p -adic L -functions on metaplectic groups

With respect to the analytic-algebraic dichotomy, the theory of Siegel modular forms of half- integral weight is lopsided; the analytic theory is strong, whereas the algebraic lags behind. In this paper, we capitalise on this to establish the fundamental object needed for the analytic side of the Iwasawa main conjecture — the p -adic L -function obtained by interpolating the complex L -function at special values. This is achieved through the Rankin–Selberg method and the explicit Fourier expansion of non-holomorphic Siegel Eisenstein series. The construction of the p -stabilisation in this setting is also of independent interest.


Introduction
Traditionally, p-adic L-functions have dual constructions -analytic and algebraic -and it is the substance of the Iwasawa main conjecture that these two are equivalent. This conjecture can be formulated for various settings; for example, over GL 1 , the conjecture asserts that the analytic construction -Kubota-Leopoldt's p-adic interpolation of the Dirichlet L-function -is equivalent to Iwasawa's algebraic p-adic L-function. The Iwasawa main conjecture for classical modular forms of integral weight is formulated over GL 2 and this has been a recent, active research area with its connections to the Birch and Swinnerton-Dyer conjecture, see [8,9]. Provided one has both the analytic and algebraic machinery, the Iwasawa main conjecture can be formulated for higher dimensional modular forms and groups, for example, [17]. The algebraic theory of half-integral weight modular forms, both classical and metaplectic, has long been inchoate due to the difficulties present in developing the 'Galois side'. Recent work by Weissmann in [18] has made progress in this regard by developing L-groups for metaplectic covers, the length and methods of which further underline the difficulties present here. The analytic theory is substantial however, and in this paper we give the analytic construction of the p-adic L-function for Siegel modular forms of half-integral weight and any degree n. In [5], we gave a similar construction when n = 1; in that case, the p-adic L-function was already known to exist by the Shimura correspondence, this is not so for general n > 1.
The proof found here is adapted from the method of Panchishkin found in [7, Chapters 2 and 3], which proves the existence of the analytic p-adic L-function for Siegel modular forms of integral weight and even degree n. This method makes critical use of the Rankin-Selberg method and reduces the question of p-adic boundedness of the L-function down to that of the Fourier coefficients of the Eisenstein series that are involved in the Rankin-Selberg integral expression. For full generality, it is assumed that p does not divide the level of the modular form f and a crucial step is to produce another form f 0 such that p does divide the level. Significant modifications to the method of [7] were required to make this work in the metaplectic casethis is Section 4. Outside of this, the success of Panchishkin's method is facilitated by the work of Shimura in developing the Rankin-Selberg integral expression in this setting, [15], and the arithmeticity of Eisenstein series, [16,. Interestingly, the p-adic boundedness of the Eisenstein series coefficients is almost immediate in this case, making the final step of the proof simpler than that found in [7].
After preliminary Sections 2 and 3, we establish the p-stabilisation in Section 4. Fairly elementary manipulations on the level of the Rankin-Selberg integral follow in Section 5. Sections 6 and 7 are devoted to transformation formulae of theta series and Fourier expansions of Eisenstein series -these are relatively well known. Finally the statement and proof of the main theorem, and the subsequent existence of the p-adic L-function, are given in Section 8.

Siegel modular forms
This section runs through the very basics of the modular forms that we study and their Fourier expansions are detailed.
For any ring R and any matrix a ∈ M n (R), note the use of the following notation: a > 0 (a 0) to mean that a is positive definite (respectively, positive semi-definite), |a| := det(a), a := | det(a)|, andã := (a T ) −1 . For any collection a 1 , . . . , a r of matrices with entries in R, let diag[a 1 , . . . , a r ] be the matrix whose jth diagonal block is a j and is zero off the diagonal.
Let A Q and I Q denote the adele ring and idele group, respectively, of Q. The Archimedean place is denoted by ∞ and the non-Archimedean places by f . If G is an algebraic group, let G A denote its adelisation. Let G ∞ := G(R), G p := G(Q p ), and denote by G f the subgroup of elements of G A whose Archimedean place is the identity of G ∞ . View G as a subgroup of G A by embedding diagonally at every place, but view G ∞ and G p as subgroups by embedding place-wise. Recall the adelic norm where x ∈ A Q , | · | denotes the usual absolute value on R, and | · | p denotes the p-adic absolute value, normalised in the sense that |p| p = p −1 . Let T denote the unit circle and define three T-valued characters on C, Q p , and A Q , respectively, by where {x} denotes the fractional part of x ∈ Q p ; if x ∈ A Q and z ∈ C, then write e ∞ (x) = e(x ∞ ) and e ∞ (z) = e(z).
