aa r X i v : . [ m a t h . N T ] F e b p -ADIC L -FUNCTIONS FOR GL × GU(1) by Daniel Disegni
Abstract . —
Let F be a totally real field and let E/F be a CM quadratic extension. We constructa p -adic L -function attached to Hida families for the group Res F/ Q GL × Res E/ Q GL . It interpolatescritical Rankin–Selberg L -values at all classical points corresponding to representations π ⊠ χ with theweights of χ smaller than the weights of π . This confirms a conjecture of the author, making relatedresults on the p -adic Be˘ılinson conjecture and its Iwasawa-theoretic variants unconditional.Our p -adic L -function agrees with a previous result of Hida when E/F splits above p , and it is newotherwise. We build it as a ratio of families of global and local Waldspurger zeta integrals, the latterconstructed using the local Langlands correspondence in families.In an appendix of possibly independent recreational interest, we give a reality-TV-inspired proof ofan identity concerning double factorials. Contents
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1. Statement of the main result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2. Idea of proof and organisation of the paper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52. p -adic modular forms and Hida families. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1. Notation and preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2. Modular forms and their q -expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3. p -adic modular forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4. Hida families. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123. Theta–Eisenstein family. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1. Weil representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2. Theta series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3. Eisenstein series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4. Theta-Eisenstein family. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224. Zeta integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1. Petersson product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2. Waldspurger’s Rankin–Selberg integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3. Evaluation of the integrals at p and ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4. Interpolation of the local zeta integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.5. The p -adic L -function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Appendix A. Reality shows and double-factorial identities. . . . . . . . . . . . . . . . . . . . . . . . . . . 34References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 This work was supported by ISF grant 1963/20.
DANIEL DISEGNI
1. Introduction
Let F be a totally real field and E/F a CM quadratic extension, and let p be a rational prime. In[Dis/b], we prove, first, the p -adic Beilinson–Bloch–Kato conjecture in analytic rank for p -ordinary,selfdual motives attached to Hilbert modular forms for F , or their twists by Hecke characters of E oflower weights; and secondly, one divisibility in an Iwasawa Main Conjecture for cyclotomic derivativesof p -adic L -functions. Both results, which are new or partly new even when F = Q and E/F splits at p (and have some new applications), depend in an essential way on having a p -adic L -function L p ( V ) with ‘maximally’ general and precise interpolation properties. The construction of L p ( V ) is the mainresult of this paper. It is new at least when E/F does not split above p ; for a discussion of previousrelated works see § 1.1.6.As for further arithmetic directions, (1) the main remaining goal is perhaps the full Iwasawa MainConjecture 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 derivativeof L p ( V ) (cf. [Dis/b, Theorem E]); in view of the p -adic Gross–Zagier formula of [Dis/b], this isequivalent to a suitable generalisation of Perrin-Riou’s main conjecture for Heegner points, which inits original form was recently proved by Burungale–Castella–Kim [BCK].Besides the specifics of our context, this work may also be viewed as a case study in: – p -adic L -functions in universal families which exactly interpolate ratios of L -values; (2) – constructing p -adic L -functions as ratios of families of global and local zeta integrals, where thelatter are interpolated using the local Langlands Correspondence in families (see [Dis20] andreferences therein). As far as the author knows, the present work is the first instance where thismethod is used for a non-abelian family. We move toward stating our main theorem, leaving a fewof of the detailed definitions of the objects involved to the body of the paper. p -adic automorphic representations . — Let G = Res F/ Q GL , H := Res E/ Q GL . If v is a place of Q , we denote by Σ v the set of embeddings F v ֒ → Q v . A (numerical) v -adic weightfor G is a tuple w := ( w , w = ( w τ ) τ ∈ Σ v ) of integers, all of the same parity, such that w τ ≥ for all τ . It is said cohomological if w τ ≥ for all τ . A weight for H is a tuple l = ( l , l = ( l τ )) of integersof the same parity. Finally, if w and l are weights for G and H , the associated contracted weight for G × H is (3) ( w + l , 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 , ( 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 .) 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 ( w ,w τ ) , where π ( w ,w τ ) is the discrete series of GL ( F τ ) of weight w τ and central character z z w . If L is p -adic,we define an automorphic representation of G( A ) of weight w over L to be a representation π of (1) This discussion has no ambition of being either a comprehensive research programme or a comprehensive survey ofthe growing literature on the subject. Moreover, we have entirely left out both the case of non-ordinary families, andthe p -adic L -function with complementary (to (1.1.5)) interpolation range introduced in [BDP13]. (2) This can be considered largely a matter of emphasis, since it is not difficult to deduce a few results of this type fromHida’s work, cf. [Hid96]. (3)
A contracted weight is the same as a weight for (G × H) := { ( g, h ) : det( g ) = N E/F ( h ) } ⊂ G × H . This is in fact thetrue group governing our constructions. -ADIC L -FUNCTIONS FOR GL × GU(1) 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. (4) To the representation π over L is attached a -dimensional representation V π of G F := Gal( F /F ) characterised by L ( s, V π,v ) = L ( s − / , π v ) (this is the ‘Hecke’ normalisation of the Langlands cor-respondence, cf. [Del73, §3.2]). We say that π is ordinary if for each v | p , the restriction V π,v of V π to a decomposition group G F v at v admits (possibly after taking a finite extension L ′ /L ) a nontrivialfiltration(1.1.1) → V + π,v → V π,v → V − π,v → such that the character α ◦ π,v : F × v → L ′× corresponding to (5) V + π,v ( − 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 , (6) andlet 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 ) = Y τ : F ֒ → L τ ′ ( t p ) ( l τ + l ) / τ ′ c ( t p ) ( − l τ + l ) / for all t p in some neighbourhood of ∈ E × p . We let V χ be the -dimensional G E -representationcorresponding to χ . L -values . — Let π , respectively χ , be a complex automorphic representation of G( A ) , respec-tively H( A ) , and let π E denote the base-change of π to E . Let V v (respectively V π,v ) be the restrictionto G E v := Q w | v G E,w (respectively G F v ) of the Galois representation associated with π E ⊗ χ (respec-tively π ) if v is finite, and the Hodge structure associated with π E,v ⊗ χ v (respectively π v ) if v |∞ . Letus also introduce the convenient notation“ V ( π,χ ) ,v := ( V π,v ⊗ Ind E v F v χ v ) ⊖ ad( V π )(1) ”(to be thought of as referring to a ‘virtual motive’).Let η : F × A /F × → {± } be the character associated with E/F , and let(1.1.2) L ( V ( π,χ ) ,v ,
0) := ζ F,v (2) L (1 / , π E,v ⊗ χ v ) L (1 , η v ) L (1 , π v , ad) · if v ∤ ∞ π − if v |∞ ∈ C , L ( V ( π,χ ) ,
0) := Y v L ( V ( π,χ ) ,v , , where the product (in the sense of analytic continuation) is over all places. These are the L -values wewill interpolate. Interpolation factors . — Let L be a finite extension of Q p , let π be an ordinary automorphicrepresentations of G( A ) over L , with a locally algebraic central character ω π : A ∞ , × F → L × , and let χ : H( F ) \ H( A ) → L × be a locally algebraic character, with ω χ := χ | A ∞ , × F . Let ι : L ֒ → C be anembedding and let ψ = Q v ψ v : F \ A F → C × be the standard additive character such that ψ ∞ ( x ) = e πi Tr F ∞ / R ( x ) .If v | p , let ad( V π,v )(1) ++ := Hom ( V − π,v , V + π,v ) , and define e v ( V ( π ι ,χ ι ) ) = Q w | v γ ( ι WD( V + π,v | G E,w ⊗ V χ,w ) , ψ E,w ) − γ ( ι WD(ad( V π,v )(1) ++ , ψ v ) − · L ( V ( π ι ,χ ι ) ,v ) − , (4) This definition is slightly different from, but equivalent to, the one adopted in [Dis/b], whose flexibility won’t beneeded here. (5)
The class field theory map F × v → G ab F v is normalised by sending uniformisers to geometric Frobenii. (6) This will only intervene in the numerical labelling of the weights.
