Dirichlet Series Expansions of p-adic L-Functions
aa r X i v : . [ m a t h . N T ] F e b DIRICHLET SERIES EXPANSIONS OF P-ADIC L-FUNCTIONS
HEIKO KNOSPE, LAWRENCE C. WASHINGTON
Abstract.
We study p -adic L -functions L p ( s, χ ) for Dirichlet characters χ .We show that L p ( s, χ ) has a Dirichlet series expansion for each regularizationparameter c that is prime to p and the conductor of χ . The expansion is provedby transforming a known formula for p -adic L -functions and by controlling thelimiting behavior. A finite number of Euler factors can be factored off in anatural manner from the p -adic Dirichlet series. We also provide an alternativeproof of the expansion using p -adic measures and give an explicit formula forthe values of the regularized Bernoulli distribution. The result is particularlysimple for c = 2, where we obtain a Dirichlet series expansion that is similarto the complex case. Introduction
Let p be a prime, let q = p if p is odd and q = 4 if p = 2, and let χ be a Dirichletcharacter of conductor f . A p -adic L -function L p ( s, χ ) for a Dirichlet character χ is a p -adic meromorphic function and an analogue of the complex L -function. Forpowers of the Teichm¨uller character ω of conductor q , one obtains the p -adic zetafunctions ζ p,i = L p ( s, ω − i ), where i = 0 , , . . . , p − i = 0 , p = 2). It iswell known that L p ( s, χ ) is identically zero for odd χ . p -adic L -functions have along history and the primary constructions going back to Kubota-Leopoldt [6] andIwasawa [3] are via the interpolation of special values of complex L -functions. Itcan also be shown that p -adic L -functions are in fact Iwasawa functions.It is well known that for Re ( s ) > − − s ) ζ ( s ) = ∞ X n =1 ( − n +1 n s and, more generally, if c ≥ − χ ( c ) c − s ) L ( s, χ ) = ∞ X n =1 χ ( n ) a c,n n s , where a c,n = 1 − c if n ≡ c and a c,n = 1 if n c . In the following,we derive similar, but slightly different, expansions for p -adic L -functions.An explicit formula for L p ( s, χ ) is given in [9] (Theorem 5.11): let F be anymultiple of q and f . Then L p ( s, χ ) is a meromorphic function (analytic if χ = 1)on { s ∈ C p | | s | < qp − / ( p − } such that Mathematics Subject Classification.
Primary: 11R23. Secondary: 11R42, 11S80, 11M41. (1) L p ( s, χ ) = 1 F s − F X a =1 p ∤ a χ ( a ) h a i − s ∞ X j =0 (cid:18) − sj (cid:19) (cid:18) Fa (cid:19) j B j . In Section 2, we will use formula (1) to derive a Dirichlet series expansion of L p ( s, χ ). p -adic L -functions can be also be defined using distributions and measures. Let χ have conductor f = dp m with ( d, p ) = 1. Choose an integer c ≥
2, where( c, dp ) = 1. Then there is a measure E ,c on ( Z /d Z ) × × Z × p (the regularizedBernoulli distribution ) such that(2) − (1 − χ ( c ) h c i − s ) L p ( s, χ ) = Z ( Z /d Z ) × × Z × p χω − ( a ) h a i − s dE ,c (see [9] Theorem 12.2). In Section 3, we give an explicit formula for the values of E ,c and derive the Dirichlet series expansion from (2).The expansion is particularly simple for c = 2, and this parameter can be usedfor p = 2 and Dirichlet characters with odd conductor. For this case we obtainsimilar results as in [1], [2], and [4]. In Section 4, we provide examples for differentparameters c . 2. Expansions of p -adic L -Functions First, we derive an approximation of L p ( s, χ ) that is close to the original defini-tion of Kubota-Leopoldt (see [6]).For r ∈ C × p we write δ ( r ) for a term with p -adic absolute value ≤ | r | . Proposition 2.1.
Let p be a prime number, χ an even Dirichlet character ofconductor f , and F a multiple of q and f . For s ∈ C p with | s | < qp − / ( p − , wehave (3) L p ( s, χ ) = 1 F s − F X a =1 p ∤ a χ ( a ) h a i − s + δ ( F/qp ) . Proof.
