The p -adic L -function for half-integral weight modular forms

. The p -adic L -function for modular forms of integral weight is well-known. For certainweightsthe p -adic L -functionformodularformsofhalf-integralweightisalsoknown to exist, via a correspondence, established by Shimura, between them and forms of integral weight. However, we construct it here without any recourse to the Shimura correspondence, allowingustoestablishitsexistenceforallweights,includingthoseexemptfromtheShimuracorrespondence.WedothisbyemployingtheRankin–Selbergmethod,andprovingexplicit p -adic congruences in the resultant Rankin–Selberg expression.


Introduction
The centrality of general p-adic L-functions to the Iwasawa main conjectures almost goes without saying, as they form the backbone of the analytic side of the conjecture. Given the diversity of L-functions and their, established or not, p-adic analogues, natural attempts can be made to formulate different versions of this. In its first form, one considers the G L 1 -case in which the Kubota-Leopoldt L-function takes centre stage as the p-adic analogue of the Dirichlet L-function. Moving up to G L 2 , we can consider modular forms and their p-adic L-functions. Should the modular form have an integral weight then the conjectures in toto are well formulated and in some cases even established (e.g. k = 2 see [21]). However, due to insufficiently developed theory the conjectures for when the modular forms are of half-integral weight are not possible to even state.
The main issue in the case where f is a modular form of half-integral weight is the difficulty of developing a 'Galois side', which forms the second and last backbone of the Iwasawa main conjectures. Such difficulty can be seen in Section 11 in the informal notes of Buzzard in [1]. Recent work of Weissman in [23] has made some serious progress in this regard by developing L-groups for metaplectic covers, the length and methods of which further underline the complications here. Nevertheless, the analytic p-adic theory of half-integral weight modular forms has been substantially developed in the thesis of [11]. So then it's germane to ask whether we can at least construct the p-adic L-function in this case.
The p-adic L-function attached to a half-integral weight modular has been constructed before using an indirect method. The Shimura correspondence, established by Shimura himself in [15], goes between modular forms of half-integral weight k = κ 2 for odd κ ∈ Z and modular forms of integer weight κ − 1 which respects the action of the Hecke operators. For large enough κ this is a bijection, in which case the well-known interpolation of the L-function for modular forms of integer weight immediately yields that of the half-integral weight L-function. We go ahead anyway and provide in this paper a direct interpolation of the L-function for half-integer weights for reasons given in the rest of this introduction.
The method employed is the Rankin-Selberg method which has been a highly successful method for p-adic interpolation e.g. see [3,10,13]. In spite of all this the recording of the method in this setting is useful for a number of reasons. Primarily, an analogy can be drawn here with the construction of the p-adic L-functions for totally real fields, for which two constructions have proven to be equally useful. On the one hand Deligne and Ribet in [5] constructed this function using constant terms of Eisenstein series, and this particular construction is vital in the proof of the Iwasawa main conjecture by Wiles in [24]. Cassou-Noguès in [2], however, gave an entirely different construction of this p-adic L-function using the Shintani decomposition, this being apposite to our analogy in that it allowed Colmez to prove the p-adic residue formula, see [4]. In our situation here we note a potentially fruitful observation, in the very final section, regarding the integrality of the p-adic measure resulting from our particular construction when contrasted to the original one for integer weight Siegel modular forms. In addition, this paper gives some impetus to extend this to half-integral weight Siegel modular forms of higher degree n > 1 for which, crucially, there is no longer a Shimura correspondence. Such p-adic L-functions are currently not known to exist. Finally, in Sect. 3 we encounter a substantial deviation in this setting from the integer weight one, which involves the construction of the p-stabilisation of our modular form. This construction has not been done previously for half-integer weight forms. Section 2 gives a well-known overview of the basic theory of half-integer weight modular forms. Some necessary results on the trace map and Hecke operators are given in Sect. 4, which allow us to reduce the level of the inner product in the integral expression of our L-function. After reducing the level of the inner product, the existence of the p-adic measure is proven by looking at the Fourier development inside the inner product. The paper is concluded in the final section by a comparison of the integer and half-integer weight p-adic measures in accordance with the Shimura correspondence, and a discussion on the integrality of this p-adic measure.

