Anticyclotomic exceptional zero phenomenon for Hilbert modular forms
aa r X i v : . [ m a t h . N T ] J a n Anticyclotomic exceptional zero phenomenon for Hilbertmodular forms
Bingyong Xie ∗ Department of Mathematics, East China Normal University, Shanghai, [email protected]
Abstract
In this paper we study the exceptional zero phenomenon for Hilbert modular forms in theanticyclotomic setting. We prove a formula expressing the leading term of the p -adic L -functionsvia arithmetic L -invariants. p -extensions Exceptional zero phenomenon was firstly discovered by Mazur, Tate and Teitelbaum in theirremarkable paper [23]. They considered elliptic modular forms .For a modular form f = P a n q n of weight 2, of level N and character ε , the attached classical L -function L ( s, f ) admits an analytic continuous to the entire complex plane, and satisfies a functionalequation which relates its values at s and k − s . The famous Birch and Swinnerton-Dyer conjecturerelates the behaviour of L ( s, f ) at s = 1 to the arithmetic of A φ , the abelian variety associated to φ by Eichler-Shimura correspondence.A p -adic analogue of the Birch and Swinnerton-Dyer conjecture is formulated in [23], whichreplaces L ( s, f ) by a p -adic L -function L p ( s, f ) attached to the cyclotomic Z p -extension Q cyc of Q . Let α be the root of X − a p X + ε ( p ) p that is a unit. Then there exists a continuous function L p ( f, χ ) on the set of continuous p -adic characters χ of Gal( Q cyc ∞ / Q ), such that when χ is of finiteorder, we have the interpolation formula L p ( f, χ ) = e p ( χ ) L (1 , f, χ )Ω f (1.1)where Ω f is a real period making L (1 ,f,χ )Ω f algebraic, and e p ( χ ) is the p -adic multiplier. Letting χ varying in {h · i s : s ∈ C p , | s | ≤ } where h · i is the Teichm¨uller character, we obtain the analyticfunction L p ( s, f ) := L p ( f, h · i s ) . ∗ This paper is supported by the National Natural Science Foundation of China (grant 11671137), and supportedin part by Science and Technology Commission of Shanghai Municipality (no. 18dz2271000). p || N and a p = 1, e p (1) = 0 and thus by the interpolation formula L p ( s, f ) vanishes at0. The exceptional zero conjecture made in [23] relates the first derivative of L p ( s, f ) at 0 to thealgebraic part of L ( f, L ′ p (0 , f ) = L ( f ) L (1 , f )Ω f , (1.2)where L ( f ), is the so called L -invariant, an isogeny invariant of A f . When A f is an elliptic curve(i.e. all of a n are rational), L ( f ) = log p ( q ( A f ))ord p ( q ( A f ))is defined by using the p -adic period q ( A f ) of A f ; in this case it is showed in [2] that L ( f ) = 0.This conjecture was proved by Greenberg and Stevens [14] using Hida’s theory of p -adic familiesof ordinary eigenforms and the 2-variable p -adic L -functions attached to them.In [15] Hida formulated a generalization of this exceptional zero conjecture. To state it we let F be a totally real field, and let f be a Hilbert modular form of parallel weight 2 over F . Correspondingto f we also have the classical L -function L ( s, f ) and the p -adic L -function L p ( s, f ) attached to thecyclotomic Z p -extension F Q cyc ∞ of F ; L p ( s, f ) satisfies an interpolation formula similar to (1.1) withthe p -adic multiplier replaced by a product Q p | p e p ( · ).Let J p be the set of primes of F above p , J the subset of prime p | p such that α p = 1, i.e. p exactly divided the level of f , and the Fourier coefficient a p at p is 1. Put J = J p \ J . Then e p (1) = 0 if and only if p ∈ J .Hida [15] conjectured that ord s =0 L p ( s, f ) ≥ r := ♯ ( J ) (1.3)and d r L p ( s, f ) ds r (cid:12)(cid:12) s =0 = r ! Y p ∈ J L p ( f ) · Y p ∈ J e p (1) · L (1 , f )Ω f (1.4)where L p ( f ) is the L -invariant of f at p . When f is attached to an elliptic curve A over F that issplit multiplicative at p with Tate period q ( A/F p ), one has L p ( f ) = log p ( N F p / Q p q ( A/F p ))ord p ( N F p / Q p q ( A/F p )) . When r = 1, Mok [24] proved (1.3) and (1.4) under some local assumption by extending themethod in [14]. In general, Spiess [25] proved (1.3) unconditionally and (1.4) under some localassumption. p -extensions To explain what an anticyclotomic Z p -extension is, let K be an imaginary quadratic field. Thereare many different Z p -extensions of K . Among them exactly two have the property that K ∞ / Q is Galois. One is the cyclotomic Z p -extension K Q cyc ∞ . The other is called the anticyclotomic Z p -extension of K and denoted by K −∞ ; K −∞ is characterized by K −∞ = S n K − n , where K − n is Galois over Q and Gal( K − n / Q ) is isomorphic to the dihedral group of order 2 p n .Bertolini and Darmon [5] proved that, when p is split in K , exceptional zero phenomenon againholds for elliptic modular forms. Now, one considers the complex L function L ( s, f / K ) associated to2he base change of f to K ; the p -adic L -adic function is replaced by the anticyclotomic one, denotedby L p ( s, f / K ) = L p ( f / K , h · i s anti ), that interpolates L (1 , f / K , χ ) with χ being anticyclotomiccharacters, i.e. characters of Gal( K −∞ / K ), instead of cyclotomic characters, i.e. characters ofGal( K Q cyc ∞ / K ). Here h · i anti is the special anticyclotomic character that plays the same role in theanticyclotomic setting as the Teichm¨uller character h · i in the cyclotomic setting. See Section 2.2for its definition. The function L p ( s, f / K ) was constructed by Bertolini and Darmon in [4], andwas showed to satisfy the interpolation formula by Chida and Hsieh in [8].Their formula says d L p ( s, f / K ) ds (cid:12)(cid:12) s =0 = L Tei ( f ) · L (1 , f / K )Ω − f , (1.5)where Ω − f is certain complex period making L (1 ,f/ K ,χ )Ω − f algebraic, and L Tei ( f ) is the L -invariantof Teitelbaum type. To define L Tei ( f ) Bertolini and Darmon [5] fixed a branch of the p -adiclogarithmic.The reader may notice that it is the first derivative appeared in (1.2), while it is the secondderivative appeared in (1.5). It is rather mysterious to the author this difference between thecyclotomic and the anticyclotomic settings.The purpose of our paper is to study the exceptional zero phenomenon for Hilbert modularforms in the anticyclotomic setting.We keep Bertolini and Darmon’s assumption that p is split in K . Let F be a totally realfield, K = K F . Let f be a Hilbert modular form of parallel weight 2 over F . Let π denote theautomorphic representation of GL ( A F ) associated to f , π K the base change of π to K .Our first task is to construct the anticyclotomic p -adic L -function L p ( f /K, χ ) that interpolatesthe special L -values of L (1 , f /K, · ) = L ( , π K , · ). This function is defined on the set (cid:8) χ : p -adic continuous characters of Gal( K −∞ F/F ) unramified outside p (cid:9) . (1.6)We need some level conditions on f . We decompose the level n of f as n = n + n − Y p | p p r p , where n + (resp. n − ) is only divided by primes that are split (resp. inert or ramified) in K , and( n + n − , p ) = 1. We assume that • r p ≤ p | p ; • n − is square-free; • the cardinal number of prime factors of n − that are inert in K has the same parity as [ F : Q ]. Theorem 1.1.
Suppose that f satisfies the above level conditions and that f is ordinary at p in thesense that α p is a p -adic unit for each prime p above p . (a) There exists a nonzero complex number Ω − f and a p -adic continuous function L p ( f /K, χ ) onthe set (1 . such that when χ is of finite order, L (1 , f /K,χ )Ω − f is algebraic and L p ( f /K, χ ) = Y p | p e p ( χ ) · L (1 , f /K, χ )Ω − f . ere e p ( χ ) is the multiplier e p ( χ ) = α p (1 − α p χ ( P ))(1 − α p χ ( ¯ P )) p F p : Q p ] if ord P χ = 0 i.e. χ is unramified at P , p n [ F p : Q p ] α n p if ord P χ = n > , where p = P ¯ P . (b) When χ varies in the analytic family {h · i s anti : s ∈ C p , | s | p < p } , we obtain an analyticfunction L p ( s, f /K ) := L p ( f /K, h · i s anti ) . Here, by abuse of notation the restriction of h · i anti to Gal( K −∞ F/F ) is again denoted by h · i anti . To construct the p -adic L -functions demanded in Theorem 1.1, we embed the anticyclotomicextension K −∞ F/F into a much larger one. For each set of places J of F above p , the union of thering class fields of conductor Q p ∈ J p n p ( n p ≥ p ∈ J ) contains a maximal subfield whoseGalois group is a free module Z p of rank ♯ (Σ J ), which we call the Γ − J -extension. See our contextfor the meaning of the notation Σ J . When J = J p , the full set of places above p , the Γ − J p -extensioncontains K −∞ F .When J = { p } is single, Hung [17] extended Chida and Hsieh’s method [8] to construct a p -adic L -function for the Γ − p -extension. In this paper we extend their method to the case of arbitrary J .For each J we obtain a multi-variable p -adic L -function denoted by L J that interpolating the specialvalues L alg ( , π K , χ ) with χ factors through Γ − J . When J = J p , we refer L J p as the total-variable p -adic L -function. The p -adic L -function in Theorem 1.1 is obtained by restricting L J p .Let J be the set of primes p above p such that α p = 1, and put J = J p \ J . By definition e p (1) = 0 if and only if p ∈ J . Hence, if J is non-empty, then L p (0 , f /K ) = 0.Comparing Spiess’ theorem and Bertolini-Darmon’s, one expects the order of L p ( s, f /K ) at 0 isat least 2 ♯ ( J ). Our second task is to show that this is indeed the case. Theorem 1.2.