We use formula (1) above and look at the series P ∞ j =0 (cid:0) − sj (cid:1) (cid:0) Fa (cid:1) j B j . Thefirst two terms are 1 + (1 − s ) − F a . We claim that the p -adic absolute value of theother terms ( j ≥
2) is less than or equal to | ( s − F /qp | . To this end, we notethat | /j ! | ≤ p ( j − / ( p − and (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − sj (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | − s | p ( j − / ( p − ( qp − / ( p − ) j − = | − s | q j − since we assumed that | s | < qp − / ( p − . Since | F | ≤ q , | a | = 1, and | B j | ≤ p , weobtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − sj (cid:19) (cid:18) Fa (cid:19) j B j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | − s | q j − q − j | F | p = | − s | | F | qp. IRICHLET SERIES EXPANSIONS OF P-ADIC L-FUNCTIONS 3
Then (1) implies L p ( s, χ ) = 1 F s − F X a =1 p ∤ a χ ( a ) h a i − s + 12 F X a =1 p ∤ a χω − ( a ) h a i − s + δ ( F/qp ) . It remains to show that the second sum can be absorbed into δ ( F/qp ). We have F X a =1 p ∤ a χω − ( a ) h a i − s = F X b =1 p ∤ b χω − ( F − b ) h F − b i − s = − F X b =1 p ∤ b χω − ( b ) h b − F i − s = − F X b =1 p ∤ b χω − ( b ) h b i − s + δ ( F/q ) . The last step can be justified by noting that h b − F i − s h b i − s = (cid:18) − Fb (cid:19) − s = 1 + ∞ X k =1 (cid:18) − sk (cid:19) (cid:18) − Fb (cid:19) k = 1 + δ ( F/q ) , since | s | < qp − / ( p − (this is the same estimate as earlier, without the presence ofthe Bernoulli number). This proves the proposition. (cid:3) Remark 2.2.
For F = f p n and n → ∞ , formula (3) gives the original definitionof L p ( s, χ ) by Kubota and Leopoldt (see [6]). Remark 2.3.
Suppose that p = 2. Then the error term in the above Proposition(as well as in the following Theorem 2.4) can be improved to δ ( F/p − ( p − / ( p − ).First we note that B j = 0 for odd j ≥
3. By the von Staudt–Clausen Theorem(see [9] 5.10), we have for even j ≥ | B j | = p iff ( p − | j , and otherwise | B j | ≤
1. Furthermore, | /j ! | = p ( j − S j ) / ( p − , where S j is the sum of the digits of j , written to the base p (see [5]). Since j ≡ S j mod ( p − j ≡ p −
1) isequivalent to S j ≡ p − | B j | = p yields S j ≥ p − | /j ! | ≤ p ( j − / ( p − p − ( p − / ( p − . This implies the above error term. We also seethat this error term cannot be further improved. ♦ Now we give the Dirichlet expansion of L p ( s, χ ). For m ∈ N , we denote by { x } m the unique representative of x mod m Z between 0 and m − Theorem 2.4.
Let p be a prime number, χ be an even Dirichlet character ofconductor f , and F a multiple of q and f . Let c > be an integer satisfying ( c, F ) = 1 . For a ∈ Z , define ǫ a,c,F = c − − {− aF − } c ∈ (cid:26) − c − , − c −
12 + 1 , . . . , c − (cid:27) . Then we have for s ∈ C p with | s | < qp − / ( p − the formula − (1 − χ ( c ) h c i − s ) L p ( s, χ ) = F X a =1 p ∤ a χω − ( a ) h a i − s ǫ a,c,F + δ ( F/qp ) . HEIKO KNOSPE, LAWRENCE C. WASHINGTON
Proof.