DANIEL DISEGNI where ι WD is the functor from potentially semistable Galois representations to complex Weil–Deligne representations of [Fon94], the inverse Deligne–Langlands γ -factor is γ ( W, ψ v ) − = L ( W ) /ε ( W, ψ v ) L ( W ∗ (1)) , (7) and ψ E,w = ψ v ◦ Tr E w /F v .Let k ∈ Z be such that the archimedean component of ω = ω π ω χ is ω ∞ ( x ) = x k . We define e ∞ ( V ( π ι ,χ ι ) ) = i k [ F : Q ] , and(1.1.3) e p ∞ ( V ( π ι ,χ ι ) ) := e ∞ ( V ( π ι ,χ ι ) ) · Y v | p e v ( V ( π ι ,χ ι ) ) Hida families . — Let U p G ⊂ G( A p ∞ ) be any open compact subgroup, and let T sph , ord U p be the p -(nearly) ordinary spherical Hecke algebra acting on ordinary p -adic modular cuspfoms for G of tamelevel 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 U p G , Q p for some U p G . It is a scheme finite flat over Spec Z p J T , . . . , T [ F : Q ]+1+ δ F,p K ⊗ Z p Q p (where δ F,p is the p -Leopoldtdefect of F ), coming with a dense ind-finite subscheme X clG ⊂ X G of classical points , and a locally free sheaf V G of rank endowed with a representation of G F :=Gal( F /F ) . To each x ∈ X cl is associated an automorphic representation π x of G( A ∞ ) over Q p ( x ) ,and the fibre V G | x ∼ = V π x . The (numerical) weight of x is defined to be the weight of π x .Let U p H ⊂ H( A p ∞ ) be an open compact subgroup. We define(1.1.4) Y H = Y H ,U p H := Spec Z p J H( Q ) \ H( A p ∞ ) /U p K ⊗ Z p Q p , where the topology on H( Q ) \ H( A p ∞ ) /U p is profinite; it comes with a universal character χ univ : H( Q ) \ H( A ∞ ) → O ( Y H ,U H ) × , identified with a G E -representation V H of rank , and adense ind-finite subscheme Y clH ⊂ Y H , whose points y correspond to U p H -invariant locally algebraicHecke 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 G × H is Y G × H := Y G ˆ × Y G := Spec T sph , ord U p G , Z p ˆ ⊗ Z p Z p J H( Q ) \ H( A p ∞ ) /U p K ⊗ Z p Q p . A Hida family for G × H is an irreducible component of Y G × H . We now isolate an interesting subspaceof Y G × H . Let ω G : F × \ A ∞ , × F → O ( Y G ) × be the character giving the action of the centre of G( A ) on p -adic modular forms, let ω H := χ univ | A ∞ , × F , and let ω := ω G ω H : F × \ A ∞ , × F → O ( Y G × H ) × . The self-dual locus Y sdG × H ⊂ Y sdG × H is the closed subspace defined by ω = . Main theorem . — Let X be a Hida family for G × H . Denote X sd := X ∩ Y sdG × H , and X cl , sd := X cl ∩ X sd .Throughout this paper, if X is a scheme over a characteristic-zero field L , we identify a point x ∈ X ( C ) with a pair ( x , ι ) where x ∈ X is a point and ι : L ( x ) ֒ → C is an embedding. (We willoften abuse notation by writing x in place of x .) Theorem A . —
Assume that X sd is non-empty. There exists a unique meromorphic function L p ( V ) ∈ K ( X ) (7) The normalisations of L - and ε -factors are as in [Tat79]. -ADIC L -FUNCTIONS FOR GL × GU(1) whose polar locus D does not intersect X cl , sd , such that for each ( x, y ) ∈ X cl ( C ) − D ( C ) whosecontracted weight ( k , w, l ) satisfies (1.1.5) | l τ | ≤ w τ − , | k | ≤ w τ − | l τ | − for all τ ∈ Σ ∞ , we have (1.1.6) L p ( V )( x, y ) = e p ∞ ( V ( π x ,χ y ) ) · L ( V ( π x ,χ y ) , . Here, if ( x , y ) ∈ X cl is the point underlying ( x, y ) and ι : Q p ( x , y ) ֒ → C is the correspondingembedding, we have denoted π x = π ιx , χ y = χ ιy , and the interpolation factor is as in (1.1.3) . The value of the interpolation factor agrees with the general conjectures of Coates and Perrin-Riou(see [Coa91]).
Remark 1.1.1 . —
This confirms [Dis/b, Conjecture ( L p ) of § 1.4], up to the possible poles. (Thepower of the discriminant in loc. cit. , as well as the precise description of the ring of rationality, arenot correct.) Previous related work . — When
E/F splits above p , Theorem A may be essentially deducedfrom the main result of [Hid91] (see also [Hid09]), cf. [Dis/b, § 7.1]. Hida’s method uses the Rankin–Selberg integral, whereas ours uses Waldspurger’s variant [Wal85] based on the Weil representation (asdiscussed below).The numerator of our L -value is a special case of the standard L -function for GL × GL over E , andwhen so considered our p -adic L -function is a multiple of the restriction to some base-change locus ofone constructed by Januszewski [Jan] (using the method of modular symbols); however that functiondoes not have the canonical interpolation properties that we seek.Finally, when F = Q and p is inert in E , a variant of L p ( V ) was recently constructed by Loefflerand Büyükboduk–Lei (see [BL, §B.4]) under various local restrictions. The proof combines the strategy of Hida[Hid91] with that of [Dis17, Proof of Theorem A], where we had constructed the ‘slices’ L p ( V )( x, − ) for x ∈ X clG of weight .We start from Waldspurger’s [Wal85] integral representation of Rankin–Selberg type(1.2.1) ( f, I ( φ, χ )) = L ( V ( π,χ ) , · Y v R v ( W v , φ v , χ v , ψ v ) where ( , ) is a normalised Petersson product, f is a form in π with Whittaker function W , the form I ( φ ) is a mixed theta-Eisenstein series depending on a certain Schwartz function φ , and the R v arenormalised 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 ) into a Y G × H -adic modular form. In § 4, we interpolate R v for v ∤ p ∞ using the local Langlands correspondence in families; we compute R v for v | p ∞ , whichyield the interpolation factors in (1.1.6); and finally, we use (1.2.1) to define L p as a quotient of theglobal 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. Acknowledgments. —
I am grateful to David Callan, Haruzo Hida, Ming-Lun Hsieh and XinyiYuan for useful correspondence. p -adic modular forms and Hida families The material of this section is largely due to Hida (see [Hid91, §§ 1–3, 7] and references therein).
DANIEL DISEGNI
The notation introduced in the present subsection (or in theintroduction) will be used throughout the paper unless otherwise noted.