Half-integral weight modular forms
Much of the theory we use here to exhibit half-integral weight modular forms is taken from [15]. First, some preliminary notation. If α ∈ M n is a matrix, then |α| = det(α). We denote by G L + 2 (R) the space of all invertible 2×2 matrices with positive determinant. The upper half-plane is H = {z ∈ C | Im(z) > 0}. The fractional linear transformation action of G L + 2 (R) on H is given by We let T denote the unit circle and let be a group whose law of multiplication is given by We also have a natural projection P : G → G L + 2 (R) and ker(P) ∼ = T. Let f : H → C be a function, then for some κ ∈ Z we can define a weight κ action of G on f by contains no elements of the form (1, t) for 1 = t ∈ T, and contains no elements of the form (−1, t) for 1 = t ∈ T should −1 ∈ P( ). Definition 2.1. Let κ be an odd integer and ≤ G 1 a Fuchsian subgroup. We say that a holomorphic function f : H → C is a modular form of weight k = κ 2 with respect to if (ii) f is holomorphic at all cusps of P( ).
The space of all such forms is denoted M k ( ).
Condition (ii), that f be holomorphic at cusps, is made precise in [15, p. 444]. Much like integral weight modular forms any f ∈ M k ( ) has a Fourier expansion of the form f = ∞ n=0 a n q n where q = e 2πiz . Then the subspace of cusp forms, denoted S k ( ), consists of all forms f the Fourier developments of which take the form f |[ξ ] κ = ∞ n=1 a n q n for all ξ ∈ G.
In this paper will always be obtained by the following types of congruence subgroup of SL 2 (Z), Therein lies the issue, the above condition of P being a bijection on double cosets is not always satisfied and causes some differences in the theory of Hecke operators, which means that we only really have operators T (m) when m is a square number. Generally for γ ∈ ∩ α −1 α we at least have for a homomorphism t : ∩ α −1 α → T. Then Proposition 1.0 in [15] says that f |[ ξ ] κ = 0 if f ∈ M k ( ) and t κ ≡ 1. Also found in [15, pp. 447-448] is the following proposition: So in particular if we take = 1 (N ) for some N divisible by 4, and a prime p N then the pth Hecke operator is given by [ ξ ] κ where ξ = 1 0 0 p , p  N )) to 0. Then T ( p) is of no interest here and we must generally consider T (m) for m a square number. In the case that m, n ∈ Z are square then T (m) and T (n) commute and one can also write the explicit actions of T ( p 2 ) on the Fourier coefficients of f , see Proposition 1.6 and Theorem 1.7 in [15].
So we have seen that T ( p) is simply the zero operator should p N , but the question remains of whether it is of interest when p | N . In this case, actually T ( p) is much like it is in the integral weight setting, it shifts the coefficients of a form. In [15, p. 448] we have the following: ) and put In particular, if p | N then the two conditions in the proposition are satisfied and f |T ( p) = ∞ n=0 a pn q n is a non-trivial action.

Complex L-functions and their integral expressions
To start with, we define the complex L-function attached to half-integral weight modular forms. With the given data of a normalised eigenform f ∈ S k ( 1 ), k = κ 2 , κ ∈ 2Z + 1, 1 = 1 (N ), and 4 | N ∈ Z we can associate for any prime p, as seen in [18, p. 46], a Satake p-parameter λ p ∈ C × . Subsequently, the local factors are given as if p N and, augmenting the given data with a character χ of some modulus, the actual L-function is with s ∈ C and which product is absolutely convergent should Re(s) > 3n 2 + 1. The SL 2 (Z)-invariant differential dμ := y −2 dxdy on H, along with the fundamental domain B( ) of \H for some congruence subgroup ≤ SL 2 (Z), are used in defining the Petersson inner product of two modular forms. If ψ is an even Dirichlet character modulo N , and if B(N ) := B( 0 (N )), then the Petersson inner which integral is convergent whenever one of f, g is a cusp form.
