$p$-adic L-functions via local-global interpolation: the case of ${\rm GL}_2 \times {\rm GU}(1)$

Let $F$ be a totally real field and let $E/F$ be a CM quadratic extension. We construct a $p$-adic $L$-function attached to Hida families for the group ${\rm GL}_{2/F}\times {\rm Res}_{E/F}{\rm GL}_{1}$. It is characterised by an exact interpolation property for critical Rankin-Selberg $L$-values, at classical points corresponding to representations $\pi \boxtimes \chi$ with the weights of $\chi$ smaller than the weights of$\pi$. Our $p$-adic $L$-function agrees with previous results of Hida when $E/F$ splits above $p$ or $F=\mathbf{Q}$, and it is new otherwise. Exploring a method that should bear further fruits, we build it as a ratio of families of global and local Waldspurger zeta integrals, the latter constructed using the local Langlands correspondence in families. In an appendix of possibly independent recreational interest, we give a reality-TV-inspired proof of an identity concerning double factorials.


Introduction
This paper is a case study in the construction of p-adic L-functions by the 'soft' method of glueing ratios of matching families of global and local zeta integrals. The local integrals are constructed and then inserted into the global context by using the local Langlands correspondence in families (see [Dis20] and references therein). The method, whose deployment seems new for non-abelian families, should be of wide applicability; we give a brief introductory description in § 1.2.
The specific context and arithmetic interest of our work is the following. Let F be a totally real field, let E/F be a CM quadratic extension, and let p be a rational prime. We construct a meromorphic function L p (V ) on Hida families for GL 2/F × Res E/F GL 1 that interpolates critical values L(1/2, π E ⊗ χ) L(1, π, ad) , for p-ordinary automorphic representations π ⊠ χ such that χ has lower weights than π. (The precise statement is Theorem A below; note that in our normalisation, the above numerators are not necessarily central values.) The function L p (V ) is new (if not surprising) at least when E/F does not split above p; for a discussion of previous related works see § 1.1.6. The interpolation property of L p (V ) holds at all classical points satisfying the weight condition and lying outside the polar locus (on which we have partial control), and it provides an entirely explicit and complete characterisation of the function, in the spirit of [Hid96]. Its generality and precision are key to some arithmetic applications in [Dis/b], which motivated our choice of case. In that paper, we prove, first, the p-adic Beilinson-Bloch-Kato conjecture in analytic rank 1 for (conjugate-)selfdual motives attached to representations π E ⊗ χ as above; and secondly, one divisibility in an Iwasawa Main Conjecture for the cyclotomic derivative d ♯ L p (V ) of L p (V ) along a selfdual locus. Both results, new or partly new even when F = Q and E/F splits at p, rely on a p-adic Gross-Zagier formula for d ♯ L p (V ). In turn, that formula is proved by analytically continuing formulas from [Dis17, Dis/a] for the central derivatives of certain cyclotomic p-adic L-functions L p (V (π,χ) , s) attached to those representations π⊠χ as above that have minimal weights. The continuation argument thus requires to exactly identify the collection {(L p (V (π,χ) , s) (π,χ) } of single-variable functions as a set of specialisations of a multivariable analytic function, which is indeed our L p (V ).
It would be interesting to extend our results to the non-ordinary case by the method of [Urb14,AI21]. As for further arithmetic directions in the ordinary case, (1) the main remaining goal is perhaps the full Iwasawa Main Conjecture for L p (V ). This was proved by Skinner-Urban [SU14] and Wan [Wan15] in the split case; in the non-split case, results toward it (when F = Q) were recently obtained by Büyükboduk-Lei [BL]. A second goal is the remaining divisibility in the Main Conjecture for the cyclotomic derivative of L p (V ) (cf. [Dis/b, Theorem E]); in view of the p-adic Gross-Zagier formula of [Dis/b], this is equivalent to a suitable generalisation of Perrin-Riou's main conjecture for Heegner points, which in its original form was recently proved by Burungale-Castella-Kim [BCK21].
1.1. Statement of the main result. -We move toward stating our main theorem, leaving a few of the detailed definitions of the objects involved to the body of the paper.
A (numerical) v 0 -adic weight for G is a tuple w := (w 0 , w = (w τ ) τ ∈Σv 0 ) of integers, all of the same parity, such that w τ ≥ 0 for all τ . It is said cohomological if w τ ≥ 2 for all τ . A weight for H is a tuple l = (l 0 , l = (l τ ) τ ∈Σv 0 ) of integers of the same parity. Finally, if w and l are weights for G and H, the associated contracted weight for G × H is (2) (w 0 + l 0 , w, l).
If w is a p-adic weight (say, for G) and ι : Q p ֒→ C is an embedding, we denote w ι := (w 0 , (w τ ) ι•τ : F ֒→C ). (In fact w ι only depends on ι |L if w is rational over the finite extension L of Q p in the sense that Gal(Q p /L) fixes w.) Let A be the ring of adèles of F . An automorphic representation of archimedean weight w is a complex automorphic representation π of G(A) such that π ∞ = π w := ⊗ τ : F ֒→R π (w0,wτ ) , where π (w0,wτ ) is the discrete series of GL 2 (F τ ) of weight w τ and central character z → z w0 . If L is p-adic, we define an automorphic representation of G(A) of weight w over L to be a representation π of G(A ∞ ) on an L-vector space, such that for every ι : L ֒→ C, the representation π ι := π ⊗ L,ι π ∞,w ι of G(A) is automorphic.
(3) To the representation π over L is attached a 2-dimensional representation V π of G F := Gal(F /F ); denoting by V π,v its restriction to a decomposition group at a place v of F , the representation V π is characterised by L(V π,v , s) = L(s + 1/2, π v ) for all v (this is the 'Hecke' normalisation of the Langlands correspondence, cf. [Del73, §3.2]). We say that π is ordinary (4) if for each place v|p of F there is a nontrivial G Fv -stable filtration such that the character α • π,v : F × v → L × corresponding to V + π,v (−1) has values in O × L . Let L be a p-adic field splitting E, suppose chosen for each τ : F ֒→ L an extension τ ′ to E, (5) and let τ ′c = τ ′ • c for the complex conjugation c of E/F . A Hecke character of H of weight l over the p-adic field L is a locally algebraic character χ : E × \A ∞,× E → L × such that χ(t p ) = τ : F ֒→L τ ′ (t p ) (lτ +l0)/2 τ ′c (t p ) (−lτ +l0)/2 for all t p in some neighbourhood of 1 ∈ E × p . We let V χ be the 1-dimensional G E -representation corresponding to χ.
1.1.2. L-values. -Let π, respectively χ, be a complex automorphic representation of G(A), respectively H(A), and let π E denote the base-change of π to E. Let us also introduce the convenient notation (to be thought of as referring to a 'virtual motive').
(2) A contracted weight is the same as a weight for (G × H) 1 := {(g, h) : det(g) = N E/F (h)} ⊂ G × H. This is in fact the true group governing our constructions.
(3) This definition is slightly different from, but equivalent to, the one adopted in [Dis/b], whose flexibility won't be needed here.
(4) The literature often adds to the definition the extra restriction that α • π,v should be unramified, and calls 'nearly ordinary' what we call 'ordinary'. (5) This will only intervene in the numerical labelling of the weights.
Let η : F × \A × → {±1} be the character associated with E/F , and let where the product (in the sense of analytic continuation) is over all places. These are the L-values we will interpolate.
1.1.3. Interpolation factors. -Let L be a finite extension of Q p , let π be an ordinary automorphic representation of G(A) over L, with a locally algebraic central character ω π : A ∞,× → L × , let χ : H(F )\H(A) → L × be a locally algebraic character, and set ω χ := χ |A ∞,× . Let ι : L ֒→ C be an embedding and let ψ = v ψ v : F \A → C × be the standard additive character such that ψ ∞ (x) = e 2πiTr F∞ /R (x) . If v|p, let ad(V π,v )(1) ++ := Hom (V − π,v , V + π,v ), and define where ιWD is the functor from potentially semistable Galois representations to complex Weil-Deligne representations of [Fon94], the inverse Deligne-Langlands γ-factor is γ(W, be any open compact subgroup, and for R = Z p , Q p , let T sph,ord be the p-(nearly) ordinary spherical Hecke R-algebra acting on ordinary p-adic modular cuspfoms for G of tame level U p G . A cuspidal Hida family X G is an irreducible component of the space Y G,U p G := Spec T sph,ord It is a scheme finite flat over Spec Z p T 1 , . . . , T [F :Q]+1+δF,p ⊗ Zp Q p (where δ F,p is the p-Leopoldt defect of F ), coming with a dense ind-finite subscheme X cl G ⊂ X G of classical points, and a locally free sheaf V G of rank 2 endowed with an O XG -linear action of G F .
To each x ∈ X cl G is associated an automorphic representation π x of G(A ∞ ) over Q p (x), and the fibre be an open compact subgroup. We define where the topology on H(F )\H(A p∞ )/U p H is profinite; it comes with a universal character identified with a G E -representation V H of rank 1, and a dense ind-finite subscheme Y cl H ⊂ Y H , whose points y correspond to U p H -invariant locally algebraic Hecke characters χ y of H over Q p (y). The weight of y is defined to be the weight of χ y .
Finally, the ordinary eigenvariety for (6) The normalisations of L-and ε-factors are as in [Tat79].