Suppose that f satisfies the same conditions as in Theorem . . (a) Put r = ♯ ( J ) . Then ord s =0 L p ( s, f /K ) ≥ r. (b) We have d r L p ( s, f /K ) ds r (cid:12)(cid:12) s =0 = ( r !) · Y p ∈ J L Tei p ( f ) · Y p ∈ J e p (1) · L (1 , f /K )Ω − f . In the above formula L Tei p ( f ) is the L -invariant of Teitelbaum type of f at p , which is defined in[9]. The branch of the p -adic logarithmic log we choose to define L Tei p ( f ) is the same as Bertoliniand Darmon’s, and thus is independent of p and f . When f is attached to an elliptic curve A , and q ( A/F p ) is the Tate period, then L Tei p ( f ) = log( N F p / Q p q ( A/F p ))ord p ( N F p / Q p q ( A/F p )) . (1.7)We obtain a result beyond Theorem 1.2. Indeed, we can determine all partial derivatives oforder 2 r of the total-variable p -adic L -function L J p .4 heorem 1.3. We have the following Taylor expansion of L J p at (0 , · · · , : L J p (( s σ ) σ ∈ Σ Jp , π K ) = L (1 , f /K )Ω − f · Y p ∈ J e p (1) · Y p ∈ J X σ ∈ Σ Jp s σ L p ,σ + higher order terms . For the definition of L p ,σ see Definition 8.8 in our context. When σ ∈ Σ p , L p ,σ is L-invariant ofTeitelbaum type; when σ / ∈ Σ p , L p ,σ is a constant independent of π f . We also obtain exceptional zero phenomenon between those L J . Theorem 1.4. If b χ is a character of Γ − J such that b χ p = 1 for each p ∈ J ∩ J , then L J (( s σ ) σ ∈ Σ J , π K , b χ ) = L J \ ( J ∩ J ) ( π K , b χ ) · Y p ∈ J ∩ J X σ ∈ Σ J s σ L p ,σ ! + higher order terms . (1.8)Note that the condition “ b χ p = 1 for each p ∈ J ∩ J ” implies that b χ comes from a character ofΓ − J ∩ J and so L J \ ( J ∩ J ) ( π K , b χ ) makes sense.In the special case that J = { p } ⊂ J and J \ ( J ∩ J ) = ∅ , Theorem 1.4 was obtained byHung [18, Theorem A and Theorem B] following Bertolini-Darmon’s method. But Hung neededthe condition on class number that p ∤ ♯ ( b F × K × \ b K × / b O × K ). This condition is removed in Theorem1.4.A result related to Theorem 1.4 was obtained by Bergunde and Gehrmann [3, Theorem 6.5].They proved a formula for leading terms of anticyclotomic stickelberger elements. Using theirresult, one may deduce a formula similar to (1.8) in the case of b χ = 1 but less precise. Indeed,the automorphic periods q S m , p in [3] are not showed to coincide with Tate’s period; these periods q S m , p are even not showed to be independent of S m . While in our formula (1.8) the leading term isprecise. We give a sketch of the proof of Theorem 1.2 and Theorem 1.3. The argument for Theorem 1.4 issimilar.The function L J p is equal to L J p , the square of another analytic function L J p . We restrict L J p to one direction. Let ( s σ ) σ ∈ Σ Jp be fixed, and put L J p ( t ; ( s σ )) := L J p ( ts σ ) σ ∈ Σ Jp . We will show that d n L J p ( t ; ( s σ )) dt n (cid:12)(cid:12)(cid:12) t =0 = r ! · Q p ∈ J P σ ∈ Σ Jp s σ L p ,σ ! · L J p \ J (0 , · · · ,
0) if n = r, n < r. (1.9)Taking ( s σ ) = (1 , · · · ,
1) we obtain Theorem 1.2. Letting ( s σ ) vary we obtain Theorem 1.3.Our approach to (1.9) is somewhat a mixture of Spiess’ and Bertolini-Darmon’s.5n [25] Spiess related the computation of d r ds r L p ( s, f ) (cid:12)(cid:12) s =0 to group cohomology. Let F × + denotethe group of totally positive elements of F . Each p ∈ J = { p , · · · , p r } is associated with anelement c ℓ p ∈ H ( F × + , C c ( F p , C p )). Then d r ds r L p ( s, f ) (cid:12)(cid:12) s =0 was showed to equal the cap-product d r ds r L p ( s, f ) (cid:12)(cid:12)(cid:12) s =0 = ( − (cid:0) r (cid:1) r !( κ ∪ c ℓ p ∪ · · · ∪ c ℓ p r ) ∩ ϑ. (1.10)Here, κ ∈ H [ F : Q ] − ( F × + , D ) is in the ([ F : Q ] − F × + with values in a distributionspace, and ϑ is the fundamental class of the quotient M/F × + of some ( r − F : Q ])-dimensionalreal manifold M with a free action of F × + .Instead of F × + , the group ∆ we use is a subgroup of F × \ K × , which is a free abelian group ofrank r .We reduce the computation of d n L Jp ( t ;( s σ )) dt r (cid:12)(cid:12) t =0 to Z l n e µ J p the integration of l n on some domain of A ∞ F \ A ∞ K for certain F × \ K × -invariant distribution e µ J p ,where l is a logarithm function depending on the direction ( s σ ) Σ Jp . In the case of n < r , we showdirectly that the integration is zero. In the case of n = r , we relate the integration of l r to groupcohomology. Being invariant by ∆, e µ J p provides an element in the 0-th cohomology group of ∆with values in some space of distributions.To each p ∈ J = { p , · · · , p r } and the logarithmic function l we attach an element [ c p ,l ] in thefirst cohomology of ∆ with values in certain function space C ≤ p . Taking cup-product we obtain anelement [ c p ,l ] ∪ · · · ∪ [ c p r ,l ] in the r -th cohomology of ∆ with values in some function space C ≤ J .Then we have an element [ e µ J p ] ∪ ([ c p ,l ] ∪ · · · ∪ [ c p r ,l ]) ∈ H r (∆ , C p )whose evaluation at the r generators of ∆ is closely related to R l n e µ J p .Similarly, to p ∈ J and the order function l we attach an element [ c p , ord ] ∈ H (∆ , C ≤ p ). Takingproduct we obtain [ e µ J p ] ∪ ([ c p , ord ] ∪ · · · ∪ [ c p r , ord ]) ∈ H r (∆ , C p )whose evaluation at the the r generators of ∆ is closely related to L J p \ J (0 , · · · , e µ J p ] ∪ ([ c p ,l ] ∪ · · · ∪ [ c p r ,l ]) = Y p ∈ J X σ ∈ Σ Jp s σ L p ,σ · [ e µ J p ] ∪ ([ c p , ord ] ∪ · · · ∪ [ c p r , ord ]) (1.11)in H r (∆ , C p ).We compare our method and Spiess’ [25]. We use the r -th cohomology group, while Spiess usedthe ( r − F : Q ])-th cohomology group. In [25] Spiess needed the element ϑ in the ([ F : Q ] − p -adic, so that wecould avoid the comparison between the p -adic world and the real world. However, our constructionof the function spaces C ≤ p is more technical than that of C c ( F p , C p ) used by Spiess.6he p -adic L -functions L J we studied are of multi-variable, while the p -adic L -function studiedin [25] is of one-variable. Such a discrepancy occurs, because the maximal abelian p -extension of K and the associated Galois group are rather large, while according to Leopoldt conjecture (whichwas proved to be true if F/ Q is abelian) the maximal abelian p -extension of F is the cyclotomicextension F Q cyc ∞ and the Galois group is just Z p .However, considering the Hida families one also has multi-variable p -adic L -functions in thecyclotomic setting. The author believes that it is possible to obtain results parallel to Theorem 1.3and Theorem 1.4 for these multi-variable p -adic L -functions by extending Spiess’ method.Our paper is organized as follows. In Section 2 we recall some basic facts on Haar measuresand construct families of p -adic anticyclotomic characters. In Section 3 we recall needed facts onautomorphic forms. We extend Chida and Hsieh’s method to construct p -adic L -functions L J inSection 4 and Section 5. In Section 6 we define Harmonic cocycle valued and cohomological valuedmodular forms, and use them as tools to define L -invariant of Teitelbaum type. In Section 7 wecompute some integrations closely related to the partial derivatives of L J . In Section 8 we definethe elements [ c p ,l ] and [ c p , ord ] in group cohomology of ∆, and then we prove (1.11). Finally weprove Theorems 1.2, 1.3 and 1.4 in Section 9. Let A be the ring of adeles of Q .For any prime number ℓ we use | · | to denote the absolute value on C ℓ such that | ℓ | = 1 /ℓ . Fora finite extension L of Q ℓ in C ℓ , and a uniformizing element ω L of L , we use | · | ω L to denote theabsolute value on C ℓ such that | a | ω L = | N L/ Q ℓ ( a ) | for a ∈ L . So | · | ω L = | · | [ L : Q ℓ ] . If q L is thecardinal number of the residue field of L , then | ω L | ω L = q − L .Let p be an odd prime number. p -adic characters and Hecke characters We fix a totally real number field F . Let Σ F be the set of all real embeddings of F , J p the set ofplaces of F above p .Let K be a totally imaginary extension of F . We fix a CM type Σ K of K , i.e. for σ ∈ Σ thereexists exactly one element of Σ K (an embedding of K into C ) that restricts to σ . By abuse ofnotation we again use σ to denote this element, and use ¯ σ to denote the other embedding K ֒ → C that restricts to σ : F ֒ → R . We fix an isomorphism of fields : C ∼ −→ C p .Let Ω p be the maximal ideal of O C p . For each σ ∈ Σ F or Σ K we put p σ = σ ( F ) ∩ − Ω p , P σ = σ ( K ) ∩ − Ω p and ¯ P σ = ¯ σ ( K ) ∩ − Ω p . Note that, for a prime p of F above p , the set of σ such that p σ = p has cardinal number [ F p : Q p ].For our convenience, we demand Σ K satisfies the following condition:If p σ = p σ , then P σ = P σ .Thus, when p is split in K , we may write p O K = PP , where P = P σ for each σ satisfying p σ = p .We will use this convention throughout this paper without mention.7he inclusion σ extends to K P σ which will be denoted by σ ′ : K P σ → C p . For ¯ σ insteadof σ , we have ¯ σ ′ : K ¯ P σ → C p . We write F σ ′ , K σ ′ and K ¯ σ ′ for σ ′ ( F p σ ), σ ′ ( K P σ ) and ¯ σ ′ ( K ¯ P σ )respectively.Let b ν : Gal( K/K ) → O × C p be a p -adic continuous character. Via the geometrically normalizedreciprocity map rec K : K × K ×∞ \ A × K ∼ −→ Gal(