For any fractional ideal r of Q, let r p denote the completion (with respect to the p-adic norm) of the localisation of r at the prime p, which is an ideal of Z p . Understand 0 N (r) ∈ Q to be the unique positive generator of r.
Write any α ∈ GL 2n (Q) as Define an algebraic group G, subgroup P G, and the Siegel upper half-space H n by A half-integral weight is an element k ∈ Q such that k − 1 2 ∈ Z; an integral weight is an element ∈ Z. The factor of automorphy of half-integral weight involves taking a square root; to guarantee consistency of the choice of root, one uses the double metaplectic cover Mp n of Sp n . The localisations M p := Mp n (Q p ) and the adelisation M A of Mp n (Q) can be described as groups of unitary transformations, respectively, on L 2 (Q n p ) and L 2 (A Q n ) with the exact sequences There are natural projections pr A : M A → G A and pr p : M p → G p , either of which will usually be denoted pr as the context is clear. On the flip side, there are natural lifts r : G → M A and r P : P A → M A through which we view G and P A as subgroups of M A .
For any two fractional ideals x, y of Q such that xy ⊆ Z, congruence subgroups are defined by the following respective subgroups of G p , G A , and G: Typically these will take the form Γ[b −1 , bc] for certain fractional ideals b and integral ideals c.
One of the key differences in the theory of half-integral weight modular forms is in the congruence subgroups one considers. The factor of automorphy involved can only be defined for a certain subgroup M M A , and any congruence subgroups Γ must therefore be contained in M. This subgroup, M, is defined via the theta series and is given by Typically we shall take b and c such that The spaces defined above interact with each other as follows. The action of Sp n (R) on H n and the traditional factor of automorphy are given by where γ ∈ Sp n (R) and z ∈ H n . If α ∈ G A , then we extend the above by α · z = α ∞ · z and For any σ ∈ M, we can define a holomorphic function h σ = h(σ, ·) : H n → C satisfying the following properties 3) The proofs for the above three properties can be found in [11, pp. 294-295].
If k is a half-integral weight, then put [k] := k − 1 2 ∈ Z; if is an integral weight, then put [ ] := . The factors of automorphy of half-integral weights k and integral weights are given as where σ ∈ M, α ∈ G A , and z ∈ H n . Given a function f : H n → C and an element ξ ∈ G A or M, the slash operator of an integral or half-integral weight κ ∈ 1 2 Z is defined by Definition 1. Let κ ∈ 1 2 Z be an integral or half-integral weight, and let Γ G be a congruence subgroup with the assumption that Γ M if κ / ∈ Z. Denote by C ∞ κ (Γ) the complex vector space of C ∞ functions f : H n → C such that f || κ α = f for any α ∈ Γ. Let M κ (Γ) ⊆ C ∞ κ (Γ) denote the subspace of holomorphic functions (with the additional cusp holomorphy condition if n = 1). Elements of M κ (Γ) are called modular forms of weight κ, level Γ; if κ / ∈ Z they are also known as metaplectic modular forms.