Its subset of classical points is
the character giving the action of the centre of G(A) on p-adic modular forms, let ω H := χ univ|A ∞,× , and let The self-dual locus Y sd G×H ⊂ Y G×H is the closed subspace defined by ω = 1. If X is a Hida family for G×H, we denote X cl := X ∩Y cl G×H , X sd := X ∩ Y sd G×H , and X cl,sd := X cl ∩ X sd . 1.1.5. Main theorem. -Throughout this paper, if X is a scheme over a characteristic-zero field L, we identify a geometric point x ∈ X (C) with a pair (x 0 , ι), where x 0 ∈ X is the scheme point image of (as a synonym, underlying) x and ι : L(x 0 ) ֒→ C is an embedding. If X is integral, we denote by K (X ) the local ring of the generic point, which we call the field of meromorphic functions on X .
If X is a Hida family for G × H, we define X cl,wt ⊂ X cl to be the subset of points (x 0 , y 0 ) whose contracted weight (k 0 , w, l) satisfies We denote X cl,sd,wt := X cl,sd ∩ X cl,wt .
Theorem A.
-Let X be a Hida family for G × H whose self-dual locus X sd is non-empty. There exists a unique meromorphic function whose polar locus D does not intersect X cl,sd,wt , such that for each (x, y) ∈ X cl,wt (C) − D(C) we have Here, if (x 0 , y 0 ) ∈ X cl is the point underlying (x, y) and ι : Q p (x 0 , y 0 ) ֒→ C is the corresponding embedding, we have denoted π x = π ι x0 , χ y = χ ι y0 , and the interpolation factor is as in (1.1.5).
The value of the interpolation factor agrees with the general conjectures of Coates and Perrin-Riou (see [Coa91]). (The notation V is meant to evoke some 'universal virtual Galois representation interpolating (1.1.3)'.) 1.1.6. Previous related work. -When E/F splits above p, Theorem A may be essentially deduced from the main result of [Hid91] (see also [Hid09]). Hida's method uses the Rankin-Selberg integral, whereas ours uses Waldspurger's variant [Wal85] based on the Weil representation (as discussed below).
The numerator of our L-value is a special case of the standard L-function for GL 2 × GL 1 over E, and when so considered our p-adic L-function is a multiple of the restriction to some base-change locus of one constructed by Januszewski [Jan] using the method of modular symbols; however that function is not uniquely characterised by its interpolation property, which involves unspecified periods.
1.2. Idea of proof, organisation of the paper, and discussion of the method. -The proof combines the strategy of Hida [Hid91] with an enhanced version of that of [Dis17, Proof of Theorem A], where we had constructed the 'slices' L p (V )(x, −) for x ∈ X cl G of weight 2.
We start from Waldspurger's [Wal85] integral representation of Rankin-Selberg type where ( , ) is a normalised Petersson product, f is a form in π with Whittaker function ⊗ v W v , the form I(φ, χ) is a mixed theta-Eisenstein series depending on a certain Schwartz function φ, and the R v are normalised local integrals.
In § 2, we discuss the general setup. In § 3, we make a judicious choice of φ v at the places v|p∞ and interpolate the ordinary projection of I(φ, χ) into a Y G×H -adic modular form. In § 4, we interpolate R v for v ∤ p∞ using sheaves of local Whittaker functions over X provided by the local Langlands correspondence in families ( § 4.4); we compute R v for v|p∞ ( § 4.3), which yield the interpolation factors in (1.1.8); and finally ( § 4.5), we use (1.2.1) to define L p (V ) as a glued quotient of the global and local (away from p∞) families of zeta integrals.
In Appendix A, we give a TV-inspired bijective proof of a combinatorial lemma occurring in § 3.3.
The method of constructing p-adic L-functions as ratios of arbitrary matching families of global and local zeta integrals should be applicable whenever an integral representation for the corresponding complex L-function is available, at least if the groups involved are products of general linear groups: for example, for Rankin-Selberg L-functions. It can be compared to the 'hard' constructions from much of the existing literature on p-adic L-functions, which rely on fine choices of local data at all places, computation of the associated integrals, and bounds on the ramification of the data (see [Hsi21] for an excellent example of the state of the art). To be sure, the two approaches should be viewed as complementary rather than alternative: while the 'soft' construction provides a flexibility useful for some applications (such as in [Dis17]), explicit choices and computations can still be plugged into it, and are likely still indispensable to address finer issues such as integrality.
For another brief general discussion of our method focused on the role of the local Langlands correspondence in families (LLCF), as well as some results on local interpolation, we refer to [Dis20, § 1.2, § 5]; (7) see also the very recent work of Cai-Fan [CF] for a related study in the context of periods attached to spherical varieties. Abelian antecedents of the construction, for which the LLCF is not needed, can be found in [LZZ18] and [Dis17].
