p-adic L-functions for Dirichlet characters Rebecca Bellovin 1 Notation and conventions Before we begin, we fix a bit of notation. We mae the following convention: for a fixed prime p, we set q = p if p is odd, and we set q = 4 if p = 2. We will always view our Dirichlet characters as primitive; since we can obtain the L-function of an imprimitive Dirichlet character by throwing away Euler factors from the L-function of its associated primitive Dirichlet character, this convention will not affect interpolation questions. It does mean, however, that the product χ 1 χ 2 of two characters is not necessarily the pointwise product. The most important Dirichlet character will be the Teichmüller character, which we will denote ω. There is a canonical isomorphism Z p = (Z/qZ) (1+qZ p ), soforanya Z p,wecanwritea = ω(a) a,whereω(a) (Z/qZ) and a is a 1-unit. We can view ω either as a Dirichlet character modulo p or as a character on Z p, via the composition Z p (Z/qZ) Z p. We also fix an embedding Q Q p. Since classical Dirichlet characters tae algebraic values, this lets us view our Dirichlet characters as valued simultaneously in Q and in C p. The Bernoulli numbers are defined by the generating function 0 B t! = t e t 1 Given a Dirichlet character χ of conductor f, we define the generalized 1
Bernoulli numbers B χ, via the generating function 0 B χ, t! = t e ft 1 f χ(a)e at The B χ, are algebraic numbers, and they live in Q(χ), the extension of Q defined by adjoining the values of χ. If χ is the trivial character, we recover the ordinary Bernoulli numbers, except at = 1. Then L(χ,1 ) = B χ, for 1. We also define the adjusted Bernoulli numbers B χ, := (1 χ(p) p 1 )B χ, Finally, we define the Bernoulli polynomials by te Xt e t 1 = n=1 B n (X) tn n! At X = 0, we recover the classical Bernoulli numbers. At X = 1, we recover the classical Bernoulli numbers, except for B 1. We record the results of three generating function calculations for later use: Proposition 1.1. B n (X) = n i=0( n i) Bi X n i. Proposition 1.2. N 1 N 1 a=0 B ( X+a N ) = B (X) Proposition1.3. Let F be anymultipleof f. ThenB χ,n = F n 1 F χ(a)b n( a F). 2 Kummer congruences Given a Dirichlet character χ : (Z/NZ) C, we have an L-function L(χ,s) := χ(n) n 1 for complex s with R(s) > 1 (here we say that χ(n) = 0 n s if (n,n) 1), which we can analytically continue to a meromorphic function on all of C. We would lie to define a p-adic analogue of this L-function. 2
That is, we would lie to view L(χ,s) as an analytic function of a p-adic variable s valued in C p. There are some immediate problems with this. First of all, there are terms in the sum with arbitrarily large powers of p in the denominator, so the sum above will diverge badly. To have any hope of defining a p-adic analogue of L(χ,s), we will need to remove the n with p n and consider L (χ,s) := (1 1 p s )L(χ,s) instead ofl(χ,s). The series n 1 p n χ(n) n s still does not converge (as each term has p-adic absolute value 1), but at least the absolute values are note blowing up. The second problem is that even if n Z p, ns is not a p-adically continuous function of s unless n 1 + pz p. So we can t do something as naive as evaluating the sum defining L (χ,s) for s 0, s Z, say that it s clearly continuous, and say that a p-adically continuous function on a dense subset of Z p uniquely interpolates to a continuous function on all of Z p. Note, however, that if we restrict s to a single residue class modulo p 1, then n s is a p-adically continuous function of s for any fixed n Z p. This suggests that if we can mae sense of L(χ,s) as a p-adic function at all, we should also expect a dependence on residue classes modulo p 1. Instead, we will interpolate special values of the analytic continuation of L (χ,s). Recall that we can evaluate the Riemann ζ-function at negative integers, and ζ(1 ) = B for 1, where B is the th Bernoulli number (and B 1 = 1 ); ζ(1 ) = 0 2 for all odd integers. More generally, for any Dirichlet character χ, and L(χ,1 ) = B χ, L (χ,1 ) = B χ, We will interpolate these special values. The classical Kummer congruences state the following p-adic continuity result about the values B : 3
