic L-function L ( s, χ) is defined as L ( s, χ) χ(c) c s x s χ(x)x de,c (x). Observe that the integral is an analytic function of s by the lemma. The term in front is analytic excet when χ(c) c s. If χ is non trivial then one can choose c such that χ(c) and therefore it is analytic also at s 0. Finally we see that this function satisfies the desired interolation roerties. Theorem 4.4. For any Z >0 we have that L (, χ),χω. Proof. It follows from this simle comutation: x χ(x)ω(x) de,c (x) x χ(x)ω(x) ( ) ω(x) de,c (x) (3.) x χ(x)ω(x) de,c (x) ( χω (c)c ),χω. Corollary 4.5. If 0 (mod ( )) and is odd then L (, χ),χ. References [Kob84] Neal Koblitz, -adic numbers, -adic analysis, and zeta-functions, second ed., Graduate Texts in Mathematics, vol. 58, Sringer-Verlag, New Yor, 984. MR 754003 (86c:086) [Lan78] Serge Lang, Cyclotomic fields, Sringer-Verlag, New Yor, 978, Graduate Texts in Mathematics, Vol. 59. MR 0485768 (58 #5578) E-mail address: xevi.guitart@gmail.com