With the complex L-function now defined the aim is to give a Rankin-Selberg expression of said L-function in terms of integrals of the form in the definition of the above inner product. To this end, we first define what our integrands shall be. One role is given by the eigenform f which we henceforth assume is a newform belonging to S k (N ) := S k (N , 1) and has Fourier coefficients a n for 1 ≤ n ∈ Z. There exists some minimal square-free positive integer t at which we have a t = 0, and normalise the form so that a t = 1. Such an integer exists by (i), (ii) of Corollary 1.8 in [15]. Furthermore by the strong multiplicity one theorem we can take t such that p t, since otherwise a q = 0 for all q = p.
be the parabolic subgroup of SL 2 (Z), for any 1 ≤ M ∈ Z we put = 0 (M), and let η be a Dirichlet character modulo M. Then the (non-holomorphic) Eisenstein series we need is of integral weight ∈ Z, level M, character η −1 , and is expressed as the following sum in the two variables z ∈ H, s ∈ C: Taking ν ∈ {0, 1} such that η(−1) = (−1) k +ν , which character has conductor c η , then we define the theta series by The p-adic L-function for half-integral weight modular forms This is of weight ν + 1 2 , character ηρ t (where ρ t is the quadratic character associated to Q(i 1 2 (2t) 1 2 )), and level 4tc 2 η , see Proposition 2.1 of [19]. The other role of the inner product is then played by a theta series multiplied by an Eisenstein series of the above kinds. Now let χ be a Dirichlet character of p-power conductor c χ = p m χ , and also put N χ := N tc 2 χ . Since c χ | N χ we can (and in the following Eisenstein series do) view χ as a character of modulus N χ in the natural way. Putting n = 1, F = Q, and decoding the notation of [19, (4.1)] yields the following integral expression: Let C p := Q p denote the completion of an algebraic closure of the p-adic numbers, and extend the p-adic norm to this field. Let ι p : Q p → C p be a fixed embedding and we henceforth work under ι p without explicitly denoting it.
We shall always assume that p = 2, that p N , and furthermore that f is p-ordinary-by which we mean that the eigenvalue of f at T ( p 2 ) is a unit at p. The Hecke operators T ( p 2 ) in [18] lack the normalising factor of ( p 2 ) are the eigenvalues in the sense of [18]. By definition of the λ p in (5.4a) of [18] we have and so as ω p is a unit at p we may assume that α p := p k−1 λ p is a unit at p. Put β p := p k−1 λ −1 p , then ω p , α p , and β p satisfy Let N 1 := N p 2 and let [T ( p; N 1 )] κ denote the (now non-zero) pth Hecke operator of level N 1 , which just shifts the coefficients of a form along. When acting on forms of level N 1 the operator [T ( p; N 1 )] κ = [T ( p)] κ is just the usual pth Hecke operator, but the notation is used to emphasise the fact that this is the operator that shifts coefficients even on forms of level N -for example f -and is not equal , and upon viewing f as a form of level N 1 Proposition 2.3 gives that f |[T ( p; N 1 )] κ = n≥1 a np q n . If η is any Dirichlet character of modulus M then we define the twist of f by η to be η(n)a n q n which, by easy generalisation of Proposition 17 (b) in [9, pp. 127-128] to halfintegral forms, is in S k (N M 2 , η 2 ). Set χ p (n) = n p which has modulus p and satisfies χ 2 , and we work mostly with f 1 whose primary benefit over f is that p now divides the level.