K/K )we may view b ν as a character of A × K that is trivial on K × K ×∞ .Fix m = P σ ∈ Σ F m σ σ with m σ nonnegative. We say that b ν is of type ( m , − m ), if the followingholds.In the case when p is inert or ramified in K , i.e. p O K = P or P , we have b ν P ( a ) = Y σ ∈ Σ K : p σ = p σ ′ (cid:16) a ¯ a (cid:17) m σ on a compact open subgroup of O × K P .In the case when p is split in K , i.e. p O K = P ¯ P and F p ∼ = K P ∼ = K ¯ P , we have b ν P ( a ) = Y σ ∈ Σ K : P σ = P σ ( a ) m σ and b ν ¯ P ( a ) = Y σ ∈ Σ K : P σ = P σ ( a ) − m σ on a compact open subgroup of O × F p .If b ν is of type ( m , − m ), we attach to b ν a complex Hecke character ν of K × \ A × K by ν ( a ) = − (cid:16)b ν ( a ) · Y σ ∈ Σ K (cid:16) ¯ σ ′ a ¯ P σ σ ′ a P σ (cid:17) m σ (cid:17) · Y σ ∈ Σ K (cid:16) a σ ¯ a σ (cid:17) m σ . Then ν is of archimedean type ( m , − m ), i.e. ν ∞ = Q σ ∈ Σ K (cid:16) a σ ¯ a σ (cid:17) m σ .Any complex Hecke character of K × \ A × K of archimedean type ( m , − m ) comes from a p -adiccharacter of type ( m , − m ) in the above way. Fix a subset J of J p . Put Σ J := { σ ∈ Σ F : p σ ∈ J } . For J = { p } we write Σ p for Σ { p } .For each p ∈ J we write K p = K ⊗ F F p . If p is inert or ramified in K , then K p is a quadraticextension of F p . If p is split in K , then K p ∼ = F ⊕ p and F p diagonally embeds into K p .Let c be an ideal of O F that is coprime to p . For each J -tuple of nonnegative integers ~n = ( n p ) p ∈ J , let O ~n, c := O F + c · Y p ∈ J p n p O K be the order of K of conductor c · Q p ∈ J p n p . Let K ~n, c be the ring class field of K of con-ductor c · Q p ∈ J p n p , and G ~n, c = Gal( K ~n, c /K ) be its Galois group. Then G ~n, c is isomorphic to K × b F × \ b K × / b O × ~n, c . Put G J ; c := lim ←−− ~n G ~n, c . Let Γ − J be the maximal Z p -free quotient of G J ; c .8ote that c := [ K × b F × \ b K × : O × K b O × F \ b O × K ] ≤ cl( K )where cl( K ) denote the class number of K . Via rec K Y p ∈ J ( O K p / O F p ) / torsionis a subgroup of Γ − J of index dividing c , andΓ − J ∼ = Y p ∈ J ( O K p / O F p ) / torsionif c is corpime to p . Here, for a discrete valuation field L let m L be the maximal ideal of O L andwe write O L = 1 + m L . When p = P ¯ P is split in K , we write O K p = O K P ⊕ O K ¯ P ∼ = ( O F p ) ⊕ .We will define an analytic family of characters of Γ − J .Let σ be a real place of F . In the case when p σ is inert or ramified in K , let ǫ σ denote thecharacter ˜ ǫ σ : O K p σ / O F p σ → C × p , a σ ′ ( a )¯ σ ′ ( a ) . In the case when p σ is split in K , let ǫ σ denote the character˜ ǫ σ : O K p σ / O F p σ = ( O K P σ ⊕ O K ¯ P σ ) / O F p σ → C × p , ( a, b ) σ ′ (cid:16) ab (cid:17) . Let µ be a sufficiently large positive integer such that for each p ∈ J , roots of unity in F p and K p (when p is inert or ramified in K ) are of order dividing p µ . Lemma 2.1. ˜ ǫ p µ σ factors through ( O K p σ / O F p σ ) / torsion and thus gives a character ( O K p σ / O F p σ ) / torsion ˜ ǫ pµσ −−→ C × p . Proof.
Obvious.We extend the additive character p µ log˜ ǫ p µ σ linearly to Γ − J ; it is independent of µ , and we denoteit by log σ . Note that the image of log σ is contained in c − p − σ ′ ( O K P σ ).Let LOG J denote the C p -vector space spanned by log σ ( σ ∈ Σ J ). For each p ∈ J we writelog p = X σ ∈ Σ p log σ . Then log p ∈ LOG J .If s ∈ C p satisfies | s | ≤ | c | p p − , we define the character ǫ sσ of Γ − J by ǫ sσ ( a ) := exp (cid:16) s · log σ ( a ) (cid:17) . J -tuple ~s = ( s σ ) σ ∈ Σ J of elements in C p with | s σ | ≤ | c | p p − for each σ ∈ Σ J , we havethe character ǫ ~s := Y σ ∈ Σ J ǫ s σ σ of Γ − J . Hence, we have an analyitc families of characters { ǫ ~s : ~s = ( s σ ) σ ∈ Σ J , | s σ | ≤ | c | p p − } . Applying the above construction to the case F = Q and K = K , we obtain the character ˜ ǫ K and an analytic family ǫ s K of characters of the group Γ −K = Gal( K −∞ / K ). The character h · i anti inour introduction is exactly ǫ K and h · i s anti = ǫ s K . Let L be a number field, d L = [ L : Q ], D L the different of L . In our application L = F or K .For each finite place v of L above a prime number ℓ , let | · | v or | · | L v denote the absolute valueof L v defined by | x | v = | N L v / Q ℓ ( x ) | ℓ . For each finite place v let ω v be a uniformizing element of O L v .For each finite place v of L we define the local zeta function ζ L v ( s ) by ζ L v ( s ) = −| ω v | sv . Forreal archimedean places we define ζ R ( s ) = π − s Γ( s ) and for complex archimedean places we define ζ C ( s ) = 2(2 π ) − s Γ( s ). Then the zeta function ζ L ( s ) is defined by ζ L ( s ) = Y v ζ L v ( s )which is convergent when Re( s ) > ψ Q be the standard additive character of A / Q such that ψ Q ( x ∞ ) = exp(2 πix ∞ ). We definethe additive character ψ of A L /L by ψ = ψ Q ◦ Tr L/ Q , where Tr L/ Q is the trace map from A L to A . We write ψ = Q v ψ v , where for each v , ψ v is anadditive character of L v .If v is finite, let d v x be the Haar measure on L v that is self dual with respect to the pairing( x, x ′ ) ψ v ( xx ′ ). Then vol( O L v , d v x ) = |D L v | v . If v is real, we define d v x to be the Lebesgue measure on R . If v is complex, we define d v x to betwice the Lebesgue measure on C .We define the Haar measure on L × v by d × v x = ζ L v (1) d v x | x | v . When v is finite, we have vol( O × L v , d × v x ) = |D L v | v . Taking product we obtain the measure on A × L . 10aking L = F or K we obtain measures on A × F and A × K . We also need the quotient measureon A × K / A × F . For our convenience we give a description of it. If v is a finite place of F that issplit in K , then K × v ∼ = F × v × F × v with F × v diagonally embedding into K × v . In this case we have K × v /F × v ∼ = F × v , ( a, b ) ab − . The quotient measure on K × v /F × v coincides with the pullback viathis isomorphism of the measure on F × v . When v is inert or ramified in K , the quotient measureon K × v /F × v is the Haar measure such that the volume of O × K v / O × F v isvol( O × K v / O × F v ) = vol( O × K v )vol( O × F v ) = |D K v | K v |D F v | v (2.1) The reference of this subsection is [20].Let v be a finite place of F . If n is an ideal of O F v , we use U ( n ) to denote the subgroup { g ∈ (cid:2) a bc d (cid:3) ∈ GL ( O F v ) : c ∈ n } . When n = ( ω v ) we write U ( ω v ) for U ( n ).For two characters µ , µ of F × v , we have an induced representation Ind( µ ⊗ µ ) of GL ( F v ).It is realized by right translation on the space of functions f on GL ( F v ) satisfying f ( (cid:2) a b d (cid:3) g ) = µ ( a ) µ ( d ) (cid:12)(cid:12)(cid:12) ad (cid:12)(cid:12)(cid:12) v f ( g )for all g ∈ GL ( F v ), a, d ∈ F × v and b ∈ F v .When µ µ − = | · | v , | · | − v , Ind( µ ⊗ µ ) is irreducible, and we denote it by π ( µ , µ ). Iffurthermore µ and µ are unramified, then there exist nonzero GL ( O F v )-invariant vectors in π ( µ , µ ). In this case, we say that π ( µ , µ ) is unraimified.When µ µ − = |·| v , Ind( µ ⊗ µ ) contains a unique irreducible proper invariant subspace denotedby σ ( µ , µ | · | − v ); σ ( µ , µ | · | − v ) is of codimension 1 in Ind( µ ⊗ µ ); we call σ ( µ , µ | · | − v ) special .If furthermore µ is unramified, then there exist nonzero U ( ω v )-invariant vectors in σ ( µ , µ | · | − v ).In this case we say that σ ( µ , µ | · | − v ) is unramified special .If π = π ( µ , µ ) is unramified and χ is an unramified character of F × v , one defines the L -function L ( s, π, χ ) = [(1 − χ ( ω v ) µ ( ω v ) | ω v | sv )(1 − χ ( ω v ) µ ( ω v ) | ω v | sv )] − . If π = σ ( µ, µ | · | − v ) is unramified special, and χ is unramified, one defines the L -function L ( s, π, χ ) = (1 − χ ( ω v ) µ ( ω v ) | ω v | sv ) − . Fix a place v of F . Let π be an admissible irreducible representation of GL ( F v ) with trivial centralcharacter, n = n π the conductor of π . 11et W ( π, ψ v ) be the Whittaker model of π attached to the additive character ψ v [20]. Recallthat W ( π, ψ v ) is a subspace of smooth functions W : GL ( F v ) → C that satisfy the following conditions: • W ( (cid:2) x (cid:3) g ) = ψ v ( x ) W ( g ) for all x ∈ F v . • If v is archimedean, then W ( (cid:2) a
00 1 (cid:3) ) = O ( | a | Mv ) for some positive number M .When π is not 1-dimensional, such a Whittaker model uniquely exists [20, Theorem 2.14]. Thespace W ( π, ψ v ) is precisely described by [19, Lemma 14.3].If v is archimedean and π is a discrete series of weight k , we take W π ∈ W ( π, ψ v ) to be W π ( z (cid:2) a x (cid:3)(cid:2) cos φ sin φ − sin φ cos φ (cid:3) ) = a k e − πa R + ( a ) · sgn( z ) k ψ ( x ) e ikφ for a, z ∈ R × , x, φ ∈ R . See [20, §
5] for the existence of such a W π .Let W π be the Whittaker newform normalized such that W π ( (cid:2) d Fv
00 1 (cid:3) ) = 1 and W π ( gu ) = W π ( g )for all u ∈ U ( n ). Here, d F v a generator of D F v . Lemma 3.1. (a) If π = π ( µ , µ ) is unramified, then W π ( (cid:2) a
00 1 (cid:3) ) = W π ( (cid:2) ad − Fv
00 1 (cid:3) ) with W π ( (cid:2) a
00 1 (cid:3) ) = µ ( aω v ) − µ ( aω ) µ ( ω v ) − µ ( ω v ) | a | v O Fv ( a ) . (b) If π = σ ( µ, µ | · | − v ) is unramified special, then W π ( (cid:2) a
00 1 (cid:3) ) = W π ( (cid:2) ad − Fv
00 1 (cid:3) ) with W π ( (cid:2) a
00 1 (cid:3) ) = µ ( a ) | a | v O Fv ( a ) . Proof.