Elements of C ∞ κ (Γ) and M κ (Γ) have Fourier expansions summing over positive semi-definite symmetric matrices, the precise forms for which are given later in this section. The subspace S κ (Γ) ⊆ M κ (Γ) is characterised by all forms f such that the Fourier expansion of f || κ σ sums over positive definite symmetric matrices, for any σ ∈ G A if κ ∈ Z, or for any where X ∈ {M, S} and the union is taken over all congruence subgroups of G (that are Take a fractional ideal b and integral ideal c and put Γ = Γ[b −1 , bc]; when κ / ∈ Z, always make the crucial assumptions that b −1 ⊆ 2Z, (2.4) bc ⊆ 2Z, (2.5) then we have Γ M in this case. By a Hecke character of Q, we mean a continuous homomorphism ψ : I Q /Q × → T. Denote the restrictions to R × , Q × p , and Q × f by ψ ∞ , ψ p , and ψ f , respectively. We have that ψ ∞ (x) = sgn(x ∞ ) t |x ∞ | iν for t ∈ Z and ν ∈ R and we say that ψ is normalised if ν = 0. For any integral ideal a, let ψ a = p|a ψ p . Now take a normalised Hecke character ψ of Q such that Modular forms of character ψ are then defined by To give the precise Fourier expansions of these forms, define the following spaces of symmetric matrices: for any fractional ideal r of Q. Take a congruence subgroup , a modular form f ∈ M κ (Γ, ψ), and matrices q ∈ GL n (A Q ), s ∈ S A . The Fourier expansion of f A is given as for some c f (τ, q) = c(τ, q; f ) ∈ C satisfying the following properties (2.10) The proof of the above expansion and properties can be found in [14,Proposition 1.1]. The coefficients c f (τ, 1) are the traditional Fourier coefficients of f in the following sense. By property (2.8), the modular form f ∈ M κ (Γ, ψ) has Fourier expansion , then it has Fourier expansion of the form where z = x + iy and the coefficients c F (τ, y) are smooth functions of y having values in C.
We finish this section with some final key definitions. Consider b fixed in the definitions of Γ = Γ[b −1 , bc], so that this group depends only on c, and let ψ be a normalised Hecke character satisfying (2.6) and (2.7). For any two f, g ∈ C ∞ κ (Γ, ψ), the Petersson inner product is defined by This integral is convergent whenever one of f, g belongs to S κ (Γ, ψ).

Complex L-function
The standard complex L-function associated to eigenforms is defined in this section, and the known Rankin-Selberg integral expression is stated. As in the previous section, take ideals b and c satisfying (2.4) and (2.5), and set For any Hecke character χ of Q, let χ * (p) = χ * (pZ) denote the associated ideal character. Although the integral expression can be stated for any half-integral weight k, we take k n + 1 to ease up on notation -we shall be making this assumption later on anyway. For a prime p, the association of the Satake p-parameters -an n-tuple (λ p,1 , . . . , λ p,n ) ∈ C n -to a non-zero Hecke eigenform f ∈ S k (Γ, ψ) is well known (see, for example, [14, p. 46]). Now set a Hecke character χ of conductor f. The standard L-function of f , twisted by χ, is then defined by The Rankin-Selberg integral expression, (4.1) in [15, p. 342], is given there in generality; we restate it now for our purposes. Fix τ ∈ N (b)S + such that c f (τ, 1) = 0 and let ρ τ be the quadratic character associated to the extension The key ingredients of the integral are three modular forms: the eigenform f , a theta series θ χ , and a normalised non-holomorphic Eisenstein series E(z, s). To define the theta series, take any 0 < τ ∈ S and define an integral ideal t by the relation h T (2τ ) −1 h ∈ 4t −1 for all h ∈ Z n . Take μ ∈ {0, 1} and a Hecke character χ such that χ ∞ (x) n = sgn(x ∞ ) nμ . The theta series is then the sum where we understand (χ ∞ χ * )(0) = 1 if f = Z and as zero otherwise. This has weight n 2 + μ, level Γ[2, 2tf 2 ] determined by [15, Proposition 2.1], character ρ τ χ −1 , and coefficients in Q(χ).
The Eisenstein series of weight κ ∈ 1 2 Z is now defined in a little more generality. Let Γ = Γ[x −1 , xy] be a congruence subgroup, contained in M if κ / ∈ Z, and let ϕ be a Hecke character satisfying (2.6) with y in place of c, and also such that ϕ ∞ (x) = sgn(x ∞ ) [κ] (note that this is a more stringent condition than the usual (2.7)). The Eisenstein series is defined by where recall Δ(z) = det(Im(z)), and we have z ∈ H n , s ∈ C. This sum is convergent for Re(s) > n+1 2 and can be continued meromorphically to all of s ∈ C by a functional equation with respect to s → n+1 2 − s. This series belongs to C ∞ κ (Γ, ϕ −1 ) and is normalised by a product of Dirichlet L-functions as follows. Let a be any integral ideal and define The normalised Eisenstein series is given by E(z, s; κ, ϕ, Γ ) := Λ n,κ y (s,φ)E(z,s; κ, ϕ, Γ ). Set η := ψχρ τ . In this setting, the integral expression of [15, (4

p-stabilisation
Fixing a prime p, the initial key ingredient in our construction of the p-adic L-function is the replacement of an eigenform f with its so-called p-stabilisation f 0 . The form f 0 is also an eigenform away from p, whose eigenvalues there coincide with f , however, it has the key property that p divides the level of f 0 and is an eigenform for the operator U p -the Atkin-Lehner operator that shifts Fourier coefficients. Thus, the L-functions of f and f 0 are easily relatable and so for full generality we can begin with an eigenform f , assume that p c does not divide the level, and then pass to f 0 . In [5], we constructed f 0 explicitly in the case n = 1 which was possible through explicit formulae on the action of the Hecke operators involved on the Fourier coefficients. For general n, we modify the method of [7], which involves abstract Hecke rings, the Satake isomorphism, and certain Hecke polynomials; at the end of this section, however, we show how all this abstract Hecke yoga reduces to the explicit form found in [5], when n = 1.