The local-global approach may in principle introduce poles coming from zeros of the families of local integrals. In our specific setup, the Waldspurger local integrals are not easy to control (at least for this author) away from the selfdual locus. This is why Theorem A, while sufficient for the arithmetic applications in [Dis/b], is not as strong as it could be: one may at least expect that the condition that X sd be non-empty is superfluous, and that the polar locus of L p (V ) should not intersect X cl,wt . As noted by a referee, approaching L p (V ) via the well-understood Rankin-Selberg integrals for GL 2 ×GL 2 would likely yield such a strengthening.
2.1.1. General notation. -The following notational choices are largely standard.
-The fields F and E are as fixed in the introduction unless specified otherwise; if * denotes a place of Q or a finite set thereof, we denote by S * the set of places of F above * ; -we denote by D F , D E and D E/F , respectively, the absolute discriminants of F and E and relative discriminant of E/F ; for a finite place v of F we denote by d v ∈ F v a generator of the different ideal of F and by D v ∈ F v a generator of the relative discriminant ideal; -we denote by < the partial order on F given by x < y if and only if τ (x) < τ (y) for all τ ∈ Σ ∞ ; we denote R + := {x ∈ R | x > 0} and F + := {x ∈ F | x > 0} ⊂ F × ; -A is the ring of adèles of F ; if S is a finite set of places of a number field F , we denote by ∈S F v , and F S := v∈S F v ; when S consists of the set of places of F above some finite set of places of Q (for instance the place p) we use the same notation with those places of Q instead of S (for instance, we denote by ψ : F \A → C × the standard additive character as in § 1.1.3; -if R/R 0 is a ring extension, A is an R 0 -algebra, and X is an R 0 -scheme, we denote we denote by G K the absolute Galois group of a field K; -if K is a finite extension of F , its class number is denoted by for a place v of F , we denote by ̟ v a fixed uniformiser at v, by q F,v the cardinality of the residue field; we denote q F,p := (q F,v ) v∈Sp ; -the class field theory isomorphism is normalised by sending uniformisers to geometric Frobenii; for K a number field (respectively a local field), we will then identify characters of G K with characters of K × \A × K (respectively K × ) without further comment; -If I is a finite index set and x = (x i ) i , y = (y i ) are real vectors, we define (xy) i = x i y i and x y := i x yi i whenever that makes sense. Moreover, we often identify an integer w 0 with the constant vector (w 0 ) i∈I ∈ Z I . 2.1.2. Subgroups of GL 2 and special elements. -We denote by Z, A, and N respectively the centre, diagonal torus, and upper unipotent subgroup of G = GL 2/F ; we let P = AN and P 1 := P ∩ SL 2/F . We define a map a : GL 1/F → G by a(y) := y 1 .
We denote by or its image in GL 2 (R) for any F -algebra R. (The context will prevent any confusion with the notation for the weights of G.) For r ∈ Z Sp ≥1 , we define as well as a sequence of compact subgroups For θ ∈ (R/2πZ) S∞ , we denote r θ : It carries an involution arising from the map g → g −1 on the group G.
For S a finite set of places of F disjoint from S p ∪ S ∞ , we define the ordinary Hecke algebra H ord U Sp := H U Sp ,Zp ⊗ Z p [A p ] over Z p , which will act on spaces of ordinary modular forms (here and in the rest of the text, a subscript Sp is shorthand for S ∪ S p ). It is endowed with the involution deduced from (2.1.1) and (2.1.2). If are open compact subgroups, and S, respectively S p , is the set of places such that U v is not maximal, we define The polynomial W f,a (y) only depends on the class of ay modulo a −1 (U ), so that defining for a ∈ A + , for all a ∈ F + and y ∈ A + we have W f,a (y) = W f (ay). We say that f is cuspidal if W 0 (y) = 0 for all y.
If f is nearly holomorphic of degree 0 (that is, ≤ (0, . . . , 0)), we simply say that f is a (holomorphic) modular form. If R ⊂ C is any subring, we denote by respectively the spaces of cuspidal forms and holomorphic forms of level U and weight w, and of nearly holomorphic forms of level U , weight w, and degree ≤ m = (m τ ), such that for all a ∈ A + , the polynomials W f (a) have coefficients in R. We write N w (U, R) := lim − →m N ≤m w (U, R), and (R) := lim − →U (U, R) if stands for the notation for any of the spaces of forms defined above (or below). Finally, we define the space S a w (U, C) of antiholomorphic cuspforms to be the C-vector space image of S w (U, C) under complex conjugation. The formula acts, as usual, by right translation) defines C-linear bijections from S w (U, C) to S a w (U, C) and viceversa.
2.2.2. Twisted modular forms. -A twisted nearly holomorphic (Hilbert) modular form of weight w, level U , degree ≤ m = (m τ ), is a function satisfying the following two conditions: 2. there is a Whittaker-Fourier expansion for all x ∈ A and y, u ∈ A × such that (uy) ∞ > 0, where: If R ⊂ C is any subring, we denote by M tw w (U, R) ⊂ N tw,≤m w (U, R) the spaces of holomorphic and nearly holomorphic forms of level U , weight w, and degree ≤ m = (m τ ), such that all the polynomials W f,a (y, u) have coefficients in R.