Assertion (a) is just Macdonald’s formula [21, 22]. See also [6, Theorem 4.6.5]. Note thatthe conductor of our ψ v is D F v while the conductor of ψ v in loc. cit. is O F v .For (b) we observe that W π ( (cid:2) au
00 1 (cid:3) ) = W π ( (cid:2) a
00 1 (cid:3) ) (3.1)for u ∈ O × F v , since W π is right invariant by the group { (cid:2) u
00 1 (cid:3) : u ∈ O × F v } ⊂ U ( ω v ). Let U v be theHecke operator defined by ( W π | U v )( g ) = X b ∈O Fv /ω v O Fv W π ( g (cid:2) ω v b (cid:3) ) . Then W π is an eigenvector of U v . Write W π | U v = α v W π . Then α v W π ( (cid:2) ω mv
00 1 (cid:3) ) = X b ∈O Fv /ω v O Fv W π ( (cid:2) ω mv
00 1 (cid:3)(cid:2) ω v b (cid:3) )= X b ∈O Fv /ω v O Fv W π ( (cid:2) ω mv b (cid:3)(cid:2) ω m +1 v
00 1 (cid:3) )= X b ∈O Fv /ω v O Fv ψ v ( ω mv b ) W π ( (cid:2) ω m +1 v
00 1 (cid:3) )= ( m < ord v d F v , | ω v | − v W π ( (cid:2) ω m +1 v
00 1 (cid:3) ) if m ≥ ord v d F v . α v = µ ( ω v ) | ω v | − v and obtain the desiredexpression of W π . Remark . If further π = σ ( µ, µ | · | − v ) is of trivial central character, then µ = | · | v and α v = ± W ∈ W ( π, ψ v ) and each continuous character χ : F × v → C the local zeta integral isdefined by Ψ( s, W, χ ) = Z F × v W ( (cid:2) a
00 1 (cid:3) ) χ ( a ) | a | s − v d × a, ( s ∈ C ) . Then Ψ( s, W, χ ) converges when Re( s ) is sufficiently large, and has a meromorphic continuation tothe whose C . Proposition 3.3.
In the case when v is finite, if π is unramified or unramified special, and if χ isunramified, then Ψ( s, W π , χ ) = L ( s, π ⊗ χ ) · χ ( D F v ) |D F v | sv Compare this with [17, (2.1)] and note that the formula in loc. cit. misses a factor.
Proof.
We have Ψ( s, W π , χ ) = Z F × v W ( (cid:2) ad − Fv
00 1 (cid:3) ) χ ( a ) | a | s − v d × a = Z F × v W ( (cid:2) a
00 1 (cid:3) ) χ ( ad F v ) | ad F v | s − v d × a = | d F v | s − v χ ( d F v ) Z F × v W ( (cid:2) a
00 1 (cid:3) ) χ ( a ) | a | s − v d × a. In the unramified special case π = σ ( µ, µ | · | − v ) we have Z F × v W ( (cid:2) a
00 1 (cid:3) ) χ ( a ) | a | s − v d × a = vol( O × L v , d × v x ) X i ≥ µ ( ω v ) i χ ( ω v ) i | ω v | isv = | d F v | / v − µ ( ω v ) χ ( ω v ) | ω v | sv = | d F v | / v L ( s, π ⊗ χ ) . In the unramified case π = π ( µ , µ ) we have Z F × v W ( (cid:2) a
00 1 (cid:3) ) χ ( a ) | a | s − v d × a = vol( O × L v , d × v x ) X i ≥ µ ( ω v ) i − µ ( ω v ) i µ ( ω v ) − µ ( ω v ) χ ( ω v ) i | ω v | isv = | d F v | / v µ ( ω v ) − µ ( ω v ) (cid:18) µ ( ω v )1 − χ ( ω v ) µ ( ω v ) | ω v | sv − µ ( ω v )1 − χ ( ω v ) µ ( ω v ) | ω v | sv (cid:19) = | d F v | / v L ( s, π ⊗ χ ) , as expected. 13ne defines the GL ( F v )-invariant pairing b v : W ( π, ψ v ) × W ( π, ψ v ) → C by b v ( W , W ) = Z F × v W ( (cid:2) a
00 1 (cid:3) ) W ( (cid:2) − a
00 1 (cid:3) ) d × a. Proposition 3.4. ( [17, Lemma 4.2] )(a) If v is archimedean and π is a discrete series of weight k , then b v ( W π , π ( (cid:2) − (cid:3) ) W π ) = (4 π ) − k Γ( k ) . (b) If v is finite and π is unramified, then b v ( W π , W π ) = ζ F v (1) ζ F v (2) L (1 , Ad π ) |D F v | / v . (c) If v is finite and π = σ ( µ, µ | · | − v ) is unramified special, then b v ( W π , π ( (cid:2) − ω v (cid:3) ) W π ) = ǫ (1 / , π, ψ v ) L (1 , Ad π ) |D F v | / v . Here, Ad π is the adjoint representation associated to π [13]. Now, let π be an irreducible admissible representation of GL ( A F ). By tensor product theorem(see [6, Theorem 3.3.3]) π has a decomposition π = ⊗ v π v , where for each v , π v is an admissiblerepresentation of GL ( F v ). Then π has a Whittaker model W if and only if for each place v , π v hasa Whittaker model W v (see [6, Theorem 3.5.4]); in this case W is unique and consists of functionsof the from W ( g ) = Q v W v ( g v ), where W v ∈ W v and W v = W π v for almost all v .We write n , the conductor of π , in the form n = n + n − Y p | p p r p , (3.2)where n + (resp. n − ) is only divided by primes that are split (resp. inert or ramified) in K , and( n + n − , p ) = 1. We assume that n − is square-free. Let B be a definite quaternion algebra over F with discriminant n − b | n − . Fix an Eichler order R of B of level n + .As ramified primes of B are not split in K (i.e. n − b | n − ), K is isomorphic to a subfield of B .We embed K into B such that O K = K ∩ R . Write B = K ⊕ KI with I = β ∈ F × . We maytake I such that β is totally negative and β v ∈ ( O × F v ) for each finite place v | p nn − b n ν ; β v ∈ O × F v if v | N K/F D K . Here n ν is the conductor of ν . We will assume that ( n ν , n − b ) = 1.14e fix an isomorphism B ⊗ F K ∼ = M ( K ) such that a ⊗ (cid:2) a
00 ¯ a (cid:3) (for a ∈ K ), and I ⊗ → (cid:2) − β − (cid:3) . For each σ ∈ Σ, we have an inclusion i σ : B ֒ → B ⊗ F K ∼ = M ( K ) σ −→ M ( C ) . and extend it to B σ .Fix an element ϑ ∈ K such that { , ϑ v } is a basis of O K v over O F v for each v | p nn − b n ν . For sucha v we define the isomorphism i v : B v ≃ M ( F v ) by i v ( ϑ p ) = (cid:2) T( ϑ v ) − N( ϑ v )1 0 (cid:3) , i v ( I ) = p β (cid:2) − ϑ v )0 1 (cid:3) , where T( ϑ v ) = ϑ v + ¯ ϑ v and N( ϑ v ) = ϑ v ¯ ϑ v .For each finite place v ∤ p nn ν that is split in K we fix an isomorphism i v : B v ≃ M ( F v ) suchthat i v ( O K ) ⊂ M ( O F v ) and is diagonally in M ( O F v ), and such that i v ( I ) ∈ ( F × v GL ( O F v ) if val v ( β ) is even ,F × v (cid:2) ω v
00 1 (cid:3) GL ( O F v ) if val v ( β ) is odd . Here val v is the valuation on F × v such that val v ( ω v ) = 1.Let G be the algebraic group over Q such that for any Q -algebra A , G ( A ) = ( A ⊗ Q B ) × . Let Z be the center of G . Then Z ( A ) = ( A ⊗ Q F ) × . For each even integer h ≥ A let V h ( A ) denote the set of homogenous polynomialsof degree h − A . Write V h ( A ) = ⊕ − h 00 1 (cid:3) GL ( O F l ) = a b l ,i GL ( O F l )we define the Hecke operator T l on S B k (Σ) by(T l f )( g ) = X i f ( gb l ,i ) . Assume that i p ( U p ) ∼ = U p, := Q p | p U ( ω p ). Then for each p | p using the double coset decomposition U p (cid:2) ω p 00 1 (cid:3) U p = a b p ,i U p we define U p by U p f ( g ) = X i f ( gb p ,i )We define the Atkin operator w p by ( w p f )( g ) = f ( g (cid:2) ω p (cid:3) ) . Let T U be the Hecke algebra generated by these T l and U p , w p .For v ∈ V ~k ( C ) and f ∈ M B~k ( C ) we can attach to v ⊗ f an automorphic form Ψ( v ⊗ f ) on G ( A )by Ψ( v ⊗ f )( g ) := h f ( g ∞ ) , ρ ~k ( g ∞ ) v i . Every automorphic form arises in this way. Indeed, if π ′ = π ′∞ ⊗ π ′∞ is an automorphicrepresentation of G ( A ) with π ′∞ ∼ = ρ ~k , then π ′∞ appears in M B~k ( C ). p -adic modular forms on definite quaternion algebras