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 placeof 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 relativediscriminant of E/F ; for a finite place v of F we denote by d v ∈ F v a generator of the differentideal 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 > } and F + := { x ∈ F | x > } ⊂ F × ; – A is the ring of adèles of Q ; if S is a finite set of places of a number field F , we denote by A SF = Q ′ v / ∈ S F v , and F S := Q v ∈ S F v ; when S consists of the set of places of F above some finiteset of places of Q (for instance the place p ) we use the same notation with those places of Q instead of S (for instance, F p = F S p ). We denote A + F := A ∞ F × F + . – we denote by ψ : F \ A F → C × the standard additive character as in § 1.1.3; – we denote by G F the absolute Galois group of a field F ; – if F is a number field, its class number is denoted by h F := | F × \ A ∞ , × F / b O × F | – for a place v of F , we denote by ̟ v a fixed uniformiser at v , by q F,v the cardinality of the residuefield; we denote q F,p := ( q F,v ) v ∈ S p ; – the class field theory isomorphism is normalised by sending uniformisers to geometric Frobenii;for F a number field (respectively a local field), we will then identify characters of G F withcharacters of A × F /F × (respectively F × ) without further comment; – If I is 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 := Q i x y i i whenever that makes sense. – we denote by [ · ] the { , } -valued function on logical propositions such that [ φ ] = 1 if and onlyif φ is true; – if R/R is a ring extension, A is an R -algebra, and X is an R -scheme, we denote A R := A ⊗ R R, X R := X × Spec R R. Subgroups of G and special elements . — We denote by Z , A , and N respectively the centre,diagonal torus, and upper unipotent subgroup of G ; we let P = AN and P := P ∩ Res F/ Q SL . Wedenote by w := − ! ∈ GL ( F ) or its image in GL ( A ) for any F -algebra A . For r ∈ Z S p ≥ , we define w r v ,v := − ̟ r v v ! ∈ GL ( F v ) , w r,p := Y v | p w r v ,v ∈ GL ( F p ) , as well as a sequence of compact subgroups U v,r v := U ( ̟ r v v ) := { (cid:0) a bc d (cid:1) ∈ GL ( O F,v ) : a − ≡ d − ≡ c ≡ ̟ r v v ) } ⊂ GL ( F v ) U p,r := Y v ∈ S p U v,r v . For θ ∈ ( R / π Z ) S ∞ , we denote r θ := cos θ v sin θ v − sin θ v cos θ v !! v ∈ SO(2 , F ∞ ) -ADIC L -FUNCTIONS FOR GL × GU(1) Hecke algebras . — Let S ⊂ S ′ be finite sets of non-archimedean places of F and let U S = Q v / ∈ S U v ⊂ GL ( A S ∞ F ) be an open compact subgroup. For each finite set of finite places S , we definethe Hecke algebra H S ′ U S := C c ( U S ′ \ GL ( A S ′ ∞ F ) /U S ′ , Q ) = Y v / ∈ S H U v . It carries an involution(2.1.1) T T g arising from the map g g − on the group G .Let A p := A ( Q p ) ⊂ G( Q p ) be the diagonal torus, and let A + p be the set of t = (cid:0) t t (cid:1) such that v ( t ) ≥ v ( t ) for all v | p . The involution(2.1.2) t t g := det( t ) − t preserves A + p . For S a finite set of places of F disjoint from S p ∪ S ∞ , we define the ordinary Heckealgebra H ord U Sp := H SpU Sp ⊗ Q p [ A p ] over Q p , which will act on spaces of ordinary modular forms. It is endowed with the involution g deduced from (2.1.1) and (2.1.2). Measures . — We use the same notation and conventions for Haar measures and integration asin [YZZ12, § 1.6] and [Dis17, § 1.9]. In particular, we have a regularised integration functional Z ∗ E × \ A × E / A × F f ( t ) dt, which satisfies the following. Lemma 2.1.1 . —
Let f be a smooth function on A × E that is invariant under E ×∞ . Let µ ⊂ O × E be afinite index subgroup fixing f (under the scaling action). Then Z ∗ E × \ A × E / A × F X x ∈ E × f ( xt ) dt = 2 L (1 , η ) h E [ O × E : µ ] Z A ∞ , × E X α ∈ µ f ( αt ) d • t, where d • t is the Haar measure giving volume to b O × E .Proof . — Let U ⊂ A ∞ , × E be any compact open subgroup fixing f . Since both sides are independentof µ , we may assume that µ = O × E ∩ U . By [YZZ12, (1.6.1) and following paragraphs], we have(2.1.3) Z ∗ E × \ A × E / A × F f ( t ) dt = vol( E × \ A × E / A × F ) − Z E × \ A × E / A × F f ( t ) dt = vol( E × \ A × E / A × F ) | E × \ A ∞ , × E /U | X t ∈ E × \ A ∞ , × E /U f ( t ) . Now by a coset identity, | E × \ A ∞ , × E /U | = h E | b O × E /U | [ O × E : µ ] and by [YZZ12, §1.6.3], vol( E × \ A × E / A × F ) = 2 L (1 , η ) . Hence (2.1.3) equals L (1 , η ) h E [ O × E : µ ] | b O × E /U | X t ∈ E × \ A ∞ , × E /U f ( t ) If we compose with the operator f ( · ) P x ∈ E × f ( x · ) = P x ∈ µ \ E × P α ∈ µ f ( αx · ) , we obtain L (1 , η ) h E [ O × E : µ ] | b O × E /U | X t ∈ A ∞ , × E /U X α ∈ µ f ( αt ) = 2 L (1 , η ) h E [ O × E : µ ] Z A ∞ , × E X α ∈ µ f ( αt ) d • t. DANIEL DISEGNI q -expansions. — Let h ⊂ C be the upper half-plane. We view G( R ) as acting on h Σ ∞ by Möbius transformations, and identify C + ∞ := ( R + SO(2 , R )) Hom ( F, R ) ⊂ G( R ) with the neutral connected component of the stabiliser of i := ( √− , . . . , √− ∈ h Hom ( F, R ) . Nearly holomorphic modular forms . — A complex nearly holomorphic (Hilbert) modular form of weight w , level U , degree ≤ m = ( m τ ) , is a function f : GL ( A F ) → C satisfying the following two conditions:1. for all g ∈ G( A ) , γ ∈ G( Q ) , k ∈ U C + ∞ , f ( γgk ) = j w ( k ∞ , i) − f ( g ) , where for z ∈ h Hom ( F, R ) , j w ( (cid:0) a bc d (cid:1) τ , z τ ) := ( ad − bc ) ( w − w ) / ( cz + d ) w .
2. There is a Whittaker–Fourier expansion(2.2.1) f y x !! = | y | X a ∈ F W ♯a ( y )( Y ) q a for all y ∈ A + F , x ∈ A F , where: – we have W ♯ ( y ) = y ( − w + w − / ∞ W ( y ) , W ♯a ( y ) = ( ay ∞ ) ( w + w − / W a ( y ) for polynomials W a ( y ) = W f,a ( y ) ∈ C [( T τ ) τ : F ֒ → R ] of degree ≤ m τ in the variables T τ , evaluated at Y := ( Y τ ) τ : F ֒ → R with Y τ = (4 πy τ ) − ; – we denote q a = q ay ∞ := ψ ( ax ) ψ ∞ (i ay ∞ ) . The polynomial W a ( y ) only depends on the class of ay modulo U ∩ (cid:16) A ∞ , × F (cid:17) , so that we may write W f,a ( y ) = W f ( ay ) for W f ( a ) := W f, ( a ) . We say that f is cuspidal if W ( y ) = 0 for all y .If f is nearly holomorphic of degree (that is, ≤ (0 , . . . , ), we simply say that f is a (holomorphic)modular form. If R ⊂ C is any subring, we denote by S w ( U, R ) ⊂ M w ( U, R ) ⊂ N ≤ mw ( U, R ) respectively the spaces of cuspidal forms and holomorphic forms of level U and weight w , and of nearlyholomorphic forms of level U , weight w , and degree ≤ m = ( m τ ) , such that for all a ∈ A + F , thepolynomials W f ( a ) have coefficients in R . We write N w ( U, R ) := lim −→ m N ≤ mw ( U, R ) . Twisted modular forms . — A twisted nearly holomorphic (Hilbert) modular form of weight w ,level U , degree ≤ m = ( m τ ) , is a function f : GL ( A F ) × A × F → C satisfying the following two conditions:1. for all g ∈ G( A ) , γ ∈ G( Q ) , k ∈ U C + ∞ , f ( γgk, det( γ ) − u ) = j w ( k ∞ , i) − f ( g, u ); -ADIC L -FUNCTIONS FOR GL × GU(1)