Proof. To prove this we note that It is easy to see (a)-(b) by virtue of Proposition 2.3. To prove (c) we make use of Corollary 1.8 in [15] points (i) and (ii), which gives for If p 2 | n then write n = p 2m t for p 2 t and (ii) then gives a np 2 = ω p a n − p 2k−2 a n p 2 . We get n p a n q n + p 2 |n ω p a n − p 2k−2 a n p 2 noting that n p = 0 if p 2 | n anyway. Now that we have proven (a)-(c) then the lemma follows since using ω p = α p + β p and p 2k−2 = α p β p . Let q = p, then to see that the eigenvalue ω q,1 of f 1 is equal to ω q of f , we need the fact that for any ( , q) = 1. This can be seen easily by considering the coset decomposition, found in [15, p. 451], of To show that f χ p also has the same eigenvalue as f we make use of Theorem 1.7 from [15] then these are given as b n = a nq 2 + −1 q k q k −1 n q a n + q 2k−2 a n q 2 where we understand a n q 2 = 0 if q 2 n. Using this same construction for f χ p we get the nth coefficient of Using the definitions of the L-function and f 1 we get the following relations: To see why this is true we really need to see how the numbers λ p are obtained in [18, p. 46]. For any prime q the number λ q satisfies where we recall that ω q = q k−2 λ(q), and let ω q,1 , λ 1 (q), λ q,1 denote the corresponding numbers for f 1 . Then our job is to compare λ q with λ q,1 . Suppose that q = p, then f |[T (q 2m )] κ = f 1 |[T (q 2m )] κ by Lemma 3.1 and so λ q = λ q,1 in this case. Assume now that q = p and for ease of notation label δ p := pλ p and γ p := pλ −1 p . We claim that and we use induction to see why this is true. First multiply both sides of (3.2) by the denominator and compare coefficients of t m for m ≥ 2 to get Using λ( p) = δ p + γ p , p 2 = δ p γ p , and the induction hypthesis, this gives From the now established identity of (3.3) above, we have 2) in the last line. Hence λ p,1 = λ p . In the Euler factors this becomes Notice that the tth coefficient of f 1 is given by Since α p is a p-adic unit and α p = β −1 p p 2k−2 , we cannot have β −1 So the tth coefficient of f 1 is also non-zero, and we normalise f 1 by this value.
Proof. Just re-use [ [19], (4.1)] as we did before, noting that the only thing changing by replacing f with f 1 is the level by a factor of p, and since p | c χ this has no real bearing on (4.1) in [19] as for example, N χ = lcm(N , 4c 2 χ ) = lcm(N 1 , 4c 2 χ ).
Our aim in this section is twofold. As p | N χ , the pth Hecke operator [T ( p)] κ of level N χ is non-zero on M k ( 1 (N χ )). We first wish to express S χ in terms of T ( p), and secondly to find the adjoint of T ( p) of level N 1 . This allows us to reduce the inner product in (3.4) to be over N 1 instead of N χ .
to prove the proposition we can now work with multiplication in G. We claim that in G we have Multiplication of the matrices is easy, using N 1 = N χ p 2−2m χ . So to prove (4.1) we only need show multiplication on the part of the functions. Using the rule (2.1) on the left-hand side of (4.1), as well as Eq. (2.2), nets us and on the right-hand side of (4.1) the functions multiply to give precisely the same So the claim is true and using it, and the fact that [( p 2−2m χ I 2 , 1)] κ is the identity operator, we have By routine calculation on the Fourier expansion it is easy to see that the operator p This concludes the first of our aims, and we now calculate the adjoint of [T ( p)] κ of level N 1 . ξ = (α, ϕ), and ξ 0 = (α 0 , ϕ 0 ) ∈ G whose functions satisfy the definition of G with respective unit circular elements t and t 0 , and whose matrices are α = a b c d and 0) ) with ϕ (0) satisfying the definition of G with unit circular elements t (0) , then t = t and t 0 = t 0 for all . For any f ∈ S k ( ) and g ∈ M k ( ) we have

Proof. (a) Expanding out the action in the product on the left-hand side gives
making the change of variables z → α 0 z, noting that f (αα 0 z) = f (z), and that B(α −1 α) maps to B( ).
(b) As we are just taking = * the projection P is a bijection between ξ and α , and so we have α = α . Let be arbitrary, then for any γ 3 ∈ there exist γ 1 , γ 2 ∈ such that and from the first we have |α | = |α|. By the law of multiplication in G the second gives j (γ 1 , ξγ 2 z) ϕ(γ 2 z) j (γ 2 , z) = j (γ 3 , ξ z) ϕ(z). Squaring both sides and using the known cocycle relation on j gives and that t = t follows because both |α| = |α | and γ 1 ξγ 2 = γ 3 ξ . Now that we know t = t and t 0 = t 0 for all the result follows using part (a) on the decompositions of both ξ and ξ 0 .