For each σ we put k σ ′ = k σ . We form two vector spaces V ~k ( C p ) = V ~k ( C ) ⊗ C , C p and L ~k ( C p ) = L ~k ( C ) ⊗ C , C p over C p . So there is a pairing L ~k ( C p ) × V ~k ( C p ) → C p . Via we have algebraicrepresentations of Q σ GL ( C p ) on V ~k ( C p ) and L ~k ( C p ). Precisely, for g p ∈ GL ( C p ) and f ∈ L ~k ( C p ) we have g p · f = (ˇ ρ ~k ( − ( g p )) − ( f ));the same holds for the algebraic representation on V ~k ( C p ). By abuse of notation we again use ˇ ρ ~k and ρ ~k to denote the resulting algebraic representations.16n Section 3.3 we fix an isomorphism . For each place p of F above p , using the isomorphism B ⊗ F K ∼ = M ( K ) given in Section 3.3 we obtain an inclusion i p : B p ֒ → M ( K p );when p = P ¯ P is split in K , from this inclusion we obtain an isomorphism i p : B p ∼ = M ( K P ) ∼ = M ( F p ) . Lemma 3.6. In the case of p split in K , i p i − p = Ad( ~ p ) with ~ p = (cid:2) √ β √ β − (cid:3)(cid:2) − ϑ ¯ P − ϑ P (cid:3) ∈ GL ( F p ) . Proof. This follows from a simple computation.For each σ ∈ Σ F we put i σ ′ = σ ′ ◦ i p σ and i σ ′ = σ ′ ◦ i p σ . Write i p = ( i σ ′ ) σ ′ . Definition 3.7. Let U = U p U p be a compact open subgroup of G ( A ∞ ). A p -adic modularform on B × , of trivial central character, weight ~k = ( k σ ′ ) and level U , is a continuous function b f : b B × → L ~k ( C p ) that satisfies b f ( zγbu ) = ˇ ρ ~k ( i p ( u − p )) b f ( b )for all γ ∈ B × , u ∈ U , b ∈ b B × = G ( A ∞ ) and z ∈ b F × . Denote by M B~k ( U, C p ) the space of suchforms. Lemma 3.8. If b f is a p -adic modular form on B × (of trivial central character, weight ~k = ( k σ ′ ) and level U ), then the image of b f lies in an O C p -lattice of L ~k ( C p ) .Proof. By definition, for each a ∈ b B × , B × aU and aU have the same image by the map b f . It iscompact since U is compact. By the theorem of Fujisaki (see [26]) we know that the double coset B × \ b B × /U is finite, which implies our lemma.For each f ∈ M B~k ( U, C ) we can attach to it a p -adic modular form b f ∈ M B~k ( U, C p ) defined by b f ( b ) = ˇ ρ ~k ( i p ( b − p )) ( f ( b )) . Then we get an isomorphism M B~k ( U, C ) → M B~k ( U, C p ) , f b f . (3.3)Let E be a finite extension of Q p in C p that contains all embeddings of K σ ′ for all σ ∈ Σ K .With L ~k ( E ) = O σ ∈ Σ K L k σ ( E )instead of L ~k ( C p ) we have the space M B~k ( U, E ) of E -valued modular form on B × of trivial centralcharacter, weight ~k = ( k σ ′ ) and level U . We also have the notations V k σ ( E ) ( σ ∈ Σ K ) and V ~k ( E ).17ssume that U p = U p, . Using notations at the end of Section 3.3 we define operators T l , U p and w p by (T l b f )( g ) = X i b f ( gb l ,i ) , U p b f ( g ) = X i ˇ ρ ~k ( i p i − p ( b p ,i )) b f ( gb p ,i )and ( w p b f )( g ) = ˇ ρ ~k ( i p i − p (cid:2) ω p (cid:3) ) b f ( g (cid:2) ω p (cid:3) ) . Let T U be the Hecke algebra generated by these T l and U p , w p . Then the map (3.3) is T U -equivariant. L -function In this section and the next one we extend the result in [17] in two directions. One is that weallow much bigger field extensions; such an extension is essential for our work. The other is thatwe weaken a local condition in [17]. Let π be an unitary cuspidal automorphic representation on GL ( A F ) with trivial central character.We assume that π satisfies the following conditions: • For each σ ∈ Σ F , π σ is a discrete series of even weight k σ . • If we write the conductor of π as in (3.2), then r p ≤ p | p . • If v | n − , then π v is a special representation σ ( µ v , µ v | · | − v ) with unramified character µ v .We will fix a decomposition n + O K = N + ¯ N + .Put W π = Q v W π v . Let ϕ π be the normalized new form in π defined by ϕ π ( g ) := X α ∈ F W π ( (cid:2) α 00 1 (cid:3) g ) . Let τ n = Q v τ n v be the Atkin-Lehner element defined by τ n σ = (cid:2) − (cid:3) if σ ∈ Σ F , and τ n v = (cid:2) − ̟ ord v n v (cid:3) . if v is finite.Let d t g be the Tamagawa measure on GL . Put h ϕ π , ϕ π i GL = Z A × F GL ( F ) \ GL ( A F ) ϕ π ( g ) ϕ π ( gτ n ) d t g, and || ϕ π || v := ζ F v (2) ζ F v (1) L (Ad π v , b v ( W π v , π ( τ n v ) W π v ) . roposition 4.1. ( Waldspurger formula [28, Proposition 6] ) We have h ϕ π , ϕ π i GL = 2 L (1 , Ad π ) ζ F (2) Y v || ϕ π || v . Put || ϕ π || Γ ( n ) = (cid:16) ζ F (2) · N n · Y v | n (1 + | ̟ v | v ) (cid:17) h ϕ π , ϕ π i GL . Corollary 4.2. One has L (1 , Ad π ) = || ϕ π || Γ ( n ) · | k | d − N n · Y v ∤ n : finite N D / F v · Y v | n − ǫ ( 12 , π v , ψ v )N D / F v · Y v | nn − | ̟ v | v ) || ϕ π || v . Proof. This follows from Proposition 4.1 and Proposition 3.4. Indeed, by Proposition 3.4 one has || ϕ π || v = − k σ − if v is archimedean , N D / F v if v is finite and v ∤ n , N D / F v ǫ ( , π v , ψ v ) if v | n − . For archimedean places σ one uses the fact L (1 , Ad π σ ) = 2 − k σ π − ( k σ +1) Γ( k σ ). Let B and G be as in Section 3.3.Let J be a set of places of F that are above p .Let p be a prime above p . If p / ∈ J and p is inert or ramified in K , we put ς p = 1. If p / ∈ J and p = P ¯ P is split in K , we put ς p = (cid:2) ϑ P ϑ ¯ P (cid:3) . In this case we have for any t = ( t , t ) ∈ K p = K P ⊕ K ¯ P , ς − p i p ( t ) ς p = i p ( t ) = (cid:2) t t (cid:3) . When p ∈ J , for each integer n we define ς ( n ) p ∈ G ( F p ) as follows. When p = P ¯ P splits in K ,we put ς ( n ) p = (cid:2) ϑ − 11 0 (cid:3)(cid:2) ̟ n p 00 1 (cid:3) ∈ GL ( F p ) . When p is inert or ramified in K , we put ς ( n ) p = (cid:2) − (cid:3)(cid:2) ̟ n p 00 1 (cid:3) . When n = 0 we also write ς p for ς (0) p .Let J ′ be a subset of J p such that J ⊂ J ′ ⊂ J p . For each J -tuple of integers ~n = ( n p ) p ∈ J we put ς ( ~n ) J,J ′ = Q p ∈ J ς ( n p ) p · Q p ∈ J ′ \ J ς p . When J ′ = J p we just write ς ( ~n ) J for ς ( ~n ) J,J p . We also write ς J = ς ( ~ J .19 .3 p -stabilizer Let π ′ = ⊗ v π ′ v be the unitary cuspidal automorphic representation on G ( A F ) with trivial centralcharacter attached to π via Jacquet-Langlands correspondence. Then π ′∞ is isomorphic to thealgebraic representation ρ ~k of G ( R ). We may assume that π ′ v = π v ◦ i v for finite place v ∤ n − b . When v | n − b , as π v = σ ( µ v , µ v | · | − v ) is special, π ′ v = ( µ v | · | − / v ) ◦ N B v /F v is 1-dimensional, where N B v /F v is the reduced norm of B v . For each v ∤ n , π ′ v is unramified. We may assume that π ′ v = π v ◦ i v foreach finite place v ∤ n − b .For each finite place v we define a new vector ϕ v ∈ π ′ v by the following • if v | n − b , then ϕ v is a basis of the one dimensional representation π ′ v ; • if v | n − but v ∤ n − b , then ϕ v is fixed by U ( n ) v ; • if v ∤ n , ϕ v is the normalized spherical vector of π ′ v that corresponds to W π v in the Whittakermodel.Realize π ′∞ in S B~k ( C ) so that ϕ ∞ := ⊗ v :finite ϕ v is an element of S B~k ( C ). Then we put ϕ π ′ = Ψ( v m ⊗ ϕ ∞ ).Next we define p -stabilization of ϕ v .For p ∈ J , if r v = 1, we put ϕ † p = ϕ p . If r p = 0, we choose a Satake parameter a p of π p anddefine α p = a p | ω p | − / p ; then we put ϕ † p = ϕ p − α p π ( (cid:2) ̟ p (cid:3) ) ϕ p .If φ ∈ π ′ satisfies φ p = ϕ p (i.e. φ p = ϕ p for each p above p ), we put φ † J = φ − (cid:16) Y p ∈ J : r p =0 α p π ( (cid:2) ̟ p (cid:3) ) (cid:17) φ. From now on, when the notation φ † J appears, we always means that φ satisfies φ p = ϕ p and φ ∞ = ϕ ∞ .Define the Atkin-Lehner element τ n ,Bv of G ( F v ) by τ n ,Bv = I if v is archimedean or v | n − b , (cid:2) − ̟ ord v n v (cid:3) if v | nn − b , v ∤ n . For φ , φ ∈ A ( π ′ ), we define the G ( A F )-equivalent pairing h φ , φ i G := Z G ( F ) Z ( A F ) \ G ( A F ) φ ( g ) φ ( g ) dg where dg is the Tamagawa measure on G/Z . For each place v as π ′ v is self-dual, there exists anon-degenerate pairing h· , ·i v : π ′ v × π ′ v → C . The pairing is unique up to a nonzero scalar. We have h· , ·i G = C π ′ O v h· , ·i v (4.1)for some constant C π ′ that only depends on π .By Casselman’s results [7, Theorem 1] we have h ϕ π ′ , π ′ ( τ n ,B ) ϕ π ′ i G = 020nd h ϕ v , π ′ ( τ n ,Bv ) ϕ v i v = 0for each v .For g ∈ G ( F v ) and a character χ v : K × v → C × one defines the local toric integral for φ v by P ( g, φ v , χ v ) = L (1 , Ad π v ) L (1 , τ K v /F v ) ζ F v (2) L ( , π K v ⊗ χ ) Z K × v /F × v h π ′ ( tg ) φ † J v , π ′ ( Ig ) φ † J v i v h ϕ v , π ′ ( τ n ,Bv ) ϕ v i v χ v ( t ) dt. Let χ : K × A × F \ A × K → C × be a Hecke character of archimedean weight ( m , − m ), m = P σ ∈ Σ F m σ σ ∈ Z [Σ F ].For φ ∈ A ( G ) and g ∈ G ( A F ) one defines the global toric period integral by P ( g, φ, χ ) := Z K × A × F \ A × K φ ( tg ) χ ( t ) dt, where dt is the Tamagawa measure. Proposition 4.3. For any φ such that φ p = ϕ p , we have P ( ς ( ~n ) J , φ † J , χ ) h ϕ π ′ , π ′ ( τ n ,B ) ϕ π ′ i G = ζ F (2) L ( , π K ⊗ χ )2 L (1 , Ad π ) · Y p ∈ J P ( ς ( ~n ) p , φ p , χ p ) · Y p | p, p / ∈ J P ( ς p , φ p , χ p ) · Y v ∤ p P (1 , φ v , χ v ) . Proof. Set φ = π ′ ( ς ( ~n ) J ) φ † J , φ = π ′ ( Iς ( ~n ) J ) φ † J , φ = ϕ π ′ and φ = π ′ ( τ n ,B ) ϕ π ′ . By Waldspurger’sformula (see [29, Theorem 1.4]), noting that in [29] {h· , ·i v } v are taken such that C π ′ = 1, we have P (1 , φ , χ ) P (1 , φ , χ − ) h φ , φ i G = ζ F (2) L ( , π K ⊗ χ )2 L (1 , Ad π ) · Y p ∈ J P ( ς ( ~n ) p , φ p , χ p ) · Y p | p, p / ∈ J P ( ς p , φ p , χ p ) · Y v ∤ p P (1 , φ v , χ v ) . Our coefficient ζ F (2) L ( ,π K ⊗ χ )2 L (1 , Ad π ) is different to the coefficient ζ F (2) L ( ,π K ⊗ χ )8 L (1 ,τ K/F ) L (1 , Ad π ) in [29, Theorem1.4] is due to the reason that our measure on K × A × F \ A × K is the Tamagawa measure so thatvol( K × A × F \ A × K ) = 2 L (1 , τ K/F ), while in [29] one has vol( K × A × F \ A × K ) = 1.Clearly P (1 , φ , χ ) = P ( ς ( ~n ) J , φ † J , χ ), and we have P (1 , φ , χ − ) = Z K × A × F \ A × K φ † J ( tIς ( ~n ) J ) χ ( t − ) dt = Z K × A × F \ A × K φ † J ( I ¯ tς ( ~n ) J ) χ (¯ t ) dt = Z K × A × F \ A × K φ † J ( tς ( ~n ) J ) χ ( t ) dt = P ( ς ( ~n ) J , φ † J , χ ) . Here, χ ( t − ) = χ (¯ t ) since χ is anticyclotomic. 21e fix an anticyclotomic character ν of archimedean type ( m , − m ) and will apply Proposition4.3 to the characters χ such that • χ is of archimedean type ( m , − m ); • χν − comes from a character of Γ − J .Note that the second condition implies that χν − is unramified outside of J .We assume that ν is of conductor n ν O K , where n ν is an ideal of O F coprime to p . By technicalreason we assume that all prime factors of n ν are inert or ramified in K , and that n ν is coprime to n − b . Lemma 4.4. (a) If v ∤ n ν is inert in K , then ν v = 1 . (b) For each v / ∈ J inert or ramified in K , we have χ v = ν v .Proof. When v ∤ n ν is inert in K , ν v is unramified. From K × v = F × v O × K v we obtain ν v = 1. If v / ∈ J is inert in K , then χ v · ν − v is unramified and thus we must have χ v · ν − v = 1.If v / ∈ J is ramified in K , as [ K × v : F × v O × K v ] = 2, we have either χ v · ν − v = 1 or χ v · ν − v is oforder 2. However, χ v · ν − v comes from a character of a pro- p abelian group. Since p > 2, we musthave χ v ν − v = 1.We compute the terms P ( ς p , φ p , χ p ) ( p | p ) and P (1 , φ v , χ v ) ( v ∤ p ).For each p ∈ J put¯ e p ( π, χ ) = χ is ramified;(1 − α − p χ ( P ))(1 − α − p χ ( ¯ P )) if χ is unramified, p = P ¯ P is split;1 − α − p if χ is unramified, p is inert;1 − α − p χ ( P ) if χ is unramified , p = P is ramified.If χ p has conductor p s , we put˜ e p ( π, χ ) = ¯ e p ( π, χ ) − ord p n · (cid:26) α p | p | p if s = 0 , | p | s p if s > . Then we define the p -adic multiplier e p ( π, χ ) by e p ( π, χ ) = ( α p | p | p ) − s ˜ e p ( π, χ ) . In [17] the p -adic multiplier e p ( π, χ ) is defined to be our ¯ e p ( π, χ ). But the author think thatit is better to define it as above since it instead of ¯ e p ( π, χ ) appears in the interpolation formula inTheorem 5.9.Let d K p be a generator of D K p . Note that | d K p | p = | d K p | / p O K . Proposition 4.5. ( [17, Proposition 4.11] ) Let p be in J . Suppose that χ p has conductor p s . Put n = max { , s } . Then we have P ( ς ( n ) p , ϕ π, p , χ p ) = ˜ e p ( π, χ ) · || ϕ || p · (cid:26) | d F p | p ζ F p (1) if p is split in K, | d K p | p L (1 , τ K p /F p ) if p is inert in K. For p / ∈ J above p , χ p is unramified. Proposition 4.6. Let p be a prime of F above p , p / ∈ J . If p is split in K , and if π ′ p is a unramified principal series, then P ( ς p , ϕ p , χ p ) = | d F p | p . (b) If p is split in K , and if π ′ p is a unramified special representation, then P ( ς p , ϕ p , χ p ) = ǫ ( , π p , ψ p ) || ϕ p || p χ P ( ω P ) | d F p | p . (c) If p is inert or ramified in K , and if π ′ p is a unramified principal series, then P ( ς p , ϕ p , χ p ) = | d F p | − p | d K p | p . (d) If p is inert in K , and if π ′ p = σ ( µ, µ | · | − p ) is a unramified special representation, then P ( ς p , ϕ p , χ p ) = | d K p | p | d F p | / p | ω p | p ( ǫ ( , π p , ψ p ) + α p )1 − | ω p | p , where α p is defined in the proof of Lemma 3.1 ( b ) . (e) If p is ramified in K , and if π ′ p is a unramified special representation, then P ( ς p , ϕ p , χ p ) = | d K p | p | d F p | / p | ω p | p ) ǫ ( , π p , ψ p ) . Proof. Assertions (a) and (b) follow from [17, Proposition 4.6, 4.9]. Assertions (c) and (e) followsfrom [16, Proposition 3.8, 3.9].We prove (d). By Lemma 4.4, χ p is trivial. Since i p ( I ) is in U ( ω p ), we have b p ( π ′ p ( g ) W p , π ′ p ( I ) W p ) = b p ( π ′ p ( g ) W p , W p ) . Note that m ( g ) = b p ( π ′ p ( g ) W p , W p )only depends on the double coset U ( p ) gU ( p ). For g = 1 we have m (1) = | d F p | / p L (1 , Ad π p ) . For g = (cid:2) (cid:3) , by the fact π ′ p ( (cid:2) − ω p (cid:3) ) W p = ǫ ( , π p , ψ p ) W p , we obtain m ( (cid:2) (cid:3) ) = ǫ ( 12 , π ′ p , ψ p ) b p ( π ′ p ( (cid:2) − ω p 00 1 (cid:3) ) W p , W p )= ǫ ( 12 , π ′ p , ψ p ) µ ( ω p ) | ω p | / p | d F p | / p L (1 , Ad π p ) . We have the decomposition K × p = F × p (1 + ω p O F p ϑ ) ⊔ F × p ( O F p + ϑ ) . a ∈ ω p O F p we have 1 + aϑ ∈ U ( p ). For any a ∈ O F p , we have a + ϑ ∈ U ( p ) (cid:2) (cid:3) U ( p ).Now we have Z K × p /F × p m ( t ) χ p ( t ) dt = m (1)vol(1 + ω p O F p ϑ ) + m ( (cid:2) (cid:3) )vol( O F p + ϑ )= m (1) | ω p | p | ω p | p | d K p | p | d F p | / p + m ( (cid:2) (cid:3) ) 11 + | ω p | p | d K p | p | d F p | / p = | d K p | p L (1 , Ad π p )1 + | ω p | p [ | ω p | p + ǫ ( 12 , π p , ψ p ) µ ( ω p ) | ω p | / p ] , and b p ( W p , (cid:2) − ω p (cid:3) W p ) = ǫ ( 12 , π p , ψ p ) | d F p | / p L (1 , Ad π p ) . When π p is unrmaified special, we have L ( 12 , π K p ⊗ χ p ) = L ( 12 , π K p ⊗ 1) = L (1 , Ad π p ) . Therefore, P ( ς p , ϕ p , χ p ) = L (1 , τ K p /F p ) ζ F p (2) | d K p | p | d F p | / p ǫ ( , π p , ψ p ) | ω p | p + µ ( ω p ) | ω p | / p | ω p | p = | d K p | p | d F p | / p ǫ ( , π p , ψ p ) | ω p | p + µ ( ω p ) | ω p | / p − | ω p | p , as expected. Corollary 4.7. Let p be a prime of O F , p / ∈ J . In the case that p is inert in K and π p is unramifiedspecial, we assume that α p = − . Then P ( ς p , ϕ p , χ p ) = 0 . Proof. The assertion for the case of p split in K and the case of π unramified follows directlyfrom Proposition 4.6. If p is inert in K , and π p = σ ( µ, µ | · | − p ) is unramified special, then ǫ ( , π p , ψ p ) = − α p = 1, then by Proposition 4.6(d) we have P ( ς p , ϕ p , χ p ) = 0 . Put c v = ord v n ν . Proposition 4.8. ( [16, Proposition 3.8, 3.9], [17, Proposition 4.6, 4.9] ) Let v be a finite placecoprime to p n − b , and satisfies the condition that if v | n − n − b and is inert in K , then v | n ν . (a) If v is split in K and v ∤ n + , put φ v = ϕ v . Then P (1 , φ v , χ v ) = (cid:26) | d F v | v if val v ( β ) is even | d F v | v · χ v (( ω v , if val v ( β ) is odd . Here ( ω v , is in K v ∼ = F v ⊕ F v . If v | n + , writing v = w ¯ w with w | N + , put φ v = π ′ v ( (cid:2) ϑ w ϑ ¯ w (cid:3) ) ϕ v . Then P (1 , φ v , χ v ) = ǫ ( , π v , ψ v ) || ϕ π || v χ w ( N + ) | d F v | v . (c) If v is inert or ramified in K and v ∤ n − , put φ v = π ′ v ( (cid:2) ω ord v n νv 00 1 (cid:3) ) ϕ v . Then P (1 , φ v , χ v ) = | d K v | v | d F v | − / v · (cid:26) if c v = 0 ,L (1 , τ K v /F v ) | ω v | c v v if c v > . (d) If v | n − n − b , put φ v = π ′ v ( (cid:2) ω ord v n ν − ord v n v 00 1 (cid:3) ) ϕ v . Then P (1 , φ v , χ v )= 1 + | ω v | v ǫ ( , π v , ψ v ) | d K v | v | d F v | / v · if c v = 0 and v is ramified in K,L (1 , τ K v /F v ) | ω v | c v v if c v > . Proof. In the first case Assertion (a) is proved in [17, Proposition 4.6]. In the second case, one onlyneeds to revise the argument in loc. cit. slightly. Proposition 4.9. If v | n − n − b , v ∤ n ν , and v is inert in K , put φ v = ϕ v . Then P (1 , φ v , χ v ) = | d K v | v | d F v | / v | ω v | v ( ǫ ( , π v , ψ v ) + α v )1 − | ω v | v . Proof. The argument is the same as the proof of Proposition 4.6 (d).Now we consider the case of v | n − b . Note that π ′ v is the representation ξ ◦ N B v /F v where N B v /F v is the reduced norm, and ξ is an unramified character of F × p such that ξ ( ω p ) = α v . Proposition 4.10. If v | n − b , take φ v = ϕ v . (a) If v is inert in K , then P (1 , φ v , χ v ) = 1 ζ F v (1) · | d K v | v | d F v | / v . (b) In the case of v ramified in K , let ˜ ω v be a unifomrmizing element in K v . If ν v (˜ ω v ) = α v ,then P (1 , φ v , χ v ) = 2 ζ F v (1) · | d K v | v | d F v | / v . If ν v (˜ ω v ) = − α v , then P (1 , φ v , χ v ) = 0 .Proof. Note that, if π v = σ ( µ, µ | · | − v ), then π ′ v is the representation ξ ◦ N B v /F v where N B v /F v isthe reduced norm, and ξ = µ | · | − / v . In particular, ξ ( ω v ) = α v .25n the case when v is inert in K , as χ v is unramified, χ v = 1. Using L (1 , Ad π v ) = L ( , π K v ⊗ P (1 , φ v , χ v ) = P (1 , φ v , 1) = L (1 , Ad π v ) L (1 , τ K v /F v ) ζ F v (2) L ( , π K v ⊗ 1) vol( F × v \ K × v ) = 1 ζ F v (1) · | d K v | v | d F v | / v . In the case when v is ramified in K , as χ v is unramified, we have χ v (˜ ω v ) = χ v ( ω v ) = 1. Thus χ v (˜ ω v ) = ± ± α v . In this case, L (1 , Ad π v ) = − µ ( ω v ) | ω v | v = ζ F v (2) and L (1 , τ K v /F v ) = 1, and L ( , π K v ⊗ χ v ) = − ξ ( ω v ) χ v (˜ ω v ) | ω v | v .We have the decomposition F × v \ K × v = O × F v \O × K v ⊔ O × F v \O × K v ˜ ω v . Thus P (1 , φ v , χ v ) = (1 − ξ ( ω v ) χ v (˜ ω v ) | ω v | v )vol( O × F v \O × K v )(1 + χ v (˜ ω v ) ξ ( ω v ))= ( ζ Fv (1) | d Kv | v | d Fv | / v if χ v (˜ ω v ) = ξ v ( ω v ) = α v χ v (˜ ω v ) = − ξ v ( ω v ) = − α v . By Lemma 4.4, χ v = ν v . We obtain the assertion in the ramified case. Proposition 4.11. For σ ∈ Σ we have P (1 , v m σ , χ σ ) = Γ( k σ ) π Γ( k σ / m σ )Γ( k σ / − m σ ) . Proof. See the proof of [17, Theorem 4.12].We make the following assumption on π , ν and B .(i) If p / ∈ J is inert in K , and if π p is unramified special, then α p = − 1. See Remark 3.2.(ii) All prime factors of n ν is inert or ramified in K , and ( n ν , n − b ) = 1.(iii) If v | n − n − b and v is inert in K (in this case π v must be unramified special), then either v | n ν or α v = − v | n − b is ramified in K , then ν v (˜ ω v ) = α v . Remark . If n − b is the product of the prime factors of n − that are inert in K , then (iii) and (iv)automatically hold. Remark . In [17] it demands (ii), (iv) and a stronger version of (iii) that if v | n − n − b , then v | n ν .These conditions are assumed or implicitly assumed in [17, Theorem 4.12]. So we weaken Hung’sconditions on ν . It is not harm for us to assume (iv), since in the branches with ν v (˜ ω v ) = − α v forsome v | n − b ramified in K , the p -adic L -functions we will construct are identically zero.Now, we fix our choice of φ . If v | p , we take φ v = ϕ v . At other places let φ v be as defined inPropositions 4.8, 4.9 and 4.10. 