2. there is a Whittaker–Fourier expansion(2.2.2) f y x !! = | y | X a ∈ F W ♯a ( y, u )( Y ) q a for all x ∈ A F and y, u ∈ A F such that ( uy ) ∞ > , where: W ♯ ( y, u ) = y ( w + w − / ∞ W ( y, u ) , W ♯a ( y, u ) = ( ay ∞ ) ( w + w − / W a ( y, u ) for polynomials W a ( y, u ) = W a,f ( y, u ) ∈ C [( T τ ) τ : F ֒ → R ] of degree ≤ m τ in the variables T τ , evaluated at Y := ( Y τ ) τ : F ֒ → R with Y τ = (4 πy τ ) − .If R ⊂ C is any subring, we denote by M tw w ( U, R ) ⊂ N tw , ≤ mw ( U, R ) the spaces of holomorphic andnearly 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 U F ⊂ b O × F , let(2.2.3) µ U F := F × ∩ U F , w U F := |{± } ∩ µ U F | , c U F = w U · [ F : Q ] h F [ O × F : µ U F ] . Let ϕ : A × F → C be a Schwartz function, invariant under a subgroup of the form µ U ′ F ⊂ F × as above.Then the sum(2.2.4) ⋆ X u ∈ F × ϕ ( u ) := c U F X u ∈ µ UpF \ F × ϕ ( u ) is well-defined independently of U F ⊂ U ′ F , and for any such choice the support of the sum is finite.If f , f are twisted nearly holomorphic forms, we may thus define a (plain) nearly holomorphicform f ⋆ f by(2.2.5) f ⋆ f ( g ) := ⋆ X u ∈ F × f ( g, u ) f ( g, u ) . Differential operators . — We attach to a nearly holomorphic (genuine or twisted) form f thefunction f h : G( A ∞ ) × h Hom ( F, R ) → C ( g ∞ , z = g ∞ i) j w ( g ∞ , i) f ( g ∞ ); the map f f h is injective.The Maass–Shimura differential operators on functions on h Hom ( F, R ) are defined as follows. For τ : F ֒ → R , k ∈ Z , let δ τ, h k := 12 πi (cid:18) k iy τ + ∂∂z τ (cid:19) , d τ := 12 πi ∂∂z τ , a differential operator on the upper half-plane h . For w, k ∈ Z Hom ( F, R ) ≥ , let δ k, h w := Y τ δ τ, h w τ +2 k τ ◦ · · · ◦ δ τ, h w τ +2 ◦ δ τ, h w τ , d k := Y τ ( d τ ) k τ . Then for any ring Q ⊂ R ⊂ C , this operator defines a map δ kw : N (tw) ≤ mw ( U, R ) → N ≤ m + k (tw) ,w +(0;2 k ) ( U, R ) such that δ kw ( f ) h = δ k, h w ( f h ) . (For a proof of the intuitive fact that the archimedean operator δ rw indeedpreserves the rationality properties of finite Whittaker–Fourier coefficients, see [Hid91, Proposition 1.2],whose calculations also apply to the twisted case.) DANIEL DISEGNI
By [Shi81, (1.16)], for all k ∈ Z Σ ∞ ≥ we have(2.2.6) δ kw = X ≤ j ≤ k Y τ ∈ Σ ∞ (cid:18) k τ j τ (cid:19) Γ( w τ + k τ )Γ( w τ + j τ ) ( − πy τ ) j τ − k τ d j . If w ≥ m + 1 , any f ∈ N (tw) , ≤ m w ( U, R ) can be written uniquely as f = X ≤ r ≤ m δ rw − r f r with f r ∈ M (tw) w +(0; − r ) ( U, R ) . (The proof in [Shi76, Lemma 7] carries over to our context.) Thus thelinear map(2.2.7) e hol : N (tw) , ≤ m w ( U, R ) → M (tw) w ( U, R ) f f is well-defined. p -adic modular forms. —2.3.1. Arithmetic q -expansion . — The q -expansion map f ( a W ,f ( a )) sends S w ( U, C ) to C A + F /U F F + ∞ , where U F = U ∩ A ∞ , × F . 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 ) .If R is any ring admitting embeddings into C , we denote by S • ( U, R ) ⊂ R A + F /U F F + ∞ the set of those sequences f = ( W a ) a such that for any ι : R ֒ → C , the sequence ( ιW a ) a is the q -expansion of a cuspform f ι ∈ S • ( U, C ) = M w S w ( U, C ) . By [Hid91, Theorem 2.2 (i)] (together with a consideration of Galois actions mixing the weights), forany such ring R we have S • ( U, R ) = S • ( U, Z ) ⊗ R . For more general rings, the previous equality istaken to be the definition of S • ( U, R ) .It is also convenient to attach to f ∈ S • ( U, R ) and an embedding ι as above the anti-holomorphiccusp form f ι, a := − ∞ ∞ ! f ι . p -adic modular forms . — Let L be a finite extension of Q p splitting F . A p -adic L -valued(cohomological) weight w = ( w , ( w τ ) τ : F ֒ → L ) is a tuple of integers, all having the same parity, suchthat w τ ≥ for all τ : F ֒ → L . If ι : L ֒ → C is an embedding, the associated complex weight is w ι = ( w , ( w τ ) ι ◦ τ ) . If w is a p -adic weight valued in L , 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,ψ has weight w ι . The p -adic q -expansion of f = ( W a ) a ∈ S w ( U, L ) is the sequence f ( p ) = ( W ( p ) a ) , W ( p ) a := a ( w + w − / p W a , so that(2.3.1) W ♯a ( y ) := ( ay ) ( w + w ι +2) / ∞ ι (cid:16) ( ay p ) ( − w − w − / W ( p ) a (cid:17) is the Whittaker–Fourier coefficient of f ι as in (2.2.1). -ADIC L -FUNCTIONS FOR GL × GU(1) We denote(2.3.2) S ( p ) w ( U, L ) := { f ( p ) | f ∈ S ( p ) w ( U, L ) } ⊂ L A + F /U F F + ∞ , S ( p ) w ( U p , L ) := lim −→ n S ( p ) w ( U p U p,n , L ) . Let U pF = U p ∩ A p ∞ , × F . The space of cuspidal p -adic modular forms S ( U p , L ) ⊂ L A ∞ , × F /U pF is the completion of S ( p ) w ( U p , L ) for the norm || ( W ( p ) a ) a || := sup a | W ( p ) a | , for any w . By a fundamentalresult of Hida (see [Hid91, paragraph after Theorem 3.1]), the space S ( U p , L ) is independent of thechoice of w . In particular, if L is a 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 ) ⊗ Q p L for a space S ( U p , Q p ) . Nearly holomorphic forms as p -adic modular forms . — We may attach a p -adic q -expansion toa 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 f = ( W ( p ) a ) ∈ L A + F /F + ∞ is a p -adic nearly holomorphic form of weight w and level U p ⊂ GL ( A p ∞ F ) if the following conditionholds. For each ι : L ֒ → C , there exists a nearly holomorphic form f ι ∈ N ≤⌊ ( w +1) / ⌋ w ι ( U p U p,n , C ) for some n ∈ Z S p ≥ , whose Whittaker–Fourier polynomials have constant terms satisfying(2.3.3) W f ι ,a (0) = ι (cid:16) a ( − w − w − / p W ( p ) a (cid:17) . The notion of a p -adic twisted nearly holomorphic form is defined similarly by the identity W f ι ,a ( y, u )(0) = ι (cid:16) ( ay ) ( − w − w − / p W ( p ) a ( y, u ) (cid:17) . Proposition 2.3.1 . — If f is a p -adic nearly holomorphic cuspform over L of level U p , then itbelongs to the space S ( U p , L ) of p -adic modular cuspforms of level U p .Proof . — This is the first assertion of [Hid91, Proposition 7.3]. Hecke operators and ordinary projection . — The space N w ( U, C ) is endowed with the usualaction of H U . By writing down the effect of this action on Whittaker–Fourier coefficients of cuspforms,this extends to a bounded action of H U p on S ( p ) ( U p , Q p ) , hence on S ( U p ) .For t ∈ A + p or y ∈ Q v | p O F,v − { } , and any n ∈ Z S p ≥ , define the double coset operators(2.3.4) U t := [ U p,n t U p,n ] , U ◦ ,wt := t − w det( t ) ( − w + w − / U t , U y := U( y ) , U ◦ ,wy := y ( − w − w − / U y If L is a finite extension of Q p , then for all y ∈ Q v | p O F,v − { } we also define the operator on S ( U p )U ◦ y : S ( U p , Q p ) → S ( U p , Q p ) W U ◦ y f ( c ) := W f ( cy ) . This is compatible with the previous definition in the following sense (see [Hid91, (2.2b)], where U y isdenoted by T ( y ) ): if f is a p -adic nearly holomorphic form of weight w over L , then for all ι : L ֒ → C we have (U ◦ y f ) ι = U ◦ ,w ι y f ι . DANIEL DISEGNI
By abuse of notation, the superscript w will be omitted when understood from context. The ordinaryprojector is(2.3.5) e ord := lim n →∞ (U ◦ p ) n ! ⊂ End ( S ( U p , L )) for any tame level U p and p -adic field L . Its image is denoted by S ord ( U p , L ) := e ord S ( U p , L ) . If π is an automorphic representation of G( A ) over L , then π ord := lim −→ n e ord π U p,n is the space ofordinary forms.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 . Differential operators after ordinary and holomorphic projections . — Let L be a finite extensionof Q p , and let f , f be p -adic twisted nearly holomorphic forms. For any ι : L ֒ → C , we have(2.3.6) [ e ord ( f ⋆ d r f )] ι = e ord ,ι [ e hol ( f ι ⋆ δ r f ι )]; the proof of [Hid91, Prop. 7.3] carries over to the twisted case. We gather the fundamental notions concerning Hida families and the asso-ciated sheaves of modular forms.