Contracted product. -For any open compact subgroup
is well-defined independently of U F ⊂ U ′ F , and for any such choice the support of the sum is finite. If f 1 , f 2 are twisted nearly holomorphic forms, we may thus define a (plain) nearly holomorphic form f 1 ⋆ f 2 by .

Differential operators.
-We attach to a nearly holomorphic (genuine or twisted) form f the function The Maass-Shimura differential operators on functions on h Σ∞ are defined as follows. For τ : Then for any ring Q ⊂ R ⊂ C, this operator defines a map (For a proof of the intuitive fact that the archimedean operator δ k w indeed preserves the rationality properties of finite Whittaker-Fourier coefficients, see [Hid91, Proposition 1.2], whose calculations also apply to the twisted case.) The subscript w will be omitted if it is clear from the context.
2.3. p-adic modular forms. -We study the completions of spaces of modular forms for certain p-adic norms.
. By the q-expansion principle (see [Dis17, Proposition 2.1.1] for a version in our setting), the map is injective. We denote its image by S w (U, C) and view the map S w (U, C) → S w (U, C) as an identification.
If R is any ring admitting embeddings into C, we denote by such that for any ι : R ֒→ C, the sequence f ι := (ιW f (a)) a is the q-expansion of a cuspform (In (2.3.1), the notation W f can be thought of as simply synonymous to f; it is introduced in order to match the identification of the previous paragraph.) By [Hid91, Theorem 2.2 (i)] (together with a consideration of Galois actions mixing the weights), for any such ring R we have For more general rings, the previous equality is taken to be the definition of S • (U, R).
is a tuple of integers, all having the same parity, such that w τ ≥ 1 for all τ : F ֒→ L. As in § 1.1.1, if w is an L-valued weight and ι : L ֒→ C is an embedding, we define the complex weight be a compact open subgroup, and let w be an L-valued weight. We define S w (U, L) to be the set of q-expansions f such that for every ι : L ֒→ C the expansion f ι belongs to S w ι (U, C).
is the Whittaker-Fourier coefficient of f ι as in (2.2.1). (In other words, we have two embeddings The space of cuspidal p-adic modular forms is the completion of S w (U p , L) for the norm ||f|| := sup a |W f (a)|, for any w. By a fundamental result of Hida (see [Hid91, paragraph after Theorem 3.1]), the space S(U p , L) is independent of the choice of w. In particular, if L is Galois over Q p , this space is stable by the action of Gal(L/Q p ) and so it is of the form S(U p , Q p ) ⊗ Qp L for a space S(U p , Q p ).

2.3.3.
Nearly holomorphic forms as p-adic modular forms. -We may attach a p-adic q-expansion to a nearly holomorphic form with coefficients in a p-adic subfield of C. Let L be a finite extension of Q p and let w be a p-adic L-valued weight. We say that is a p-adic nearly holomorphic cuspform of weight w and level U p ⊂ G(A p∞ ) if the following condition holds. For each ι : L ֒→ C, there exists a cuspidal nearly holomorphic form for some n ∈ Z Sp ≥1 , whose Whittaker-Fourier polynomials have constant terms satisfying The notion of a p-adic twisted nearly holomorphic cuspform is defined similarly by the identity W f ι ,a (y, u)(0) = ι (ay) Proof. -This is the first assertion of [Hid91, Proposition 7.3].