Proof. First we claim that W (N 1 ) normalises 1 (N 1 ). On the part of matrices this is the well-known and easy matrix multiplication β −1 Of particular use will be To finish we use Lemma 4.3. In the notation of that Lemma we have α 0 = p 0 0 1 and ϕ 0 (z) 2 = p − 1 2 , where we chose t 0 = 1. By (b) in Lemma 4.3 the adjoint is then W (N 1 ), and the matrix multiplication is easy. To show that the functions match up we show W (N 1 )ξ 0 = ξ W (N 0 ), and the law of G-multiplication gives p 1 4 N 1 (−i z) 1 2 immediately from the right-hand side, whereas the left-hand side gives

Fourier expansion of Eisenstein series
This section involves material all of which has long been well-known, however we include it here anyway for two reasons. The first is for further clarity in the calculation of Fourier coefficients in the next section, but secondly that it gives a nice motivation, outside of algebraicity, for the choice of special values to be interpolated. We calculate the explicit Fourier expansion of the non-holomorphic Fricke-involuted Eisenstein series of an integral weight ∈ Z, level 1 ≤ M ∈ Z, and character χ −1 . For the variables z ∈ H and s ∈ C the Eisenstein series is defined as in Sect. 2. We can choose a set of representatives for (P ∩ )\ as follows for certain values of s that we specify later. We replicate the calculation found in [16], which there is done for half-integral weight Eisenstein series. Using that Throwing the z − from the action of W (M) into the sum we get so it just remains to figure out the Fourier expansion of E (z, s). Writing b = d j +m where j ∈ Z and 1 ≤ m ≤ d, this then takes the form which is amenable to the following lemma: The p-adic L-function for half-integral weight modular forms Lemma 5.1. [16, p. 84] If α, β ∈ C with Re(α), Re(β) > 0 and Re(α + β) > 1 then for and if Re(β) > 0 we have which we can continue analytically to the whole β-plane as in [16].
Define a divisor sum function by σ ,χ (n) = d|n χ(n/d)d . Applying the above lemma with α = + s Explicit expressions of the τ n for certain values of s are now deduced. Suppose that s = m where m is a negative even integer, and s > − + 1. We claim that in this situation we have τ n (y, + s 2 , s 2 ) = 0 if n ≤ 0 and is non-zero for n > 0. The easiest of these is if n = 0 in which case ( + s − 1)(4π y) 1−s− and since s 2 is a negative integer, there is a pole at ( s 2 ) which won't be cancelled in the numerator as s > − + 1. So τ 0 (y, + s 2 , s 2 ) = 0. Suppose that n < 0, then e −2π |n|y σ (4π |n|y, s 2 , + s 2 ).
Now + s 2 > 0 is a positive integer, so that σ is here defined by the integral in Lemma 5.1 and is finite. The pole in the denominator at ( s 2 ) is then still not cancelled out and we again obtain τ n (y, + s 2 , s 2 ) = 0. The difficulty is in showing that τ n is non-zero for n > 0, and then actually finding its explicit expression. We have and since s 2 < 0 we need to make use of the analytic continuation of σ to proceed, which essentially cancels out the pole in this case. This analytic continuation is given in [16, p. 83] as where we are integrating over the key-hole contour going to +∞ on the real axis and positively oriented about the origin. More specifically, recalling that s 2 is a negative integer and using the binomial theorem we have The Gamma function has a well-known meromorphic continuation, which is given by (e 2πis − 1) (s) = (0+) ∞ t s−1 e −t dt, and which gives The p-adic L-function for half-integral weight modular forms By the choice of s we have − s 2 < + s 2 − 1, which gives . Now we obtain the the following Fourier expansion for E (z, s): Recalling that E * (z, s) = i M − s+ 2 y s 2 E (z, s) then we obtain our final Fourier development of this, whenever 0 ≥ s > − + 1 is an even integer, to be A function f : H → C is said to be nearly holomorphic of weight k = κ 2 for even or odd κ, and of level , if it satisfies f |[ξ ] κ = f for all ξ ∈ and if it has a Fourier expansion of the form f = r j=0 (π y) − j ∞ n=0 a n q n where r ∈ Z. Denote this space by N k ( ) and we can analogously define the spaces N k (N , ψ) if 4 | N ∈ Z and ψ is a character modulo N . By our calculation above we have E * (z, s) ∈ N (M, χ −1 ) for any even 0 ≥ s > − + 1.