Weight space . — Let U ◦ F,p = Q v | p U ◦ F,v ⊂ O × F,p be a compact open subgroup (which will befixed once and for all in § 2.4.4). Let U pF ⊂ A p ∞ , × F be a compact open subgroup, and consider thetopological groups (with the profinite topology) [ Z ] U pF := Z ( F ) U pF \ Z ( A ∞ F ) , [ Z ] U pF × U ◦ F,p ; the latter is isomorphic to ∆ × Z F : Q ]+ δ F,p p , where ∆ is a finite group and δ F,p is the p -Leopoldtdefect of F . It is embedded into A ( A ∞ F ) by ( z, y p ) zy p z ! . The weight space (of tame level U pF ) is(2.4.1) W = W U pF := Spec Z p J [ Z ] U pF × U ◦ F,p K Q p . A point κ ∈ W is identified with the pair of characters(2.4.2) (cid:16) κ := κ | [ Z ] UpF , κ = κ | U ◦ F,p (cid:17) . We have an involution defined by κ g ( t ) := κ (det t ) − κ ( t ) . If k is a p -adic weight for G , we say that κ is classical of weight k if for all v | p , κ sm0 ( z p ) := κ ( z p ) z − k p , κ sm v ( y ) := κ v ( y ) Y τ | v τ ( y ) ( − k − k τ − / are smooth characters of F × p , respectively U ◦ F,v , where the product runs over the τ ∈ Σ p inducing theplace v ∈ S p . For a classical weight κ , we define(2.4.3) κ ′ := κ sm κ sm , − ,p = κ g , sm , a smooth character of U ◦ F,p . -ADIC L -FUNCTIONS FOR GL × GU(1) We denote by W cl ⊂ W 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 , k ) , then κ g is classical of weight k ∨ = ( − k , k ) . We let W cl , ≥ bethe set of classical points satisfying k ≥ . Hida families . — Let S ord , a ( U p ) be the space S ord ( U p ) endowed with the dualised action T.f := T g f of H ord U p . Since for every f ∈ S ( p ) ( U p , Q p ) and ι : Q p ֒ → C we have T f ι = ( T g f ) ι , the space S ord , a ( U p ) may be identified with the completion of the space of anti-holomorphic ordinarycuspforms. Let T sph , ord U p ⊂ T ord U p ⊂ End ( S ord , a ( U p , Q p )) be the images of H sph , ord U p , H ord U p . We let Y G ,U p := Spec T sph , ord U p be the ordinary eigenvariety for G , which by construction carries a (coherent) sheaf S U p whose module of global sections is S ord , a ( U p ) . (8) The space Y G ,U p , called the ordinary eigenvariety for G of tame level U p , is a union of finitely many irreducible components, called Hida families of tamelevel (dividing) U p .Letting U pF := U p ∩ Z ( A p ∞ ) , we have a weight-character map κ G : Y G ,U p → W U pF which, when identified with a pair ( κ G , , κ G ) of O ( Y G ) × -valued characters as in (2.4.2), is κ G , ( z ) = T ( z ) , κ G ( y p ) = U ◦ y p . The weight map is finite and flat and it intertwines the involutions g .The set of classical points of Y G is Y clG := Y G × W W cl , ≥ ⊂ X . If x ∈ Y clG , we denote by π x the automorphic representation of G( A ) on which H sph U p acts by the Q p ( x ) -character corresponding to x . If x ∈ Y clG ( C ) corresponds to ( x ∈ Y clG , ι : Q p ( x ) ֒ → C ) , wedenote π x := π ιx . Hida schemes . — In light of the examples of W and Y G , it will be convenient to introduce asuitable category of spaces. (9) Define the category of
Hida rings to consist of finite flat Z p J X , . . . , X n K -algebras A ◦ (for some n ) and continuous maps, and the category of Hida algebras to be the image ofHida rings under the functor ⊗ Z p Q p . Define the category of affine Hida schemes to be dual to thecategory of Hida algebras. A Hida scheme is an open subset of an affine Hida scheme. If A ◦ i are Hidarings (for i = 1 , ) and X i = Spec ( A ◦ i ⊗ Z p Q p ) (for i = 1 , ), we define X ˆ × X := Spec ( A ◦ ˆ ⊗ A ◦ ) Q p , where ˆ ⊗ is the completed tensor product. Weight-character map for H . — Let Y H = Y H ,U p H := Spec Z p J H( Q ) \ H( A p ∞ ) /U p K ⊗ Z p Q p (8) We have constructed Y G based on anti-holomoprhic forms for convenience reasons when considering the interpolationof Petersson products in § 4.1. (9) The treatment proposed here is minimal and somewhat ad hoc, but it will be sufficient for our purposes. We believethat a more systematic treatment of the geometry of Hida theory should be based on the theory of uniformly rigid spacesdeveloped in [Kap12]. DANIEL DISEGNI as in (1.1.4). A (Hida) family for H is a connected component of Y H .Fix a sufficiently small open compact subgroup U ◦ , √ F,p = Q v | p U ◦ , √ F,v ⊂ O × F,p and an injective grouphomomorphism j ′′ : ( U ◦ , √ F,p ) → O × , E,p := { t ∈ O × E,p | N E p /F p ( t ) = 1 } , and let U ◦ F,p := ( U ◦ , √ F,p ) ⊂ O × F,p , which is now fixed as promised in § 2.4.1. Let √ : U ◦ F,p → U ◦ , √ F,p be the square root, and let and let j, j ′ : U ◦ F,p → O × E,p be the maps (10) (2.4.4) j ′ ( a ) := j ′′ ( √ a ) / √ a, j ( a ) = j ′ ( a ) a. For any open compact U pE ⊂ A p ∞ , × E and U pF := U pE ∩ A p ∞ , × F , define a map(2.4.5) κ H : Y H ,U pE → W U pF y κ H ( y ) = (cid:16) κ = χ y | [ Z ] UpF , κ := χ y ◦ j (cid:17) . The classical points is Y clH := Y H × W W cl . Note that if y ∈ Y H is a classical point such that κ H ( y ) has weight ( l , l ) , then χ y has weight ( l , l ) asdefined in the introduction. Hida families for G × H . — These are defined as in § 1.1.4. Universal automorphic sheaf on a Hida family . — Let X be a Hida family for G (of arbitrarytame level), and let X cl := X ∩ Y clG . For each sufficiently small U p , we may view X ⊂ Y G ,U p andwe define Π U p = Π U p X := S U p | X . For each x ∈ X cl , by Hida’s Control Theorem (see for instance [Hid91, Corollary 3.3]) and the theoryof newforms, we have an isomorphism of H U p -modules,(2.4.6) Π U p | x ∼ = π U p , ord x := e ord π U p x . Let U p X be minimal such that X is a component of Y G ,U p X . By [Hid91, §3], there is a unique(2.4.7) f = f , X ∈ Π U p X X (the normalised primitive form over X ) whose first coefficient is ( f ) = 1 . Any f ∈ Π U p can bewritten as f = T f for some Hecke operator T supported at the places v ∤ p ∞ such that U p is notmaximal. Universal Galois sheaf on a Hida family and local-global compatibility . — Let X G be a Hidafamily for G . By results of Hida and Wiles (see [Dis/b, Proposition 3.2.4]), there exist an open subset X ′ G ⊂ X G containing X clG and a locally free sheaf V = V X ′ G of rank , endowed with a Galois action G F → End O X ′ G ( V ) such that for all x ∈ X clG , the fibre V | x is the Galois representation attached to π x by the globalLanglands 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 of X G is maximal at v . We define Π Sp X G := lim −→ U S Π U Sp U S X G , (10) If v | p splits in E , then for a ∈ U ◦ F,v we have j ( a ) = ( a, under some isomorphism E × v ∼ = F × v × F × v . -ADIC L -FUNCTIONS FOR GL × GU(1) which is a finitely generated O X G [GL ( F S )] -module. On the other hand, [Dis20, Theorem 4.4.1]attaches to the restriction V v := V | G Fv an O X ′ G [GL ( F S )] -module Π( V v ) , which is torsion-free and co-Whittaker in the sense of [Dis20, Definition 4.2.2]. Proposition 2.4.1 . —
After possibly replacing X ′ G with another open subset X ′′ G ⊂ X ′ G still con-taining X clG , there exists a line bundle Π ◦ X ′ G over X ′ G with trivial GL ( F S ) -action, such that Π | X ′ G ∼ = Π ◦ X ′ G ⊗ O v ∈ S Π( V v ) . 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 clG and all places v , the GL ( F v ) -representation π x,v corresponds, under local Langlands, to the Weil–Deligne representation V x,v attached to V | x | G Fv .Then the result follows from [Dis20, Theorem 4.4.3]. Z -adic modular forms . — Let Z be a Hida scheme endowed with a map ϕ : Z → Y G . Ameromorphic cuspidal ordinary Z -adic modular form is a sequence f ∈ O ( Z ) A + F /F + ∞ ⊗ O ( Z ) K ( Z ) , such that for all z in some dense set of closed points Σ ⊂ Z , the specialisation f ( z ) is an ordinary p -adic modular form in S g | ϕ ( z ) ⊗ Q p ( ϕ ( z ) Q p ( z ) . Lemma 2.4.2 . —
The inclusion j of S g ⊗ O Y G K ( Z ) into the space of meromorphic ordinary Z -adic forms is an isomorphism.Proof . — Let f be a meromorphic Z -adic ordinary form. Up to multiplying by an element of O ( Z ) ,we may assume that the coefficients of f belong to a Hida ring A over B := T sph , ord Z p . Let p z ⊂ A bethe ideal corresponding to z ∈ Σ . Let n ∈ N and let N range among finite subsets of Σ ; the filteredsystem of ideals I n,N := ( p n ) + \ z ∈ N p z forms a fundamental system of neighbourhoods of ∈ A , i.e. A = lim ←− n,N A/I n,N . Now by assumption,the image of f in ( A/I n,N ) A + F /F + ∞ belongs to S ord Z p ⊗ B ( A/I n,N ) , where S ord Z p is the B -module of integralordinary p -adic modular forms. It follows that the inclusion j , restricted to integral Z -adic formswithout poles, is surjective modulo I n,N for all n, N , hence surjective.
3. Theta–Eisenstein family
In this section, we define the kernel of the Rankin–Selberg convolution giving the p -adic L -function. We recall the definition of the Weil representation for groups of simil-itudes; this subsection is largely identical to [Dis17, §3.1].