Let k be a half-integral weight, (b −1 , bc) ⊆ 2Z × 2Z be ideals, and ψ be a Hecke character satisfying (2.6) and (2.7) If (Δ, Ξ) is a Hecke pair, in the sense of [1, pp. 77-78], then the abstract Hecke ring R(Δ, Ξ) denotes the ring of formal finite sums ξ c ξ ΔξΔ, where c ξ ∈ C and ξ ∈ Ξ. Each double coset has a finite decomposition into single right cosets, and the law of multiplication is given in [1, pp. 78-79]. Consider the Hecke ring R(V, W ) defined in [14, p. 39] and let R denote the factor ring of R(V, W ) defined in [14, p. 41] or analogously to (4.1) below -this is the adelic Hecke ring which acts on forms in M k (Γ, ψ), and it is factored in order to give the Satake isomorphism. We need the use of a slightly different Hecke ring and we define this more explicitly. Let D 0 := D ∩ P A and Γ 0 := Γ ∩ P ; define Now define the Hecke ring S := R(V 0 , W 0 ), which differs from R(V, W ) of [14] in allowing denominators of p into the matrices r defining Y 0 (contrast with the definition of Z 0 ), and is therefore analogous to the Hecke ring also has a well-defined action on f ∈ M k (Γ, ψ) and f A . The action of the double coset , for example, is given by first decomposing into single cosets where α ∈ G ∩ D diag[q, q]D and then summing over the actions of α on f by the slash operator involving an extended factor of automorphy J k (α, z) -see [14, Sections 2, 3, and 4] for the details here.
. Then the local rings R p and S 0p are the spaces generated by A q and A r , respectively, where now q ∈ X p and r ∈ GL n (Q p ∩ Q).
Assume p c. Let W n be the Weyl group of transformations generated by the transformations pp. 41-42] through the composition of two maps  Lemma 2.4] for the precise definition and characterisation of J(α)). By [14,Lemma 4.3], the map Φ p is injective.
The map ω 0p . Note that any coset O p d with d ∈ GL n (Q p ) contains an upper triangular matrix of the form ⎛ with a di ∈ Z, and then define Through the decomposition O p xO p = d O p d and C-linearity, we extend this to obtain ω 0p . By multiplying out elements of diag[r, r](D 0 ) p , for r ∈ GL n (Q p ∩ Q), we see that (D 0 ) p diag[r, r](D 0 ) p also has a single coset decomposition of the form (4.2). Thus, we can analogously define Φ p : S 0p → GL n (O p , GL n (Q p )) and . The map Φ p , and therefore ω p , is no longer necessarily injective. There is a local embedding ε 0p : R p → S 0p and we have ω p = ω p • ε 0p . There exists U p ∈ S 0p -called the Frobenius element -defined by If n = 1, it is well known that U p corresponds to the pth Hecke operator when p | c; for general, n > 1, this is no longer true. Note that ω p (U p ) = p n(n+1) 2 x 1 · · · x n . Let C := {A ∈ S 0p | U p A = AU p } denote the centraliser of U p in S 0p . The map Φ p is injective when restricted to C by the following argument.