2.3.4.
Hecke operators and ordinary projection. -The space N w (U, C) is endowed with the usual action of H U . By writing down the effect of this action on Whittaker-Fourier coefficients of cuspforms, we may descend it to a bounded action of H U p ,L on S w (U p , L), hence on S(U p , L), for any p-adic field L.
. This is compatible with the previous definition in the following sense (see [Hid91,(2 The superscript w will be omitted when understood from the context. The ordinary projector is for any tame level U p and p-adic field L. Its image is denoted by The operator e ord preserves S w (U p , L), and we denote S ord . If f C is a complex modular form arising as f C = f ι for a form f ∈ S(L) for some finite extension L of Q p and some ι : L ֒→ C, we define e ord,ι (f C ) := (e ord f ) ι .
2.3.5. Differential operators after ordinary and holomorphic projections. -Let L be a finite extension of Q p , and let f 1 , f 2 be p-adic twisted nearly holomorphic forms over L. For any ι : L ֒→ C and k ∈ Z Σ∞ ≥0 , we have the proof of [Hid91, Proposition 7.3] carries over to the twisted case.
2.4. Hida families. -We gather the fundamental notions concerning Hida families and the associated sheaves of modular forms.
the latter is isomorphic to ∆ × Z A point κ ∈ W is identified with the pair of characters We have an involution defined by If k is a p-adic weight for G, we say that κ is classical of weight k if for all v|p, , and the product runs over the τ ∈ Σ p inducing the place v ∈ S p . For a classical weight κ, we define κ sm : the set of points of classical weight, which has the structure of an ind-étale ind-finite scheme over Q p . If κ is classical of weight k = (k 0 , k), then κ is classical of weight k ∨ = (−k 0 , k). We let W cl,≥2 be the set of classical points satisfying k ≥ 2.
be the images of the Hecke algebras H sph,ord be the ordinary eigenvariety for G of tame level U p . (The subscript U p will be omitted when unimportant or understood from the context.) The space Y G,U p is a union of finitely many irreducible components, called Hida families of tame level (dividing) U p . It carries an involution deduced from the one on H sph,ord The treatment proposed here is minimal and somewhat ad hoc, but it will be sufficient for our purposes. We believe that a more systematic treatment of the geometry of Hida theory should be based on the theory of uniformly rigid spaces developed in [Kap12].
that, when identified with a pair (κ G,0 , κ G ) of O(Y G ) × -valued characters as in (2.4.2), is κ G,0 (z) = the Hecke operator acting by right translation by z on modular forms, κ G (y p ) = U • yp . The weight map is finite and flat and it intertwines the involutions .
The set of classical points of Y G is For each x ∈ Y cl G,U p of weight w, there exists a unique (up to isomorphism) ordinary automorphic representation π the isomorphism is unique up to scalars. This defines a bijection between Y cl G,U p (Q p ) and the set of isomorphism classes of ordinary automorphic representation π of G(A) over Q p with π U p = 0.
characterised by the property that for every κ ∈ W cl , every closed point z ∈ ϕ −1 (κ), and every a ∈ where the right-hand side is the p-adic q-expansion coefficient of the classical modular form Moreover, the image of (2.4.4) equals the space of those sequences (W (a)) a for which there exists a set of closed points Proof. -It suffices to construct (2.4.4) for Z = W as the general case follows by base-change. Let be the space of ordinary forms with Z p -coefficients; this is an A • -module and a Z p -lattice in S U p W (W) = S ord (U p , Q p ). For κ ∈ W cl , let p κ ⊂ A • be the corresponding prime ideal. Let n ∈ N and let M range among finite subsets of W; the filtered system of ideals and after taking projective limits, the desired map S ord (U p , It is injective by the q-expansion principle and the preservation of injectivity under inverse limits. We now consider the second statement. It is clear that, for any fixed Σ as in the lemma, the spacẽ For z ∈ Σ, let p z ⊂ B • be the corresponding prime ideal. Let n ∈ N and let N range among finite subsets of Σ; then the filtered system of ideals J n,N := (p n ) + z∈N p z forms a fundamental system of neighbourhoods of 0 ∈ B • . By assumption, for each z ∈ Σ the map (2.4.6) is an isomorphism modulo p z ; hence it is an isomorphism modulo J n,N for all (n, N ), hence an isomorphism.
We call elements of S U p Z (respectively S U p Z ⊗ O Z K (Z )) Z -adic ordinary modular cuspforms (respectively meromorphic Z -adic ordinary modular cuspforms) of weight ϕ : Z → W.
The set of classical points is Y cl H := Y H × W W cl . Note that if y ∈ Y H is a classical point such that κ H (y) has weight (l 0 , l), then χ y has weight (l 0 , l) as defined in the introduction. 2.4.7. Universal automorphic sheaf on a Hida family. -Let X G be a Hida family for G, and let X cl G := X G ∩ Y cl G . For each sufficiently small U p , we may view X ⊂ Y G,U p and we define (9) If v|p splits in E, then for a ∈ U • F,v we have j(a) = (a, 1) under some isomorphism For each x ∈ X cl G , by Hida's Control Theorem (see for instance [Hid91, Corollary 3.3]) and the theory of newforms, we have an isomorphism of H U p -modules, . By [Hid91,§3], there is a unique (the normalised primitive form over X G ) such that W f0 (1) = 1 ∈ O(X G ) for the q-expansion map deduced from (2.4.5). Any f ∈ Π U p can be written as f = T f 0 for some Hecke operator T supported at the places v ∤ p∞ such that U p is not maximal.
such that for all x ∈ X cl G , the fibre V G|x is the Galois representation attached to π x by the global Langlands correspondence.
Let S be a finite set of finite places of F , disjoint from S p , such that for all v / ∈ S the tame level as O X ′ G [G(F S )]-modules. Proof. -By the local-global compatibility of the Langlands correspondence for Hilbert modular forms (see [Car86] or [Dis/b, Theorem 2.5.1]), for all x ∈ X cl G and all places v, the G(F v )-representation π x,v corresponds, under local Langlands, to the Weil-Deligne representation V x,v attached to V G|x|GF v . Then the result follows from [Dis20, Theorem 4.4.3].