Local case . — Let V = ( V, q ) be a quadratic space of even dimension over a local field F ofcharacteristic not 2. Fix a nontrivial additive character ψ of F . For simplicity we assume V has evendimension. For u ∈ F × , we denote by V u the quadratic space ( V, uq ) . We let GL ( F ) × GO( V ) act onthe usual space of Schwartz functions (11) S ′ ( V × F × ) as follows (here ν : GO( V ) → G m denotes thesimilitude character): – r ( h ) φ ( x, u ) = φ ( h − x, ν ( h ) u ) for h ∈ GO( V ) ; – r ( n ( b )) φ ( x, u ) = ψ ( buq ( x )) φ ( x, u ) for n ( b ) ∈ N ( F ) ⊂ GL ( F ) ; (11) The notation is only provisional for the archimedean places, see below. DANIEL DISEGNI – r a d !! φ ( x, u ) = χ V u ( a ) | ad | dim V φ ( at, d − a − u ) ; – r ( w ) φ ( x, u ) = γ ( V u ) ˆ φ ( x, u ) for w = − ! . Here χ V = χ ( V,q ) is the quadratic character attached to V , γ ( V, q ) is a fourth root of unity, and ˆ φ 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 e.g. [JL70]): χ V is trivial if V is aquaternion algebra over F or V = F ⊕ F , and χ V = η if V is a separable quadratic extension E of F with associated character η . Fock model and reduced Fock model . — Assume that F = R and V is positive definite. Thenwe will prefer to consider a modified version of the previous setting. Let the Fock model S ( V × R × , C ) be the space of functions spanned by those of the form H ( u ) P ( x ) e − π | u | q ( x ) , where H is a compactly supported smooth function on R × and P is a complex polynomial functionon V . This space is not stable under the action of GL ( R ) , but it is so under the restriction of theinduced ( gl , R , O ( 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 ( uq ( x )) + sgn( u ) P ( uq ( x ))) e − π | u | q ( x ) where P , P are polynomial functions with rational coefficients.By [YZZ12, §4.4.1, 3.4.1], there is a surjective quotient map(3.1.1) S ( V × R × , C ) → S ( V × R × ) ⊗ Q C Φ φ ( x, u ) = Φ( x, u ) = Z R × − Z O ( V ) r ( ch )Φ( x, u ) dh dc. 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 S ( V × F × ) = S ( V × F × ) := S ′ ( V × F × ) . Global case . — Let ( V , q ) be an even-dimensional quadratic space over the adèles A = A F ofa totally real number field F , and suppose that V ∞ is positive definite; we say that V is coherent ifit has a model over F and incoherent otherwise. Given an b 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 thespherical elements φ v ( x, u ) = V v ( x ) ̟ nvv ( u ) , if ψ v has level n v . We call such φ v the standard Schwartz function at a non-archimedean place v . Wedefine similarly the reduced space S ( V × A × ) , which admits a quotient map S ( V × A × ) → S ( V × A × ) (3.1.2)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 ) × GL ( A ∞ ) × ( gl ,F ∞ , O ( V ∞ )) is the restricted tensor product ofthe local representations. The quadratic spaces of interest . — For a quadratic space V = ( V , q ) over A F , we define ε ( V ) = +1 (respectively − ) if and only if there exists (respectively does not exist) a quadratic space V over F such that V ⊗ F A F = V .In this paper, we will consider the quadratic spaces V = ( B , q ) , where B is a quaternion algebraover A F , definite at all the archimedean places and split at p , and endowed with an A F -embedding -ADIC L -FUNCTIONS FOR GL × GU(1) A E ֒ → B , and q : B = V → A F is its reduced norm. It has a decomposition V = V ⊕ V where V = A E (on which the restriction of q coincides with N E/F ) and V is the q -orthogonalcomplement. Thus ε ( V ) = ε ( V ) . We denote by r the restriction of r to a representation of A × E =GO( V ) on S ( V × A × F ) .For each place v , we define(3.1.3) ε ( B v ) = ε ( V v ) = +1 if B v ∼ = M ( F v ) − if B v is a division algebra.We have ε ( V ) := Q v ε ( V v ) = ( − [ F : Q ] Q v ∤ p ε ( V v ) . Let φ ∈ S ( V × A × F ) . We define a twisted automorphic form(3.2.1) θ ( g, u, φ ) := X x ∈ E r ( g ) φ ( x, u ) . It satisfies(3.2.2) θ ( zg, u, φ ) = η ( z ) θ ( g, u, r ( z − , φ ) for all z ∈ A × .For a complex weight l for H , let(3.2.3) φ ,l, ∞ := ⊗ v |∞ φ ,l,v ,φ ,l,v ( t, u ) := R + ( u ) t l v | u | ( − l + l v ) / e − πuq ( t ) if l v ≥ t ) − l v | u | ( − l − l v ) / e − πuq ( t ) if l v ≤ , and let θ ( g, u, φ ∞ ; l ) := θ ( g, u, φ ∞ φ ,l, ∞ ) . Define | l | := ( l , ( | l v | ) v ) . Lemma 3.2.1 . —
The series θ ( g, u, φ ∞ ; l ) is a twisted holomorphic form of weight (0 ,
1) + | l | .Proof . — The usual proof that classical theta series are automorphic shows that our θ is twistedautomorphic. The archimedean component of the central character is easy to determine by (3.2.2).The weight is computed as in [Xue07, §A1 on p. 350].The Whittaker–Fourier expansion of θ ( l ) is standard: for all g = ( y x ) ∈ GL ( A F ) with y ∈ A + F ,(3.2.4) θ ( g, u, φ ∞ ; l ) = X a ∈ F × X x ∈ E × : uq ( x )= a r ( g ) φ ( x, u ) = η ( y ) | y | y l | l | ∞ X a ∈ F × X x ∈ E × uq ( x )= a φ ∞ ( yx, y − u ) q a . The following expansion result will be used in § 3.4.
Lemma 3.2.2 . —
Let χ : E × \ A × E → C × be a locally algebraic character of weight l , and let E ( g, u ) beany twisted modular form such that E ( g, u ∞ u ) = E ( g, u ) for all u ∞ ∈ F + ∞ . Suppose that φ ∞ (0 , u ) = 0 for all u . Then for all g = y x ! ∈ GL ( A F ) , we have Z ∗ E × \ A × E / A × F χ ( t ) θ ( g, u, r ( t ) φ ∞ ; l ) ⋆ E ( g, q ( t ) u ) dt = 4 | D E/F | / X a ∈ F × F + ∞ ( y − ∞ a ) η ( y ) | y | / y | l | + l ∞ Z A ∞ , × E χ ∞ ( t ) r ( t ) φ ∞ ( y, y − a ) E ( g, q ( t ) a ) d • t q a . DANIEL DISEGNI
Proof . — We may assume that U F is small enough that E ( u ) is invariant under u ∈ U F and w U F = 1 .Taking fundamental domains for µ U F \ F × , the expression of interest is c U F η ( y ) | y | / Z ∗ E × \ A × E / A × F χ ( t ) X u ∈ µ UF \ F × X a ∈ µ UF \ F × X x ∈ E × φ ( t − xy, y − q ( t ) u ) [ uq ( x ) = a ] E ( q ( t ) u, g ) dt Since the integrand is invariant under E ×∞ , by Lemma 2.1.1 with µ = µ U F and a change of variables a = uq ( x ) , this equals L (1 , η ) c U F h E [ O × E : µ U F ] η ( y ) | y | / Z A ∞ , × E X a ∈ µ UF \ F × X α ∈ µ UF χ ∞ ( tα ) φ ∞ ( t − α − y, y − aq ( t ) α )) F + ∞ ( y − ∞ a ) y | l | + l ∞ E ( g, q ( tα ) a ) q α a d • t. By the invariance properties under U F , this can be brought into the desired expression by a change ofvariables a ′ = α a and the calculation L (1 , η ) c U F h E [ O × E : µ U F ] = L (1 , η )[ O × E : O × F ] h E /h F = 4 | D E/F | / , which follows from the definition of c U F = (2.2.3) and the class number formula Let V be a -dimensional quadratic space over A F , totally definite atthe archimedean places. Let φ ∈ S ( V × A × F ) be a Schwartz function, and let ξ : F × \ A × F → C × be alocally algebraic character such that ξ ∞ ( x ) = x k for some integer k and for all x ∈ F + ∞ . Define theautomorphic Eisenstein series (12) E r ( g, u, φ , ξ ) = L ( p ∞ ) (1 , ηξ ) L ( p ∞ ) (1 , η ) X γ ∈ P ( F ) \ SL ( F ) δ ξ,r ( γgw r,p ) r ( γg ) φ (0 , u ) where (with s ∈ C ) δ ξ,r ( g ) := δ ξ,r, ( g ) δ ξ,r,s ( g ) := ξ ( d ) − | a/d | s/ ψ ( k θ ) if g = (cid:0) a bd (cid:1) h with h = h ∞ r θ ∈ U p,r K ∞ if g / ∈ P K ( p r ) K ∞ . (The defining sum is absolutely convergent for ℜ ( s ) sufficiently large, and otherwise it is interpretedby analytic continuation.) It satisfies E r ( zg, u, φ , ξ ) = ηξ − ( z ) E r ( g, u, r ( x, φ , ξ ) . Schwartz function at ∞ . — Let P k ,k ∈ R [ X ] be the (rescaled) Laguerre polynomial(3.3.1) P k ,k ( X ) := (2 πi ) − k (4 π ) − k ( k + k )! k X j =0 (cid:18) kj (cid:19) ( − X ) j j ! . For k = ( k , ( k v )) ∈ Z × Z Hom ( F, R ) ≥ such that k v + k ≥ for all v , define E r ( g, u, φ ∞ , ξ, k ) = E r ( g, u, φ ∞ φ ǫ , ∞ ,k , ξ ) where φ , ∞ ,k = ⊗ v |∞ φ ,v,k v with(3.3.2) φ ,v,k v ( x, u ) = R + ( u ) P k ,k v (4 πuq ( x )) e − πuq ( x ) . The series E r ( ξ, k ) belongs to N ≤ k ( − k ,k + k ) . (12) For k = 0 , this is L ( p ∞ ) (1 , ηξ ) /L ( p ∞ ) (1 , η ) times the series defined in [Dis17]. -ADIC L -FUNCTIONS FOR GL × GU(1) Whittaker–Fourier expansion . — The following standard result is essentially [Dis17, Proposi-tion 3.2.1].