Proposition 4.1. Any A ∈ C is a linear combination of double cosets Proof. This is essentially the second statement of [1, Proposition 2.1.1] with δ = 0 (in the notation of Andrianov). To prove it, define U − p := Γ 0 (  Proof. By the previous proposition, if A r ∈ C, then r ∈ M n (Z p ) ∩ GL n (Q p ). We therefore have the decomposition x δ1 1 · · · x δn n z), It has an immediate decomposition of the form  Proof. Denote the sum on the left-hand side by Y , this belongs to S 0p . It is easy to check that R n (z) = (p n(n+1) 2 z) 2 n R n ((p n(n+1) z) −1 ) so, immediately from (4.3), we have For any 0 m 2 n , define Proposition 4.4. The Hecke polynomial R(z) can be factorised as Proof. By definition V 0 = 1 and by Lemma 4.3, V 2 n −1 U p = −T 2 n . For the rest, 1 m 2 n − 2, we have Expanding the right-hand side of (4.5) therefore gives the factorisation (4.3), which concludes the proof.
Definition 2. Let f ∈ M k (Γ, ψ) be a non-zero Hecke eigenform with Satake p-parameters (λ p,1 , . . . , λ p,n ), assuming p c. Set Then the p-stabilisation of f is defined by The action of α ∈ C on f is considered the scalar one, that is, f |α = αf . The second property is then given by the calculation
For q = p, the qth Hecke operator commutes with V m,p . Therefore, f 0 and f share the same eigenvalues away from p, and we then have the following corollary. In [5] we showed, if n = 1, that the p-stabilisation of f takes the form where, for any Dirichlet character ϕ of conductor F , denotes the twist of f by ϕ. This satisfies , ψ) and this matches the first part of Proposition 4.5. By definition, we have V 1,p = U p − T 1 = U p − T p in this case, so the abstract definition of f 0 in Definition 2, when we set n = 1, becomes denotes the eigenvalue of f under T p . By [5, Lemma 3.1(c)], this is precisely the form of (4.7) above.
Non-vanishing of f 0 . It is not clear from the above method that f 0 = 0 if f = 0. That f 0 may vanish is entirely possible, as is remarked in [7, p. 50].
Suppose that Λ : R → C is a homomorphism defining the eigenvalues of f , that is for all 1 m 2 n , we have f |T m = Λ(T m )f . By the definition in (4.6) and of V m,p , we get Assume that f = 0, so that we can take τ ∈ S + such that c f (τ, 1) = 0. Using the fact that The above formula may be used as a method of checking, computationally, whether one has c f0 (τ, 1) = 0 as well. Given the formula in (4.8) above, it seems unlikely that c f0 (τ, 1) should vanish for all τ outside of a few special cases. As an example, consider the n = 1 case and assume that c f (τ, 1) = 0 for some 0 < τ ∈ Z such that p 2 τ . By (4.7), the coefficient c f0 (τ, 1) = 0 only if pλ p, This becomes less trivial a situation if c f (τ, 1) = 0 only for p 2 | τ . As things become significantly more complex for general n, we acknowledge that this does not constitute a particularly strong argument, but it is hopefully enough to convince the reader that there should exist eigenforms f = 0 for which f 0 = 0 as well. In [2, Section 9], Böcherer and Schmidt give an alternative construction for the p-stabilisation of a Siegel modular form of integral weight, which does guarantee that f 0 = 0. Although this is perhaps stronger than our construction, one still needs to make an assumption that such a non-zero f 0 should exist and this is incorporated into Böcherer-Schmidt's definition of p-regular [2, p. 1431]. Their construction takes two Andrianov-type identities of Dirichlet series for f and f 0 and uses them to compare their Satake parameters directly. It has a fairly simple generalisation to the present setting by using the identity of [14,Corollary 5.2]. Indeed this identity becomes almost exactly the same as that of [2, Proposition 9.1] by putting [|x|Z] = Y ordp(|x|) and [v] = Y in the notations found in [14], as well as in the definition of D(τ, p; f ) in [14,Theorem 5.1]. All that remains is to manipulate the lattice sum, the far righthand component of [14,Corollary 5.2], and express it as a sum of the U (π i ) Hecke operators (defined as the double coset Γ 0 diag[π i , π i ]Γ 0 and π i = diag[pI i , I n−i ]). This was done for the Hermitian modular forms in [3,Section 7], but remains the same for our case.

Tracing the Rankin-Selberg integral
Given the relationship, established in Corollary 4.6, between L(s, f, χ) and (s, f 0 , χ), the focus can be shifted to the latter. The level, y, of the Rankin-Selberg integral (3.2) will depend on χ, which dependence we naturally seek to avoid. This is achieved in this section by making crucial use of the behaviour of f 0 under U p .