Theorem 2.1. Let m,n be positive even integers with m n (mod p a 1 (p 1)) and n 0 (mod p 1). Then B m /m and B n /n are p-integral and (1 p m 1 ) B m m (1 pn 1 ) B n n (mod p a ) This tells us that on p 2 of the p 1 residue classes for Z(p 1)Z, we can interpolate the Riemann zeta function to a p-adically continuous function, which gives us p 2 distinct p-adic zeta functions. Rather amazingly, there is a stronger result refining the Kummer congruences. Theorem 2.2. Suppose χ 1 is a power of the Teichmüller character. Then if m n (mod p a 1 ), we have (1 χω m (p)p m 1 ) B χω m,m m (1 χω n (p)p n 1 ) B χω n,n n (mod p a ) In other words, twisting χ by appropriate powers of the Teichmüller character has eliminated the requirement that m and n be in the same residue class modulo p 1, as well as the requirement that they not be divisible by p 1. Thus, granting these refined congruences, we can define a p-adic L-function as follows: For s Z p, choose a sequence of positive integers i converging to 1 s p-adically. Then we set L p (χ,s) = lim i (1 χω i (p)p i 1 ) B χω i, i i This defines a continuous function on Z p which interpolatates special values of several classical L-functions. We actually have the following stronger result: Theorem 2.3. Let χ be a Dirichlet character of conductor f. Then there is a p-adic meromorphic function L p (χ,s) on {s C p s < qp 1/(p 1) } such that L p (χ,s) = (1 χω n (p)p n 1 ) B χω n,n = L (χω n,1 n) n If χ is not the trivial character, then L p (χ,s) is analytic. If χ = 1, then the only pole of L p (χ,s) is at s = 1, where the residue is 1 1/p. 4
In fact, we have an explicit formula. If F is any multiple of q and f, then L p (χ,s) = 1 F 1 s 1 F p p χ(a) a 1 s j 0 ( 1 2 j ) (B j ) ( ) j F a A proof (including a proof of the explicit formula) can be found in [3]. We will give a different proof of analyticity after we explain the source of the twistedness of the interpolation. 3 Weight space Rather than view p-adic L-functions as analytic (or meromorphic) functions on Z p, we can view them as functions on the weight space. Definition 3.1. The weight space X is the set of continuous characters Hom cont (Z p,c p ). To understand this set of characters, first note that we can rewrite Z p as while we can rewrite C p as Z p = (Z/qZ) (1+qZ p ) = (Z/qZ) Z p C p = pq W U 1 where W is the group of roots of unity of C p of order prime to p, and U 1 = {x C p x 1 p < 1}. Any continuous map Z p C p must send µ p 1 (Z p ) to W and U 1 to U 1. Therefore, we can specify any character χ by a pair (i,s), where i Z/(p 1)Zands U 1 (s isthe imageofsome fixed topological generatorof 1+qZ p ). After fixing a topological generator γ of 1+qZ p, we write χ s for the character sending γ to s. We can embed Z in X by sending to the character ψ : x x. However, this map Z X is not p-adically continuous, because for a fixed x, x is generally not a p-adically continuous function of. To get a continuous map Z X, we need to ill the Z/(p 1)Z part of the map. For example, we could instead send to ψ ω =. 5