Theta-Eisenstein family
In this section, we define the kernel of the Rankin-Selberg convolution giving the p-adic L-function.
Here χ V = χ (V,q) is the quadratic character attached to V , γ(V, q) is a fourth root of unity, andφ denotes Fourier transform in the first variable with respect to the self-dual measure for the character ψ u (x) = ψ(ux). We will need to note the following facts (see for instance [JL70]): where H is a compactly supported smooth function on R × and P is a complex polynomial function on V . This space is not stable under the action of GL 2 (R), but it is so under the restriction of the induced (gl 2,R , O 2 (R))-action on the usual Schwartz space (see [YZZ12, §2.1.2]). We will also need to consider the reduced Fock space S(V × R × ) spanned by functions of the form φ(x, u) = (P 1 (uq(x)) + sgn(u)P 2 (uq(x)))e −2π|u|q(x) where P 1 , P 2 are polynomial functions with rational coefficients. By [YZZ12, §4.4.1, 3.4.1], there is a surjective quotient map We let S(V × R × ) ⊂ S(V × R × , C) be the preimage of S(V × R × ). For the sake of uniformity, when F is non-archimedean we set 3.1.3. Global case. -Let (V, q) be an even-dimensional quadratic space over the adèles A of a totally real number field F , and suppose that V ∞ is positive definite; we say that V is coherent if it has a model over F and incoherent otherwise. Given an O F -lattice V • ⊂ V, we define the space S(V × A × ) as the restricted tensor product of the corresponding local spaces, with respect to the spherical elements We call such φ v the standard Schwartz function at a non-archimedean place v. We define similarly the reduced space S(V × A × ), which admits a quotient map defined by the product of the maps (3.1.1) at the infinite places and of the identity at the finite places. The Weil representation of GO(V) × G(A ∞ ) × (gl 2,F∞ , O(V ∞ )) is the restricted tensor product of the local representations.
For a quadratic space V = (V, q) over A, we define ε(V) = +1 (respectively −1) if and only if there exists (respectively does not exist) a quadratic space V over F such that V ⊗ F A = V.
3.1.4. The quadratic spaces of interest. -Let us go back to our usual notation: thus F is our chosen totally real field and E its chosen CM quadratic extension. In this paper, we will consider the quadratic spaces V = (B, q), where B is a quaternion algebra over A, definite at all the archimedean places and split at p, and endowed with an A-embedding A E ֒→ B, and q : B = V → A is its reduced norm. It has a decomposition V = V 1 ⊕ V 2 where V 1 = A E (on which the restriction of q coincides with N E/F ) and V 2 is the q-orthogonal complement. Thus ε(V) = ε(V 2 ). We denote by r 1 the restriction of r to a representation of For each place v, we have
Lemma 3.2.2. -Let χ : E × \A × E → C × be a locally algebraic character of weight l, and let E(g, u) be any twisted modular form such that E(g, u ∞ u) = E(g, u) for all u ∞ ∈ F + ∞ . Suppose that φ ∞ 1 (0, u) = 0 for all u. Then for all g = y x 1 ∈ G(A), we have * Proof. -We may assume that U F is so small that E(u) is invariant under u ∈ U F and ν UF = 1. Taking fundamental domains for µ 2 UF \F × , the expression of interest is Since the integrand is invariant under E × ∞ , by Lemma 2.1.1 with µ = µ UF and a change of variables a = uq(x), this equals By the invariance properties under U F , this can be brought into the desired expression by a change of variables a ′ = α 2 a and the calculation which follows from the definition of c UF = (2.2.5) and the class number formula 3.3. Eisenstein series. -Let V 2 be a 2-dimensional quadratic space over A, totally definite at the archimedean places. Let φ 2 ∈ S(V 2 × A × ) be a Schwartz function, and let ξ : F × \A × → C × be a locally algebraic character such that ξ ∞ (x) = x k0 for some integer k 0 and for all x ∈ F + ∞ . Define the automorphic Eisenstein series (10) (The defining sum is absolutely convergent for ℜ(s) sufficiently large, and otherwise it is interpreted by analytic continuation.) It satisfies E r (zg, u, φ 2 ; ξ) = ηξ −1 (z)E r (g, u, r(x, 1)φ 2 , ξ). (10) For k 0 = 0, this is L (p∞) (1, ηξ)/L (p∞) (1, η) times the series defined in [Dis17].
The series E r (φ ∞ 2 ; ξ, k) belongs to N ≤k tw,(−k0,k+k0) (C). Here (Note that the functions W a,r (φ 2 , ξ) correspond to the W C Er (φ2,ξ),a of § 2.2.2. We prefer to use lighter notation in this section.) We choose convenient normalisations for the local Whittaker functions: let γ u,v = γ(V 2,v , uq) be the Weil index, and for a ∈ F × v set Then for the global Whittaker functions we have

.5)
A simple calculation shows that for all v and a = 0, We will sometimes drop φ 2,v from the notation. The following sufficient condition for cuspidality will simplify matters a little later on.  for all u. Then for all g = ( y x 1 ) with y ∈ A + , x ∈ A, we have W 0,r (g, u, φ 2 ; ξ) = 0.
Proof. -This is a special case of [YZZ12, Proposition 6.10].
Therefore, when u > 0, The integral is the same one appearing in [YZZ12, bottom of p. 55] with d = 2 + 2k + 2k 0 . By [YZZ12, Proposition 2.11] (whose normalization differs from ours by L(1, η v ξ v ) = πi), we find if a, u > 0, as well as simpler formulas implying the desired ones in the other cases.
Proof. -After recalling the definition of P k0,k in (3.3.2), by Lemma 3.3.3 and (3.3.6) we find the asserted vanishing and that for ay, uy > 0 we have (dropping subscripts v): which is equal to the asserted formula.
3.3.5. Non-archimedean Whittaker functions. -We study the functions W • a,v and the q-expansion of E r . Proposition 3.3.6. -Let v be a non-archimedean place of F .
where d u x 2 is the self-dual measure on (V 2,v , uq) and (When the sum is infinite, it is to be understood in the sense of analytic continuation from characters ξ| · | s with s > 0.) 2. For all finite places v, |d| , and for almost all v we have Proof which gives the asserted value.
Let φ p∞ 2 satisfy (3.3.7), so that by Lemma 3.3.2, the corresponding Eisenstein series is cuspidal. For ξ a locally algebraic p-adic character of A × of weight k 0 , consider the (bounded) sequence of coefficients in Q p (ξ) 1, η). This is the p-adic q-expansion attached to E r . Analogously to Corollary 3.3.5, we have

p-adic interpolation of Whittaker functions
, the spaces of characters of F × v and respectively E × v . We say that a meromorphic function Φ on an integral scheme has poles controlled by the (nonzero) meromorphic function Φ ′ if Φ/Φ ′ is regular.
for all ξ v ∈ Y v (C) whose underlying scheme point is not a pole.
3.4. Theta-Eisenstein family. -Fix a compact open subgroup U p ⊂ G(A p∞ ) (which will be usually omitted from all the notation), and let U p F := U p ∩ A p∞,× . Let φ p∞ ∈ S(V p∞ × A p∞,× ) be a Schwartz function fixed by U p . Let ξ : F × \A × → C × be a locally algebraic character fixed by U p F and such that ξ(x ∞ ) = x k0 ∞ for some k 0 ∈ Z, and let k ∈ Z Σ∞ ≥0 satisfy k v + k 0 ≥ 0 for all v. We fix a choice of a Schwartz function in S(V 2,p × F × p ) as follows. Let U • F,p ⊂ O × F,p be as fixed in § 2.4.5. For r ∈ Z Sp ≥1 and κ ′ 1 : U • F,p → C × a smooth character, we define (3.4.1) where the product ⋆ is (2.2.7), and E r (·) = (3.3.12).
Fix a compact open subgroup U p H ⊂ A p∞,× E (which will be omitted from all the notation). Let l be a complex weight for H, let χ : E × \A × E → C × be a locally algebraic character of weight l fixed by U p H , and assume that for all w|v|p, the integer r v ≥ 1 is greater than the conductors of χ w , ξ v , κ ′ 2,v . Then we define which does not depend on the choice of r; here κ ′ 1,χ,p is as in (2.4.3), namely Lemma 3.4.1. -For each c ∈ A + satisfying v(c) ≥ 1 for some v|p, the c th Whittaker-Fourier coefficient of I(φ p∞ ; χ, ξ, κ ′ 2 , k) is Here, taking Proof. -We lighten some of the notation. The assumptions of Lemma 3.2.2 are satisfied, therefore for c ∈ F + and g = ( y x 1 ) with y ∈ A + , x ∈ A, Now for λ = −ε(V 2 )/L (p) (1, η), by Corollary 3.3.7 we have E(g, q(t)ay 2 ) = λ·|y| 1/2 ηξ −1 (y) W 0 (y, u) for some coefficients W C I,c (y), which we now explicitly calculate if c satisfies v(c) ≥ 1 for some v|p. Under this condition, we have where we have noted that, since we have assumed v(c) ≥ 1 and the choice of φ 1,p implies v(a) = 0, the constant term (corresponding to a = c) of the Eisenstein series does not contribute. Finally, we rewrite the resulting formula with c in place of cy.