Use (3) with cF in place of F , and subtract χ ( c ) h c i − s times (3) with F , toobtain (1 − χ ( c ) h c i − s ) L p ( s, χ ) = 1 cF s − cF X a =1 p ∤ a χ ( a ) h a i − s − F s − F X a =1 p ∤ a χ ( ac ) h ac i − s + δ ( F/qp ) . (4)Let 0 < a < F with ( a , p ) = 1. Since we assumed ( c, F ) = 1 and p | F ,there is a unique number of the form a c with 0 < a c < cF and ( a c, p ) = 1 ineach congruence class modulo F relatively prime to p . The first sum in (4) can bewritten as1 cF s − F X a =1 p ∤ a χ ( a c ) h a c i − s cF X a =1 a ≡ a c mod F (cid:28) a − a ca c (cid:29) − s = 1 cF s − F X a =1 p ∤ a χ ( a c ) h a c i − s cF X a =1 a ≡ a c mod F − s ) a − a ca c + δ ( F/q ) . Note that | a − a ca c | ≤ | F | , so this is the same type of estimate used in the proof ofProposition 2.1. Subtracting the second sum in (4) yields(1 − χ ( c ) h c i − s ) L p ( s, χ )= − cF F X a =1 p ∤ a χ ( a c ) h a c i − s cF X a =1 a ≡ a c mod F a − a ca c + δ ( F/qp )= − c F X a =1 p ∤ a χω − ( a c ) h a c i − s cF X a =1 a ≡ a c mod F a − a cF + δ ( F/qp ) . We compute the inner sum. Let b = { a c } F . Then a c = b + {− F − b } c F , since thelatter sum is congruent to b modulo F and congruent to 0 modulo c . If a satisfies a ≡ a c mod F and 0 < a < cF , then a = b + jF with 0 ≤ j < c . Hence cF X a =1 a ≡ a c mod F a − a cF = c − X j =0 ( j − {− F − b } c ) = c ǫ b,c,F . Since b ≡ a c mod F , we have χω − ( b ) h b i − s = χω − ( a c ) h a c i − s + δ ( F/q ) by thesame estimate as earlier, so − (1 − χ ( c ) h c i − s ) L p ( s, χ ) = F X b =1 p ∤ b χω − ( b ) h b i − s ǫ b,c,F + δ ( F/qp ) . This completes the proof. (cid:3)
IRICHLET SERIES EXPANSIONS OF P-ADIC L-FUNCTIONS 5
We can take the limit of F = f p n as n → ∞ and obtain: Corollary 2.5.
Let p be a prime number, χ an even Dirichlet character of conduc-tor f , and c > an integer satisfying ( c, pf ) = 1 . Then we have for s ∈ C p with | s | < qp − / ( p − , − (1 − χ ( c ) h c i − s ) L p ( s, χ ) = lim n →∞ fp n X a =1 p ∤ a χω − ( a ) ǫ a,c,fp n h a i s . The next Theorem shows that a finite number of Euler factors can be factoredoff in a similar way as in [8], where a weak Euler product was obtained. The mainstatement is that the remaining Dirichlet series has the expected form, similar tothe complex case.
Theorem 2.6.
Let p be a prime number and let χ be an even Dirichlet characterof conductor f . Let S be any finite (or empty) set of primes not containing p andset S + = S ∪ { p } . Let F be a multiple of q , f and all primes in S . Let c > bean integer satisfying ( c, F ) = 1 . Then we have for s ∈ C p with | s | < qp − / ( p − theformula − (1 − χ ( c ) h c i − s ) · Y l ∈ S (1 − χω − ( l ) h l i − s ) · L p ( s, χ ) = F X a =1( a,S + )=1 χω − ( a ) ǫ a,c,F h a i s + δ ( F/qp ) . Proof.