We define the Shimura-Maass differential operators for any λ ∈ R and for any 0 ≤ a ∈ Z as in [17, p. 812] to be where the product is with respect to composition, and we then have δ a k N k ⊆ N k+2a .
Lemma 5.2. [17, p. 813] Any g ∈ N k can be written uniquely as where g μ ∈ M k−2μ for some r ≤ k 2 , and g 0 is known as the holomorphic projection of g. Moreover, if g ∈ N k (M, ψ) for an integer 4 | M, then f, g = f, g 0 for any f ∈ S k (M, ψ).

Interpolation
We are now in a position to construct our p-adic L-function, and this is achieved by constructing a p-adic measure. Here we are actually constructing two families of measures, one for each ν ∈ {0, 1}.
Let m ∈ 1 2 Z\Z be any half-integer satisfying 0 ≥ m − k > −k + 3 2 and that m − k + ν ∈ 2Z. If p N and f ∈ M k (N ) is a p-ordinary normalised eigenform, there exist unique measures μ (ν) f,m on Z × p such that if χ is a Dirichlet character whose conductor is c χ = p m χ for 1 ≤ m χ ∈ Z, then for the trivial character it gives and the constant D (ν) = D (ν) (k, m, N ) will be given later.
To prove the above we first take our integral expression from Sect. 3 and manipulate the inner product using the results of Sect. 4, reducing it from level N χ to N 1 . Removing the dependence of the inner product on χ allows us to bring sums over Dirichlet characters inside the inner product, in particular to the second argument. The main theorem is then proved by giving the precise Fourier development of this sum given in Sect. 5, noting it is p-integral and rational, and then using finite dimensionality of such forms to deduce boundedness of the measure. Proposition 6.2. Let χ have conductor c χ = p m χ where 1 ≤ m χ ∈ Z. If m χ ≤ r ∈ Z, then we have For some character η we modify the divisor sum function σ at p to give the p-modified power divisor sum In Chapter 7 of [7] this is p-adically interpolated, a fact that will essentially yield our interpolation. Lemma 6.4. For any integer , any Dirichlet character η, and positive integer n we have In particular, if the modulus of η is divisible by p then σ ,η ( p 2 n) = p 2 σ ,η (n).
Before going on to prove this theorem with a series of lemmas, first note that by the previous section we have H (ν) χ is nearly holomorphic and so can be written in the form n, j (χ )q n for some r . Lemma 6.7. There exists a constant 0 < ∈ Z and linear forms F n (X 0 , . . . , X r ), which belong to Z[X 0 , . . . , X r ] and are dependent only on and n, such that n,0 (χ ), . . . , c (ν) n,r (χ ))q n and, crucially, F n (X 0 , . . . , X r ) ≡ X 0 (mod n).
Proof. We make use of Lemma 5.2 and the following identity This identity is easily checked using induction and the binomial theorem, as in the integral weight case, the major differences with the identity appearing in [13, p. 211] here occuring since factorials are not even defined for half-integers. Nevertheless with this identity the rest of the proof in [13] still follows through nicely.
We now give the values of c ∈ Z for i = 1, 3 since m − k + ν is even, so that everything is well-defined, and the values are integers.
As for the theta series, it is known from [15, p. 457] that where N = N 4 . Let W n := {(n 1 , n 2 ) ∈ N 2 | tn 2 1 N + n 2 = n, p n 1 }. Then by multiplying the above expressions together along with the factor of ( p m χ −1 ) 2−k occuring in H (ν) χ we easily see that we have the following lemma. general guidance and particularly for staying patient throughout this paper's seemingly never ending development of problems, and to the anonymous referee for their careful reading and suggestions. Funding was provided by Engineering and Physical Sciences Research Council (Grant No. 000118421).
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