Let
(3.4.5) For the sake of simplicity, we momentarily introduce the assumption that the weight l of χ satisfies l ≥ 0. We will see in Corollary 3.4.5 that this does not affect our main construction.
With notation as in Proposition 3.4.3, let It is easy to verify, using only that J p (·) is a Schwartz function of a ∈ A p∞,× , that the Riemann sums on U • F,p valued in Q p (χ, κ 2 ). Then by the same argument of the standard result in [Kob77, Theorem 6 on p. 39], any continuous Q p (χ, κ 2 )-valued function is integrable for this measure. The same holds with Q p (χ, κ 2 ) replaced by O(Y H× W) and χ, κ 2 by the universal characters, J p by the universal function; for this universal situation, we will use the same notation without χ, κ 2 .
Proof. -The interpolation property (3.4.9) at the level of q-expansions follows from Proposition 3.4.3 and the previous discussion. The simplification in the argument of κ 2 in the interpolated coefficient (3.4.7) is justified by the fact that κ 2 (a − p n! c) − κ 2 (a) → 0 uniformly in a, and that the expression of interest is a bounded function of κ 2 (·). Lemma 2.4.4 then shows the existence of the Y H× W-adic modular form I ord (φ p∞ ; χ, κ 2 ).

Consider the weight map
Recycling notation (in a way that should cause no confusion), define an ordinary meromorphic (Y G× Y H )-adic modular cuspform of weight ϕ by (3.4.10) where the right-hand side is the form of Corollary 3.4.4. We denote by Corollary 3.4.5. -Assume that φ p∞ 2 satisfies (3.3.7). For all x ∈ Y cl G (C) and y ∈ Y cl H (C), of weights w, l such that for all τ ∈ Σ ∞ , where ξ = ξ x,y = ω x ω y , whose weight we denote by k 0 ; -k x,y = (w − 2 − |l| − k 0 )/2; -κ ′ 2,x,y = κ x ′ · κ ′−1 y (with notation as in (2.4.3)); -ι : Q p (x, y) ֒→ C is the embedding attached to the complex geometric point (x, y).
Proof. -The interpolation property at characters satisfying l ≥ 0 follows from Proposition 3.4.3 via Corollary 3.4.4; the same argument also goes through without the assumption l ≥ 0, since the weight of the chosen theta-Eisenstein series does not depend on l (or y) but only on x. The inequalities on the weights come from the conditions k, k + k 0 ≥ 0.

Zeta integrals
In this final section, we interpolate global and local (away from p∞) zeta integrals, compute the archimedean and p-adic integrals, and construct the p-adic L-function.
As preliminary, we recall gamma factors introduced in the introduction. Let F v and L be p-adic fields. The (inverse) Deligne-Langlands gamma factor of a potentially semistable representation ρ of Gal(F v /F v ) over L, with respect to a nontrivial character ψ v : F v → C × and an embedding ι : L ֒→ C, is defined as where ιWD is Fontaine's functor [Fon94] to complex Weil-Deligne representations. We also define γ(s, W, ψ v ) := γ(W ⊗ | · | s , ψ v ). Let Q ab p be the abelian closure of Q p . If W χ is the Weil-Deligne representation corresponding to a smooth character χ : F v → Q ab,× p , then for any σ ∈ G ab Qp corresponding to a ∈ Q × p under the reciprocity map, we have For now until the final § 4.5.3, we fix an embedding (4.0.2) ι ab : Q ab p ֒→ C, by which we identify the fixed standard character ψ ∞ : A ∞ → C with one valued in Q ab p (still denoted by ψ ∞ ).
4.1. Petersson product. -Let π be an ordinary automorphic representation of G(A) over a finite extension L of Q ab p . For v|p, let ω π,v , α π,v : F × v → L × be the central character and, respectively, .
The pairing , satisfies the following properties.
1. For all f ∈ π ord , g ∈ N w ∨ (L), we have f, g = f, e ord e hol g ; 2. If f 0 ∈ π ∞ , f ∨ 0 ∈ π ∨,∞ are ordinary forms, new at places away from p, holomorphic at the infinite places, and with first Fourier coefficients equal to 1, then for some constant c π ∞ ∈ L × depending only on the Bernstein components and the monodromy of π v for all v ∤ p∞.
Proof. Proposition 4.1.2. -Let X G ⊂ Y G be a Hida family of tame level U p , let S be a finite set of places such that U Sp is maximal, and let Π = Π U Sp XG . There is a unique O(Y G,Q ab p )-bilinear pairing , : Π ⊗ O(YG) S (Y G ) Q ab p → K (X G,Q ab p ) such that for all x ∈ X cl G,Q ab p , corresponding to an ordinary representation π = π x over Q ab p (x), and for all f ∈ Π Q ab Proof. -The construction is very similar to that of the pairing denoted by H −1 l λ in [Hid91,p. 380].
In this case, let f 0 = (2.4.10) be the normalised primitive form in Π , let U p 0 ⊂ G(A p∞ ) be a maximal open compact subgroup fixing f 0 , and let where c XG := c π ∞ for any automorphic representation π such that π U Sp ,ord ∼ = Π |x . for some x ∈ X cl G,Q ab p . Let us explain why this is well-defined independently of x. As noted before, c π ∞ only depends on the Bernstein component and the (rank of the) monodromy of π x,v for v ∤ p∞. (In plain terms, the rank of the monodromy is 1 if π x,v is a special representation and it is 0 otherwise.) The Bernstein component is an invariant of connected families. As for the rank of the monodromy, by the local-global compatibility result of Proposition 2.4.2, it is the rank of the monodromy of the Weil-Deligne representation attached to V G|x . Since the latter is pure, the desired constancy along X G follows from [Dis20, Proposition 3.3.1].
In general, we may write f = T f 0 for some Hecke operator T supported away from p. We then define f , g := f 0 , T g . The interpolation property follows from the definitions, the interpolation property proved in [Hid91,Lemma 9.3], and (4.1.2). , let χ = χ y be the corresponding character of E × \A × E , and let κ χ ∈ W cl (C) be its weight, let l be its numerical weight, and let ω χ := χ |A × . Let x ∈ X cl G (C), corresponding to a point x 0 ∈ X cl G and an embedding ι : Q p (x) ֒→ C. Let π 0 be the ordinary automorphic representation of G(A) over Q p (x 0 ) attached to x 0 , and let π = π ι 0 . We denote by κ π ∈ W cl (C), w, ω π respectively the weight, numerical weight, and central character of π. We let α = ⊗ v α v : F × p → C × be the character such that U y f ι = α(t)f ι for any f ∈ π ord 0 and t ∈ F × p . Then κ π,0 (z) = ω π (z)z w0 , κ π (t) = α |U • F,p (t)t (w+w0)/2 are the decompositions of κ π,0 and κ π into a product of a smooth and an algebraic character.