Proposition 3.3.1 . —
We have E r ( (cid:0) y x (cid:1) , u, φ , ξ ) = X a ∈ F W a,r ( (cid:0) y (cid:1) , u, φ , ξ ) ψ ( ax ) where W a,r ( g, u, φ , ξ ) = Y v W a,r,v ( g, u, φ ,v , ξ v ) with, for each v and a ∈ F v , W a,r,v ( g, u, φ ,v , ξ v ) = L ( p ∞ ) (1 , η v ξ v ) L ( p ∞ ) (1 , η v ) Z F v δ ξ,r,v ( wn ( b ) gw r,v ) r ( wn ( b ) g ) φ ,v (0 , u ) ψ v ( − ab ) db. Here L ( p ∞ ) ( s, ξ ′ v ) := L ( s, ξ ′ v ) if v ∤ p ∞ and L ( p ∞ ) ( s, ξ ′ v ) := 1 if v | p ∞ , and we use the convention that r v = 0 if v ∤ p . We choose convenient normalisations for the local Whittaker functions: let γ u,v = γ ( V ,v , uq ) bethe Weil index, and for a ∈ F × v set W ◦ a,r,v ( g, u, φ ,v , ξ v ) := γ − u,v L ( p ) (1 , η v ) W a,r,v ( g, u, φ ,v , ξ v ) . Then for the global Whittaker functions we have W a,r ( g, u, φ , ξ ) = − ε ( V ) L ( p ) (1 , η ) Y v W ◦ a,r,v ( g, u, φ ,v , ξ v ) (3.3.3)if a ∈ F × , where ε ( V ) = Q v γ u,v equals − if V is coherent or +1 if V is incoherent. We similarlydefine W ◦ ,r ( g, u, φ , ξ ) by the identity W ,r ( g, u, φ , ξ ) = − ε ( V ) L ( p ) (1 , η ) W ◦ ,r ( g, u, φ , ξ ) . (3.3.4)A simple calculation shows that for all v and a = 0 ,(3.3.5) W ◦ a,r, (( y ) , u, φ ,v , ξ v ) = ηξ − ( y ) | y | / W ◦ ay,r, (1 , y − u, φ ,v , ξ v ) . We will sometimes drop φ ,v from the notation.The following sufficient condition for cuspidality will simplify matters a little later on. Lemma 3.3.2 . —
Assume that there is a place v ∤ p ∞ , at which ξ v is unramified, such that (3.3.6) φ ,v (0 , u ) = 0 for all u . Then for all g = ( y x ) with y ∈ A + F , x ∈ A F , we have W ,r ( g, u, φ , ξ ) = 0 . Proof . — This is a special case of [YZZ12, Proposition 6.10].
Archimedean Whittaker functions . — We compute them explicitly based on our explicit choiceof Schwartz function.
Lemma 3.3.3 . —
Let v |∞ , let ξ v ( x ) = x k for some k ∈ Z , and let φ ,v ( x, u ) := R + ( u )( uq ( x )) k e − πuq ( x ) for some k ∈ Z ≥ with k ≥ − k . Let a ∈ R × . Then W ◦ a,v (1 , u, φ ) = πi ) k k !( k + k )! a k + k e − πa if a, u > if a < or u < DANIEL DISEGNI
Proof . — We drop the subscript v . We have wn ( b ) = (1 + b ) − / − b (1 + b ) − / (1 + b ) / ! r θ b with e iθ b = ( i − b ) / (1 + b ) / . Then δ ξ,s ( wn ( b )) = i k (1 + b ) − s/ (1 − ib ) − k . Since L (1 , η ) = π − and γ v = i , we have(3.3.7) i − k W ◦ a ( s, , u, Φ) := i − π − Z R δ ξ,s ( wn ( b )) r ( wn ( b ))Φ(0 , u ) ψ ( − ab ) db = π − Z R (1 + b ) − s/ (1 − ib ) − k Z C Φ( x, u ) ψ ( ubq ( x )) d u x ψ ( − ab ) db = π − Z R (1 + b ) − s/ (1 − ib ) − k Z C u k +1 q ( x ) k e − πuq ( x ) ψ ( ubq ( x )) d x ψ ( − ab ) db, where we recall that d u x = | u | d x and d x is twice the usual Lebesgue measure. The integral over C is u k +1 k X j =0 Z R Z R (cid:18) kj (cid:19) x j x k − j e − πu (1 − ib )( x + x ) dx dx . Since R R x j e − Ax dx = A − j − / Γ( j + 1 / , this equals u k k X j =0 (cid:18) kj (cid:19) Γ( j + 1 / k − j + 1 / · (2 πu ) − k − (1 − ib ) − k − = 2 − k π − k k ! (1 − ib ) − k − by the combinatorial identity (see Appendix A)(3.3.8) k X j =0 (cid:18) kj (cid:19) Γ( j + 1 / k − j + 1 /
2) = πk ! . Therefore, when u > , W ◦ a ( s, , u, Φ) = i − k π − k k ! Z R (1 + ib ) − s/ (1 − ib ) − ( s +2 k +2 k +2) / e − πiab db. The integral is the same one appearing in [YZZ12, bottom of p. 55] with d = 2 + 2 k + 2 k . By[YZZ12, Proposition 2.11] (whose normalization differs from ours by L (1 , η v ξ v ) = πi ), we find W ◦ a (0 , , u, Φ) = i k − k π − k − k ! (2 π ) k + k Γ(1 + k + k ) a k + k e − πa = 2(2 πi ) k k !( k + k )! a k + k e − πa if a, u > , as well as simpler formulas implying the desired ones in the other cases.We deduce the following. Let(3.3.9) Q k ,k ( X ) = k X j =0 (cid:18) kj (cid:19) ( k + k )!( j + k )! ( − X ) k − j , which satisfies Q k ,k (0) = 1 . Proposition 3.3.4 . —
Let v |∞ , let a ∈ R , let k ∈ Z , k ∈ Z ≥ with k ≥ − k . Then for a = 0 wehave W a,v (0 , ( y ) , u ; k ) = ( − k η v ξ − v ( y ) | y | / · ay ) k + k Q k ,k ((4 πay ) − ) e − πay if ay > , uy > , and W a,v (0 , ( y ) , u, φ ,k ) = 0 otherwise.Proof . — After recalling the definition of P k ,k in (3.3.1), by Lemma 3.3.3 and (3.3.5) we find theasserted vanishing and that for ay, uy > we have W a,v (0 , ( y ) , u, φ ,k ) = η v ξ − v ( y ) | y | / · π ) − k ( − π ) − k ( k + k )! k X j =0 (cid:18) kj (cid:19) ( k + k )!( j + k )! ( − πay ) j + k e − πay , -ADIC L -FUNCTIONS FOR GL × GU(1) which is equal to the asserted formula. Corollary 3.3.5 . —
Let ξ : F × \ A × F → C × with ξ ( x ∞ ) = x k ∞ for some k ∈ Z . For each k ∈ Z Σ ∞ ≥ with k ≥ − k , we have E r ( g, u, φ ∞ , ξ, k ) = ( − k ∞ δ k E r ( g, u, φ ∞ , ξ, . Proof . — This follows from Proposition 3.3.4 and (2.2.6).