Fix 0 < τ ∈ S + such that c f0 (τ, 1) = 0. Recall t as an integral ideal such that h T (2τ ) −1 h ∈ 4t −1 and defineτ := N (t)(2τ ) −1 ∈ M n (Z). This section involves many levels and liftings of modular forms through these levels, so first we define and clarify these schematically. Fix b and note by (2.8) that b −1 | t, so we can think of f 0 as a form of level Γ[b −2 , b 2 tc 0 ] and put The ideal y χ can be taken as the level of the integral in the Rankin-Selberg expression of L ψ (s, f 0 , χ) only if χ 2 n − 1; to avoid this condition, we generally choose higher levels. The levels involved are Γ α := Γ[b −2 , b 2 y α ], where the integral ideals y α are indexed by α ∈ {r, ∈ Z | χ r} ∪ {0}. They are defined below, arranged in order of divisibility: Later on, when we invoke the Kummer congruences, we shall take a set of Dirichlet characters of varying moduli p and we shall be considering a sum of Rankin-Selberg integral expressions of varying levels y . Then we shall take a single r 0 so that all characters in the set are defined modulo p r and therefore we can simply lift all the Rankin-Selberg integrals of varying levels to all be of the same level y r and finally we trace the Rankin-Selberg integral back down to y 0 which process is given in the rest of this section. This is so that we can treat all characters uniformly. In specific cases, that is, when we consider a single primitive Dirichlet character with = χ 2 n − 1, one need not lift up to r in the first place and such a case is given as an example at the end of this section but will not be of much use later on. Assuming that χ is a Dirichlet character of modulus p with 1, the Rankin-Selberg expression from [15, (4.1)] of L ψ (s, f 0 ,χ) is given as in which r and V r : The definition of the trace map on modular forms is well known; with b fixed, the map Tr c2 c1 for any c 2 ⊆ c 1 takes modular forms in M k (Γ 2 , ψ) down to forms in M k (Γ 1 , ψ), where Γ i = Γ[b −2 , b 2 c i ], and is defined by decomposing Γ 1 = γ Γ 2 γ and summing over all the slash operator actions by these coset representatives. If g ∈ M n 2 +μ (Γ[b −2 , b 2 y r ], χρ τ ), then put F g (z, s) := g(z)E(z, 2s−n 4 ; k − n 2 − μ,η, Γ r ) and we have Tr yr y0 (F g ) = u∈S(Z/p 2r Z) Define, for any M ∈ Z, the matrix which belongs to P ι and is therefore in M. Associate to ι M the operator W (M ), acting on any modular form h of weight κ ∈ 1 2 Z by h|W (M ) = h κ ι M .
Proposition 5.1. Let χ be of modulus p , and let g and F g be as above. If r 0 is an integer, then Proof. By the definition of the trace map and substitution of variables in the integral, we have To finish, note that W (Y 0 ) 2 = (−1) n[k] and we claim Tr yr y0 (F g )|W (Y 0 ) = H g |U r p , the proof of which, in contrast to the integral-weight case, is twofold. That the matrices corresponding to the operators match is given by the simple matrix multiplication for u ∈ S(b −2 /p 2r b −2 ) and in which we used Y r = Y 0 p r . For the claim to hold, however, we need to check that the half-integral weight factors of automorphy match up as well, for which the requisite identity is Making use of Y r = Y 0 p r and combining all of the above, observe that both sides (5.3) coincide with |Y 0 i(z − u)| 1 2 . Thus. the claim, and therefore the proposition, holds.