4.2.2.
Waldspurger's integral. -The next proposition gives an integral representation for the Lfunction we are interested in. We first define the local terms. Let f 0 ∈ π ord 0 , let f := f ι 0 , and let be the Whittaker function of f a with respect to ψ −1 . It is related to the q-expansion (2.2.1) of f by We assume that W : where δ ξ,r is as in (3.3.1). Note that the integral R r,v does not depend on r ≥ 1 unless v|p; we will accordingly simplify the notation in these cases. We also define normalised versions. For v|p∞, let Φ v be as fixed in § 4.2.1. Then we put By a result of Waldspurger (see [Dis17, Lemma 5.3.2]), for a place v such that π v and χ v are unramified, φ v is standard, and W v is unramified, we have and assume that f := f ι,a 0 has a factorisable ψ −1 -Whittaker function W = ⊗ v W v . Let φ p∞ ∈ S(V p∞ × A p∞,× ). For sufficiently large r = (r v ) v|p , we have where all but finitely many of the factors in the infinite product are equal to 1.
Proof -Let v ∤ p∞ be a place of F and let L be a field of characteristic zero. Let π v be a smooth irreducible representation of G(F v ) over L, with central character ω π,v , and let χ v : E × v → L × be a smooth character. Assume the self-duality condition ω π,v χ |F × v = 1. There exist a 4-dimensional quadratic space V v = B v over F v of the type described in § 3.1.4, uniquely determined by If moreover all the data are unramified at a place v inert in E, it is possible to choose W v and φ v = φ 1,v φ 2,v such that φ 2,v (0, u) = 0 for all u (condition (3.3.7)).
Proof. -The argument in [Dis17, Proof of Proposition 3.7.1, second paragraph] applies verbatim to prove the first statement. Let us prove the second one. We drop all subscripts v. Fix an isomorphism V 2 ∼ = E, and let us choose W to be a new vector, φ 1,v to be the standard Schwartz function, and
= for an equality up to nonzero scalars, by the Iwasawa decomposition Let U 0 (̟ r ) ⊂ U 0 := GL 2 (O F ) be the set of matrices which are upper-triangular modulo ̟ r . It is easy to verify that φ 2 is invariant under U 0 (̟ r ) for some r, and that where the last equality follows from interchanging the order of integration and observing that The last quantity equals φ • (t −1 y, y −1 q(t)) for the standard Schwartz function φ • ; therefore the integral R is a constant multiple of the unramified integral, in particular it is nonzero by (4.2.3).
Lemma 4.3.1. -Let v|p, and assume that W v is normalised by W v (1) = 1. Then for any sufficiently large r (depending on χ v , π v ) we have Proof The asserted formula follows.
Proof. -By the Iwasawa decomposition we can uniquely write any g ∈ GL 2 (R) as Dropping all subscripts v, since the weights match the integration over SO(2, R) yields 1, and we have By definition, ω π χξ −1 (z) = 1, so that the integration in d × z simply realises the map Φ → φ. Then where 2 = vol(R × \C × ).

Now the result follows from identifying
is a constant in Q × ; here, ω x = ω πx and ω y = ω χy . Note that the (base-change of the) functional f , − may be applied to I ord, (φ p∞ ), thanks to Lemma 2.4.1.
Proposition 4.5.3. -The collection of meromorphic functions on X Q ab p has the following properties.
Note that the right-hand side of (4.5.6) is the same as in (1.1.8) and independent of (f , φ p∞ ). This will enable us to glue the various L (V , f , φ p∞ ) into the sought-for p-adic L-function.
Proof. -The second statement follows from Lemma 4.2.2.
Here, we denote by ω π be the central character of π, let ω χ := χ |A × , let ω = ω π ω χ , and define γ(s, ω ′∞ , ψ ∞ ) := v∤∞ γ(s, ω ′ v , ψ v ). for all σ ∈ Gal(Q ab p /Q p ). Let X cl, ,reg Q ab p be the intersection of X cl, Q ab p with the complement of the polar locus of L p (V ). Since this set is dense in X Q ab p , it suffices to show that (4.5.8) holds for the restriction L p (V ) of L p (V ) to X cl, ,reg Q ab p ; in other words, that L p (V ) belongs to O(X cl, ,reg ).
(11) That is, wτ is independent of τ ∈ Σ∞ and so is lτ . Without this condition, we may have a slightly weaker result.