Schwartz function at p . — Let U ◦ F,p ⊂ O × F,p be as fixed in § 2.4.4, and let κ ′ : U ◦ F,p → C × be asmooth character. We define(3.3.10) φ ,κ ′ ( x, u ) := V ,p ( x ) · δ ◦ U F,p ( u ) κ ′ ( u ) , where δ ◦ U F,p ( u ) := vol( O × F,p )vol( U ◦ F,p ) · ◦ U F,p ( u ) . Let(3.3.11) E r ( φ p ∞ ; ξ, κ ′ , k ) := E r ( φ p ∞ φ κ ′ ,p φ ,k, ∞ ; ξ ) , and denote its normalised Whittaker functions by W ◦ a,r,v ( g, u, φ p ∞ ; ξ, κ ′ , k ); depending on the place v we will drop the unnecessary elements from the notation. Non-archimedean Whittaker functions . —
Proposition 3.3.6 . —
Let v be a non-archimedean place of F .1. If v ∤ p , then W ◦ a,v,r = W ◦ a,v does not depend on r , and for all a ∈ F v W ◦ a,v (1 , u, ξ ) = | d v | / L (1 , η v ξ v )(1 − ξ v ( ̟ v )) ∞ X n =0 ξ F,v ( ̟ v ) n q nF,v Z D n ( a ) Φ ,v ( x , u ) d u x , where d u x is the self-dual measure on ( V ,v , uq ) and D n ( a ) = { x ∈ V ,v | uq ( x ) ∈ a + p nv d − v } . (When the sum is infinite, it is to be understood in the sense of analytic continuation fromcharacters ξ | · | s with s > .)2. For all finite places v , | d | − / v | D v | − / W ◦ a,v (1 , u, ξ ) ∈ Q [ ξ, Φ v ] , and for almost all v we have | d | − / v | D v | − / W ◦ a,v (1 , u, ξ ) = if v ( a ) ≥ − v ( d v ) and v ( u ) = − v ( d v )0 otherwise.3. If v | p , then W ◦ a,r,v (1 , u ; ξ, κ ′ ) = | d v | / | D v | / ξ v ( − κ ′ ,v ( u ) if v ( a ) ≥ − v ( d ) and u ∈ U ◦ F,v , otherwise.Proof . — See [Dis17, Propostion 3.2.3, Lemma 3.2.4] for parts 1 and 2. For part 3, we drop subscripts v and compute δ r,ξ ( wn ( b ) w r ) = ξ ( − O F ( b ) , γ − u r ( wn ( b )) φ ,κ ′ (0 , u ) = | d v || D v | / O × F ( u ) κ ′ ( u ) (the latter if b ∈ O F ), so that W ◦ a (1 , u ; ξ, κ ′ ) = | d v || D v | / ξ ( − O × F ( u ) κ ′ ( u ) Z O F ψ ( − ab ) db, which gives the asserted value. DANIEL DISEGNI
Corollary 3.3.7 . —
For g = ( y x ) with y ∈ A + F , x ∈ A F , we have E r ( g, u, φ ∞ ; ξ, k ) = − ε ( V ) L ( p ) (1 , η ) ηξ − ( y ) | y | / · W ◦ ( y, u ) + X a ∈ F + [ F : Q ] ( ay ∞ ) k + k W ◦ , ∞ a,r (0 , , y − u, φ , ξ ) Q k ,k ((4 πay ∞ ) − ) q a ! where W ◦ , ∞ a,r (0 , , u, φ ; ξ ) = Q v ∤ ∞ W ◦ a,r,v (0 , , y − u, φ ,v ; ξ ) . Let φ p ∞ satisfy (3.3.6), so that by Lemma 3.3.2, the corresponding Eisenstein series is cuspidal.For ξ a locally algebraic p -adic character of A × F of weight k , consider the (bounded) sequence ofcoefficients in Q p ( ξ )E r ( u, φ ∞ , ξ, k ) = (cid:16) λ · ηξ − ( y )2 [ F : Q ] ( by p ) k + k | D F | / | D E | / W ◦ , ∞ by (1 , y − u, ξ ) (cid:17) b ∈ F + . where λ = − ε ( V ) / | D E/F | / L ( p ) (1 , η ) . This is the p -adic expansion attached to E r . Analogously toCorollary 3.3.5, we have(3.3.12) E( u, φ , ξ, k ) = ( − k ∞ d k E( u, φ , ξ, . p -adic interpolation of Whittaker functions . — Proposition 3.3.8 . —
Let v ∤ p ∞ . For each a ∈ F × v , y ∈ F × v , and φ ,v ∈ S ( V p ∞ × A p ∞ , × , Q p ) there is a meromorphic function W ◦ a,v ( y, u, φ ,v ) ∈ K ( Y v ) , regular if φ ,v is standard and otherwise with possible poles at those ξ v such that L (1 , η v ξ v ) = 0 ,satisfying W ◦ a,v ( y, u, φ ,v ; ξ v ) = | d v | − / | D v | − / W ◦ a,r,v (( y v ) , u, φ ,v ( ξ v ) , ξ v ) for all ξ v ∈ Y v ( C ) whose underlying scheme point is not a pole.Proof . — Part 1 is proved as in [Dis17, Lemma 3.3.1], except that we write the arbitrary φ ,v = cφ ◦ ,v + φ ′ ,v without the extra factor of equation (3.3.2) ibid. Then the argument shows that (only)when φ ′ ,v = 0 , there may be a pole controlled by L (1 , η v ξ v ) . Fix a compact open subgroup U p ⊂ B ∞× (which will be usuallyomitted from all the notation), and let U pF := U p ∩ A p ∞ , × . Let φ p ∞ ∈ S ( V p ∞ × A p ∞ , × ) be a Schwartzfunction fixed by U p . Let ξ : F × \ A × → C × be a locally algebraic character fixed by U pF and such that ξ ( x ∞ ) = x k ∞ for some k ∈ Z , and let k ∈ Z Σ ∞ ≥ satisfy k v + k ≥ for all v .We fix a choice of a Schwartz function in S ( V ,p × F × p ) as follows. Let U ◦ F,p ⊂ O × F,p be as fixed in§ 2.4.4. For r ∈ Z S p ≥ and κ ′ : F × p → C × a smooth character, we define(3.4.1) φ ,r,κ ′ ,p ( x, u ) = δ ,r,p ( x ) U ◦ F,p κ ′ ( u ) := Y v | p vol( O E,v , dt )vol(1 + ̟ r v v O E,v , d × t ) ̟ rvv O E,v ( x ,v ) U ◦ F,p ( u ) κ ′ ( u ) . For t ∈ A × E , r ≥ , and φ p ∞ = φ p ∞ ⊗ φ p ∞ , define a form in N k ( l − k , l + k +2 k ) ( C ) by I r ( t, φ p ∞ ; κ ′ , l, ξ, κ ′ , k ) := | D F | − / · θ ( u, r ( t, φ p ∞ φ ,κ ′ ,r,p ; l ) ⋆ E r ( q ( t ) u, φ p ∞ ; ξ, κ ′ , k ) . where the product ⋆ is (2.2.5).Fix a compact open subgroup U p H ⊂ A p ∞ , × E (which will be omitted from all the notation). 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 ≥ is greater than the conductors of χ w , ξ v , κ ′ ,v . Then we define(3.4.2) I ( φ p ∞ ; χ, ξ, κ ′ , k ) := Z ∗ E × \ A × E / A × F χ ( t ) I r ( t, φ p ∞ ; κ ′ ,χ,p , l, ξ, κ ′ , k ) dt, -ADIC L -FUNCTIONS FOR GL × GU(1) which does not depend on the choice of r ; here κ ′ ,χ,p is as in (2.4.3), namely(3.4.3) κ ′ ,χ,p := χ p ◦ j ′ for j ′ : U ◦ F,p → O × E,p as in (2.4.4)
Lemma 3.4.1 . —
For each c ∈ A + F satisfying v ( c ) ≥ for some v | p , the c th Whittaker–Fouriercoefficient of I ( φ p ∞ ; χ, ξ, κ ′ , k ) is W ♯I ( c ) = − ε ( V ) · [ F : Q ] L ( p ) (1 , η ) | D E/F | / | D F | / · X a/c ∈ F,
Let a, c ∈ A ∞ , × F , φ ∈ S ( V p ∞ × A p ∞ , × F ) , and for locally algebraic characters ξ : F × \ A × F → C × , χ : E × \ A × E → C × , consider the terms J v ( a, c ; χ v , ξ v ) = (3.4.4) .1. For v | p , if v ( c ) ≥ then | d v | − / | D v | − / · J v ( a, c ; χ v , ξ v , κ ′ ,v ) = ξ v ( − U ◦ F,v ( a ) κ ′ ,χ,v κ ′ ,v ( a ) . DANIEL DISEGNI 2. For all but finitely many v ∤ p , | d v | − / | D v | − / J v ( a, c, φ ,v ; χ v , ξ v ) = 1 . 3. For all v ∤ p ∞ , there is a function J v ( a, c ) ∈ O ( Y v × Y H ,v ) such that for all χ v , ξ v ∈ Y v × Y H ,v ( C ) , J v ( a, c, φ v )( χ v , ξ v ) = | d v | − / | D v | − / J v ( a, c, φ ,v ; χ v , ξ v ) = 1; Proof . — Part 1 follows from the definitions and Proposition 3.3.6.3. Part 2 follows from Proposition3.3.6.2 and a simple calculation. Finally, since the integrand in (3.4.4) is compactly supported, part 3follows from Proposition 3.3.8.We lighten the notation by occasionally dropping φ p ∞ . Let(3.4.5) λ := − ε ( V ) · [ F : Q ] | D E/F | / L ( p ) (1 , η ) ∈ Q × . For the sake of simplicity, we momentarily introduce the assumption that the weight l of χ satisfies l ≥ . We will see in Corollary 3.4.5 that this does not affect in our main construction. Proposition 3.4.3 . —