We prove the statement by induction on | S | . By Theorem 2.4, the formulais true for S = ∅ . Now assume the formula is true for S , and l = p is a prime with l / ∈ S and ( c, l ) = 1. It suffices to prove the following formula:(1 − χω − ( l ) h l i − s ) F X a =1( a,S + )=1 χω − ( a ) h a i − s ǫ a,c,F = lF X a =1( a,S + ∪{ l } )=1 χω − ( a ) h a i − s ǫ a,c,lF + δ ( F/qp ) . (5)Note that | − χω − ( l ) h l i − s | ≤ | lF | = | F | , so we can keep the error term.We can use lF in place of F and write the left side of (5) as lF X a =1( a,S + )=1 χω − ( a ) h a i − s ǫ a,c,lF − F X a =1( a,S + )=1 χω − ( la ) h la i − s ǫ a,c,F + δ ( F/qp ) . (6)Now we have ǫ la,c,lF = c − − {− la ( lF ) − } c = c − − {− aF − } c = ǫ a,c,F . HEIKO KNOSPE, LAWRENCE C. WASHINGTON
Thus (6) is equal to lF X a =1( a,S + )=1 χω − ( a ) h a i − s ǫ a,c,lF − F X a =1( a,S + )=1 χω − ( la ) h la i − s ǫ la,c,lF + δ ( F/qp )= lF X a =1( a,S + )=1 l ∤ a χω − ( a ) h a i − s ǫ a,c,lF + δ ( F/qp ) , which shows equation (5). (cid:3) Remark 2.7.
What happens if S contains more and more primes? It is wellknown that the Euler product does not converge p -adically. There are infinitelymany primes l with χω − ( l ) = 1, and we have for l = p ,1 − h l i − s = − ∞ X j =1 (cid:18) − sj (cid:19) ( h l i − j . For | s | < qp − / ( p − , the p -adic absolute value of each term of the above series isless than ( qp − / ( p − ) j p ( j − / ( p − q − j = p − / ( p − < . Hence the product of Euler factors (1 − χω − ( l ) h l i − s ) − does not converge as S picks the primes one after the other. We also see that the p -adic Dirichlet series overnumbers relatively prime to S converges to 0 as S expands to include all primes, n → ∞ and F being a multiple of p n , f and the primes in S .3. Regularized Bernoulli Distributions
Let p be a prime number and let d be a positive integer with ( d, p ) = 1. Define X n = ( Z /dp n Z ) and X = lim ←− X n ∼ = Z /d Z × Z p . Let k ≥ Bernoulli distribution E k on X is defined by E k ( a + dp n X ) = ( dp n ) k − k B k (cid:18) { a } dp n dp n (cid:19) , where B k ( x ) is the k -th Bernoulli polynomial and B k = B k (0) are the Bernoullinumbers (see [5], [7]). For k = 1, one has B ( x ) = x − . Choose c ∈ Z with c = 1and ( c, dp ) = 1. Then the regularization E k,c of E k is defined by E k,c ( a + dp n X ) = E k ( a + dp n X ) − c k E k (cid:18)n ac o dp n + dp n X (cid:19) . One shows that the regularized Bernoulli distributions E k,c are measures (see [7]).In the following, we consider only k = 1; the cases k ≥ Theorem 3.1.
Let p be a prime, c, d ∈ N , and c ≥ such that ( c, dp ) = 1 . Let X be as above, and let E ,c be the regularized Bernoulli distribution on X . For a ∈ { , , . . . , dp n − } , we have E ,c ( a + dp n X ) = c − − {− a ( dp n ) − } c = ǫ a,c,dp n . IRICHLET SERIES EXPANSIONS OF P-ADIC L-FUNCTIONS 7
Proof.
By definition, E ,c ( a + dp n X ) = E ( a + dp n X ) − cE ( c − a + dp n X ) = adp n − − c (cid:18) { c − a } dp n dp n (cid:19) + c . We give the standard representative of c − a mod dp n : { c − a } dp n = {− a ( dp n ) − } c dp n + ac Note that the numerator is divisible by c , since {− a ( dp n ) − } c dp n ≡ − a mod c .Hence the quotient is an integer between 0 and dp n −
1. Furthermore, the numeratoris congruent to a modulo dp n , and so the quotient has the desired property. Weobtain E ,c ( a + dp n X ) = adp n + c − − {− a ( dp n ) − } c dp n + adp n = c − − {− a ( dp n ) − } c which is the assertion. (cid:3) Now the Dirichlet series expansion in Corollary 2.5 follows from Theorem 3.1and the integral formula (2).4.