Thus, we get a natural embedding of D = {s C p s p < qp 1/(p 1) } into X, by sending s s. Given a Dirichlet character χ, we constructed a family of p 1 p-adic L- functions L p (χω i,s), so they define a (meromorphic) function L : X C p via L(ω i s ) = L p (χω i, s). But then B χ,n n = L (χ,1 n) = L p (χω n,1 n) = L(ω n n 1 ) = L(ψ n 1 ) So we find the special values of the classical L-function by evaluating L on a translate of the naive embedding of Z into X. 4 Mazur-Swinnerton-Dyer Fix a Dirichlet character χ of conductor p n M with p M and let Z p,m = Z/MZ Z p and Z p,m = (Z/MZ) Z p, so that χ is a character on Z p,m. Wewill construct a measure onz p,m anddefine the L-functionbyintegrating characters in the weight space over Z p,m. Definition 4.1. A measure µ on Z p,m assigns to each compact open subset U Z p,m a number µ(u) C p such that the distribution property is satisfied. That is, µ( U i ) = µ(u i ). Given a measure µ, we can integrate locally constant functions f = c i (1) Ui (here 1 is the characteristic function of U i ) by taing Z p,m fdµ = c i µ(u i ) This sum converges because Z p,m is compact. If f is a continuous function, we would lie to integrate f by approximating f on a+p m MZ p,m by f(a) and setting fdµ := lim f(a)µ(a+p m MZ p,m ) Z m p,m This will wor so long as µ is bounded, that is, there is a constant C R such that µ(u) p C for all compact open subsets U Z p,m. 6
As a first attempt, we define a family of measures µ on Z p,m by µ (a+p m MZ p,m ) = (p m M) 1B ({ a p m M }) where { a } is the fractional part of a. Then Proposition 1.2 tells us p m M p m M that this is a finitely additive measure on open sets of Z p,m. Furthermore, we can write Z p,m = a (Z/p n M) (a+p n MZ p,m ) for any n 1. Thus, if p divides the conductor of χ, we can use Proposition 1.3 to compute χdµ = χ(a)µ(a+p n MZ p,m ) Z p,m a (Z/p n M) pn M = (p n M) 1 = B χ, = B χ, If p does not divide the conductor of χ, χdµ = χ(a)µ(a+pmz p,m ) Z p,m a (Z/pM) = (pm) 1 ( pm = (1 χ(p)p 1 ) B χ, χ(a) B ( a ) pm χ(a) B ( a p n M ) = B χ, M χ(pa) B ( a ) ) M To define µ, we used {0,...,p m M 1} as distinguished integral representatives of (Z/p m MZ). If we had chosen a different set and used it to define a different measure µ, we would have µ (a+p m MZ p,m ) µ (a+p m MZ p,m ) (mod p m M) so our choices made very little difference. However, µ is not a bounded measure (because B ( We modify it as follows: Fix a u 1, u Z p,m, and define a p m M ) is p-adically large). µ,u (a+p m MZ p,m ) = µ (a+p m MZ p,m ) u µ (u 1 a+p m MZ p,m ) 7
For each m 0, there is a unique a m Z, 0 a m < pm M such that u 1 a a m (mod pm M). Then µ,u is bounded, because µ,u (a+p m MZ p,m ) = (p m M) 11 = 1 i=0 ( ( ) ( )) a a B u B m p m M p m M ( ) (B i )(p m M) i 1( a i u (a m i ) i) Every term in this sum is p-adically bounded as m gets large, except possibly the i = 0 term. But a u (a m) 0 (mod p m M) so even the i = 0 term is p-adically bounded for m large. In fact, rewriting a m as u 1 a q m (p m M), for some q m Z p,m, we get ( ( ) (p m M) 1 (a u (a m) ) = (p m M) 1 a u )(u 1 a) j ( q m p m M) j j j=0 ( ) = u (u 1 a) j ( q m ) j (p m M) j 1 j j=1 ua 1 q m (mod p m ) If we reduce our expression for µ,u (a+p m MZ p,m ) modulo p m 1 for m 0 (in case some of the B have a p in the denominator), we get (1 χ(p)p (1 ) ) ( ua 1 q m +B 1 a 1 (1 u) ) which is the same as 1 χ(p)p (1 ) a 1 µ 1,u (a+p m MZ p,m ) 1 χ(p) This shows that for f a continuous function on Z p,m, fdµ,u = x 1 f(x)dµ 1,u Z p,m Z p,m 8
Another calculation shows χdµ,u = (1 χ(u)u ) B χ, Z p,m Now we put everything together and define a function L on the weight space X by 1 L(ϕ) := χ(x)ω(x) 1 ϕ(x)dµ 1,u 1 χ(u)ϕ(u) u Z p,m This is defined for any ϕ X except 1. And for ϕ = ψ n 1, 1 L(ϕ) = χ(x)x 1 dµ 1 χ(u)u 1,u Thus, L is a meromorphic function on the weight space whose only pole is at 1 and which interpolates our special values correctly. Z p,m References [1] Neal Koblitz. p-adic numbers, p-adic analysis, and zeta-functions, 2nd ed. [2] Jay Pottharst. Many Twisted Interpolations, Part I. [3] Lawrence C. Washington. Cyclotomic Fields. 9