A transformation formula of the theta series
Transformation formulae for theta series of the form θ χ |W (Y χ ) when χ is a primitive Dirichlet character are generally well-known entities. The precise formula of this section is encompassed by the generality of both Theorem A3.3 and Proposition A3.17 of [16]; what follows is a concrete derivation and calculation of the integrals found in the aforementioned results. Theorem A3.3 of [16] gives the existence of a C-linear automorphism λ → σ λ of M on the space of 'Schwartz functions on M n (Q f )', and it gives formulae of this action by P A and the inversion ι = ( 0 −In In 0 ). This is relevant since a more general class of theta series is defined using Schwartz functions λ by gives the series θ(z, λ) = θ (μ) χ (z; τ ) of (3.1). Assume that χ is a Hecke character of conductor p χ and let ι χ = ι Yχ . Since ι χ ∈ C θ , [16,Proposition A3.17] says that and so we calculate ι −1 and so ι −1 χ λ = ι ( σ λ). Let d = n 2 2 if n is even, d = 0 if n is odd, and let d p y be the Haar measure on M n (Q p ) such that the measure of bM n (Z p ) is |b| n 2 /2 p for any b ∈ Q. Theorem A3.3 (5), and equation (A3.3) of [16], and the definition of λ in (6.1) above gives making the change of variables y → Y χ y in the last line. By the definition ofτ in (5.1) and The integral in the above equation is non-zero if and only if the integrand is a constant function in y -that is, if and only if x ∈ |N (bc) Hence, by the calculation in (6.4), the transformation formula (6.2) on theta series with Schwartz functions translates, when χ is a primitive Dirichlet character, to

Fourier expansions of Eisenstein series
The holomorphic projection map Pr : C ∞ κ (Γ) → M κ (Γ, ψ) and its explicit action on Fourier coefficients is well known when 2n < κ ∈ Z -see [7,Theorem 4.2,p. 71]. This has a simple extension to the half-integral weight case with the formulae remaining unchanged, and we did this in [6, Theorem 3.1].
Given Proposition 5.1 and the transformation formula (6.5), it will be germane to give the explicit Fourier development of , for certain values m ∈ 1 2 Z defined below. To ease up on notation, let δ := n (mod 2) ∈ {0, 1}.
The projection map is only applicable for certain values s at which the Eisenstein series satisfies growth conditions; restriction to the set of special values, Ω n,k , at which the standard L-function satisfies algebraicity results guarantees this and this set is given by Proposition 7.1. For any ς ∈ S + , define Assume that k > 2n, χ is a Dirichlet character, and m ∈ Ω n,k . For any β ∈ Z, there exists a polynomial P (σ, σ ; β) ∈ Q[ς ij , ς ij | 1 i j n], defined on σ, σ ∈ S + ; a finite subset c of primes; polynomials f σ,q ∈ Z[t], defined for each σ ∈ S + and q ∈ c, whose coefficients are independent of χ; and a factor where m + = m − n − 1 2 and m − = 0, such that if m ∈ Ω n,k \{n + 1 2 } (and m = n + 3 2 if n > 1 and (ψ * χ) 2 When k ∈ Z and n is even the above kind of result is well-known, see, for example, [7,Theorem 4.6,p. 77]. Since the definition of the projection map remains unchanged, we can obtain the above in a similar manner, by using results on the Fourier development of integral and half-integral weight Eisenstein series as follows.

p-adic measures and the main theorem
Alhough complex L-functions are defined on variables s ∈ C, they can equally be viewed as Mellin transforms of the continuous characters R >0 → C × ; y → y s . In this latter vantage point, p-adic L-functions can naturally be constructed as Mellin transforms of continuous characters on Z × p with respect to a p-adic measure. Fix a prime p c, let C p := Q p denote the completion of the algebraic closure of Q p , and fix an embedding ι p : Q → C p . The p-adic norm naturally extends to C p and its ring of integers is given by The domain of the p-adic L-function will be The discussion in [7, pp. 23-25] concerning the decomposition of X p tells us that any C panalytic function F on X p is uniquely determined by its values F (χ 0 χ) for a fixed χ 0 ∈ X p and χ ranging over non-trivial elements of X tors Since Z × p = lim ← − (Z/p i Z) × is a profinite group, taken with respect to the natural projections π ij : (Z/p i Z) × → (Z/p j Z) × for each i j, to any distribution there associates a system of functions ν i : (Z/p i Z) × → A satisfying This association works by noting that each φ ∈ LC(Z × p , C p ) factors through some (Z/p i Z) × and by The compatibility criterion of [7, p. 17] tells us when we can run the above process backwards.  So distributions are generally quite easy to define; p-adic measures arise from p-adic distributions that are p-adically bounded. Hence, defining a distribution interpolating L-values is relatively trivial and showing that these expressions are bounded is the crux of the matter. To do this, we will invoke the abstract Kummer congruences, which criterion is well known in generality and is due to Katz in [4, p. 258]; we give a specialisation of it.
The proof of this can be found in [7, pp. 19-20]; it covers C p -valued measures as well by multiplication of some non-zero constant. An easy example of these criteria is the Fourier