Expansions for different regularization parameters
We look at the coefficients ǫ a,c,dp n for different parameters c and the resultingDirichlet series expansions. The following observation follows directly from thedefinition. Remark 4.1.
The sequence of values E ,c ( a + dp n X ) = ǫ a,c,dp n for a = 0,1, 2, . . . , dp n − c . The sequence begins with c − andcontinues with a permutation of c − , . . . , − c − . If we restrict to values of n suchthat dp n lies in a fixed congruence class modulo c , then the values do not changeas n → ∞ . ♦ The measure E ,c and the Dirichlet series expansion are particularly simple for c = 2. Note that we assumed that d and p are odd in this case. If a is even, then {− a ( dp n ) − } = 0 and E , ( a + dp n X ) = ǫ a, ,dp n = 12 . If a is odd, then − a ( dp n ) − is odd, {− a ( dp n ) − } = 1 and E , ( a + dp n X ) = ǫ a, ,dp n = − . Hence E , is up to the factor equal to the following simple measure: Definition 4.2.
Let p = 2 be a prime, and let X ∼ = Z /d Z × Z p be as above. Then µ ( a + dp n X ) = ( − { a } dpn defines a measure on X . We call µ the alternating measure , since the measure ofall clopen balls is ± ♦ The corresponding integral is also called the fermionic p -adic integral (see [4]).Now we obtain the following Dirichlet series expansion from Corollary 2.5. HEIKO KNOSPE, LAWRENCE C. WASHINGTON
Corollary 4.3.
Let p = 2 be a prime number, and let χ be an even Dirichletcharacter of odd conductor f . Then we have for s ∈ C p with | s | < p ( p − / ( p − , (1 − χ (2) h i − s ) L p ( s, χ ) = lim n →∞ fp n X a =1 p ∤ a ( − a +1 χω − ( a ) 1 h a i s . For χ = ω − i and odd i = 1 , . . . , p − , we obtain the branches of the p -adic zetafunction: ζ p,i ( s ) = L p ( s, ω − i ) = 11 − ω (2) − i h i − s · lim n →∞ p n X a =1 p ∤ a ( − a +1 ω ( a ) − i h a i s Remark 4.4.
Dirichlet series expansions of p -adic L -functions were studied by D.Delbourgo in [1] and [2]. He considers Dirichlet characters χ satisfying ( p, f φ ( f )) =1 and their Teichm¨uller twists. We obtain the same expansion for c = 2 and χ = ω − i . However, we require ( c, f p ) = 1 and use other methods for the proof.Similar expansions for a slightly different p -adic L -function using a fermionic p -adic integral (i.e., c = 2) were also obtained by M.-S. Kim and S. Hu (see [4]). ♦ Example 4.5.
We look at the case c = 3. The sequence of values ǫ a, ,dp n isperiodic with period 3. If dp n ≡ , − , , . . . . If dp n ≡ , , − , . . . . Corollary 4.6.
Let p be a prime number, and let χ be an even Dirichlet characterof conductor f = dp m such that (3 , dp ) = 1 . If d ≡ mod , then define a sequence ǫ = 1 , ǫ = − , ǫ = 0 , . . . with period . Otherwise, set ǫ = 1 , ǫ = 0 , ǫ = − and extend it with period . Then we have for s ∈ C p with | s | < qp − / ( p − , − (1 − χ (3) h i − s ) L p ( s, χ ) = lim n →∞ dp n X a =1 p ∤ a χω − ( a ) ǫ a h a i s . Example 4.7.
For c = 5, we get a periodic sequence with period 5 and we have ǫ a, ,dp n = 2 for a ≡ − −
1, 0 and 1, depending on the class of dp n mod 5. Example 4.8.
Let c = 7. Then ǫ , ,dp n = 3. Now suppose, for example, that dp n ≡ dp n ) − ≡ ǫ , ,dp n = 1 , ǫ , ,dp n = − , ǫ , ,dp n = − , ǫ , ,dp n = 2 , ǫ , ,dp n = 0 , ǫ , ,dp n = − , and these are extended with period 7. Acknowledgements.
The first author thanks Daniel Delbourgo for hints to hiswork and helpful conversations.
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