26ut e Ω − J,φ = 2 L (1 , Ad π ) ζ F (2) h ϕ π ′ , π ′ ( τ n ,B ) ϕ π ′ i G · Y σ ∈ Σ P (1 , v m σ , ν σ ) · Y p ∈ J : inert || ϕ p || p | d K p | p L (1 , τ K p /F p ) · Y p ∈ J : split || ϕ p || p | d F p | p ζ F p (1) · Y p | p, p / ∈ J : p is split and r p = 1 || ϕ v || p ǫ ( , π v , ψ v ) | d F p | p · Y p | p, p / ∈ J : p is not split or r p = 0 1 P ( ς p , ϕ p , · Y v | n + || ϕ v || v ǫ ( , π v , ψ v ) | d F v | v · Y v ∤ p n + :finite and not split 1 P (1 , φ v , ν v ) · Y v ∤ p n + :finite and split 1 | d F v | v . This makes sense since almost all factors in the product are 1; e Ω − J,φ depends on J , π ′ and ν , butnot on χ ; e Ω − J,φ = 0.We can state Proposition 4.3 as follows. Corollary 4.14. Let π and χ be as above. Put χ ( N + ) = Y w | N + χ v ( ω w ) ord w N + . Then P ( ς ( n ) J , φ † J , χ ) = χ ( N + )( Y v χ v (( ω v , Y p ∈ J ˜ e p ( π, χ )) · L ( , π K ⊗ χ ) e Ω − J,φ , where v runs over the set { v | p, v / ∈ J : v is split in K and r v = 1 } ∪ { v is finite and split in K : val v ( I ) is odd } . (4.2) p -adic L -functions Fix m = P σ ∈ Σ K m σ σ , and a continuous p -adic character b ν of Gal( ¯ K/K ) of type ( m , − m ). Let ν bethe complex Hecke character attached to b ν ; we assume that ν is of conductor c = n ν O K (( n ν , p ) = 1)and that ν p = 1. We keep the assumption on π and ν at the end of Section 4.3. Definition 5.1. For a continuous p -adic character b χ of Gal( ¯ K/K ), we say that b χ lies in the J -branch of b ν if b χ · b ν − comes from a character of Γ − J .Write U p = Y v : finite and coprime to p O × K v . For an ideal a of O K (coprime to p ) let U p a be the subgroup U p a := { x = ( x v ) v ∈ A ∞ ,pK : x v ∈ b O × K v ∩ a b O K v } U p .We take φ as in Section 4.3. There exists an ideal ˜ c of O K coprime to p such that φ ∞ is U p ˜ c -invariant. Sharking ˜ c if necessary we may assume that c | ˜ c . We put˜ φ ∞ = 1[ U p c : U p e c ] X g ∈ U p c /U p e c gφ ∞ , and ˜ φ = Ψ( v m ⊗ ˜ φ ∞ ). Then ˜ φ ∞ , ˜ φ and ˜ φ † J are U p c -invariant. Lemma 5.2. The function a ˜ φ † J ( aς ( ~n ) J ) ν ( a ) on A × K is A × F K × K ×∞ b O × ~n, c -invariant.Proof. As K ×∞ acts on v m by ν − ∞ , we obtain that a ˜ φ † J ( aς ( ~n ) J ) ν ( a ) is K ×∞ -invariant.Both a ˜ φ † J ( aς ( ~n ) J ) and ν are A × F -invariant and K × b O × ~n, c -invariant. Thus the function a ˜ φ † J ( aς ( ~n ) J ) ν ( a ) is A × F K × b O × ~n, c -invariant.For any J -tuple ~n = ( n p ) p ∈ J let [ ] ~n, c be the composition of maps A ∞ , × K /K × rec K −−−→ Gal( K/K ) → G ~n, c . Note that [ ] ~n, c factors through X ~n, c := A ∞ , × F \ A ∞ , × K /K × b O ~n, c . We define the theta element Θ ~n, c by Θ ~n, c = 1 Q p ∈ J α n p p · X a ∈ X ~n, c ˜ φ † J ( aς ( ~n ) J ) ν ( a )[ a ] ~n, c . For two J -tuples ~n and ~n ′ we say that ~n ′ ≥ ~n if n ′ p ≥ n p for each p ∈ J . When ~n ′ ≥ ~n we havethe natural quotient map π ~n ′ ,~n : G ~n ′ , c → G ~n, c .The theta elements Θ ~n, c ( ~n varying) have the following compatible property. Proposition 5.3. If ~n ′ ≥ ~n , then π ~n ′ ,~n (Θ ~n ′ , c ) = Θ ~n, c . Proof. We easily reduce to the case that there exists exactly one p ∈ J such that n ′ p = n p + 1 and n ′ q = n q for q ∈ J, q = p . In this case our assertion follows from the relation ˜ φ † J | U p = α p ˜ φ † J andthe fact that ( ˜ φ † J | U p )( aς ( n p ) p ) = X u ˜ φ † J ( auς ( n p +1) p )where u runs through the coset [(1 + p n p O K p ) O × F p ] / [(1 + p n p +1 O K p ) O × F p ]. Proposition 5.4. If b χ is in the J -branch of b ν , and if b χ b ν − comes from a character of G ~n, c , then Q p ∈ J α n p p P ( ς ( ~n ) J , φ † J , χ ) = vol( b O × ~n, c / ( b O × ~n, c ∩ K × ) b O × F )Θ ~n, c ( − ( b χ b ν − )) , where χ is the complex Hecke character attached to b χ . roof. As χν − = − ( b χ b ν − ) comes from a character of G ~n, c , by Lemma 5.2 the function t ˜ φ † J ( tς ( ~n ) J ) χ ( t )is A × F K × K ×∞ b O × ~n, c -invariant. Hence, P ( ς ( ~n ) J , φ † J , χ ) = Z K × A × F \ A × K φ † J ( tς ( ~n ) J ) χ ( t ) dt = vol( b O × ~n, c / ( b O × ~n, c ∩ K × ) b O × F ) · X a ∈ A ∞ , × F \ A ∞ , × K /K × b O × ~n, c ˜ φ † J ( aς ( ~n ) J ) χ ( a )= vol( b O × ~n, c / ( b O × ~n, c ∩ K × ) b O × F ) · X a ∈ A ∞ , × F \ A ∞ , × K /K × b O × ~n, c ˜ φ † J ( aς ( ~n ) J ) ν ( a ) χν − ([ a ] J,~n ) , as wanted.By abuse of notation we use N c to denote the positive integer generating the ideal N F/ Q ( c ). Lemma 5.5. We have vol( b O × ~n, c / ( b O × ~n, c ∩ K × ) b O × F ) = 1[ O × K ∩ b O × ~n, c : O × F ] | N F/ Q D F | ∞ | N K/ Q D K | ∞ N c · Q p ∈ J | ω p | − n p p · Y v ∈ J or v | c : v inert L (1 , τ K v /F v ) · Y v ∈ J or v | c : v split ζ F v (1) . Proof. By (2.1) we havevol( b O × K / b O × F ) = Y v : finite vol( O × K v / O × F v ) = | N F/ Q D F | ∞ | N K/ Q D K | ∞ . Note that[ b O × K : b O ~n, c ] = Y v ∈ J or v | c : v inert L (1 , τ K v /F v ) − · Y v ∈ J or v | c : v split ζ F v (1) − · N c · Y p ∈ J | ω p | − n p p . Sovol( b O × ~n, c / b O × F ) = | N F/ Q D F | ∞ | N K/ Q D K | ∞ N c · Q p ∈ J | ω p | − n p p · Y v ∈ J or v | c : v inert L (1 , τ K v /F v ) · Y v ∈ J or v | c : v split ζ F v (1) . Our lemma follows.Put b v m = ( v m ). Lemma 5.6. For any a ∈ A × K we have [ ˜ φ † J ( aς ( ~n ) J ) ν ( a )] = h ρ ~k ( i p ( ς ( ~n ) J )) − b v m , \ ˜ φ ∞ , † J ( aς ( ~n ) J ) i · b ν ( a ) . roof. Both sides of the equality are K ×∞ -invariant. So we may assume a = ( a p , a p ) ∈ A ∞ , × K . Bydefinition of \ ˜ φ ∞ , † J we have ( ˜ φ ∞ , † J ( aς ( ~n ) J )) = ˇ ρ ~k ( i p ( a p ς ( ~n ) J )) \ ˜ φ ∞ , † J ( aς ( ~n ) J ) . Thus ( ˜ φ † J ( aς ( ~n ) J )) = h b v m , ˇ ρ ~k ( i p ( a p ς ( ~n ) J )) ˜ φ ∞ , † J ( aς ( ~n ) J ) i = h ρ ~k ( i p ( a p ς ( ~n ) J )) − b v m , ˜ φ ∞ , † J ( aς ( ~n ) J ) i = Y σ ∈ Σ K (cid:18) σ ′ a P σ ¯ σ ′ a ¯ P σ (cid:19) m σ · h ρ ~k ( i p ( ς ( ~n ) J )) − b v m , ˜ φ ∞ , † J ( aς ( ~n ) J ) i , where the last equality follows from the fact ρ ~k ( i p ( a p ) − ) b v m = Y σ ∈ Σ K (cid:18) σ ′ a P σ ¯ σ ′ a ¯ P σ (cid:19) m σ b v m . Now, our lemma follows from the relation b ν ( a ) = Y σ ∈ Σ K (cid:18) σ ′ a P σ ¯ σ ′ a ¯ P σ (cid:19) m σ · ( ν ( a ))for a ∈ A ∞ , × K .Put b Θ ~n, c := Θ ~n, c . The following is a direct consequence of Lemma 5.6. Corollary 5.7. We have b Θ ~n, c = 1 ( Q p ∈ J α n p p ) · X a ∈ X ~n, c h ρ ~k ( i p ( ς ( ~n ) J )) − b v m , \ ˜ φ ∞ , † J ( aς ( ~n ) J ) i b ν ( a )[ a ] ~n, c . Let o V ~k be the O C p -lattice in V ~k ( C p ) generated by { b v ~s : k σ − ≤ s σ ≤ k σ − } . Lemma 5.8. There is a constant c ∈ C × p such that ρ ~k ( i p ( ς ( ~n ) J )) − ( o V ~k ) ⊆ c Y p ∈ J p n p P σ ∈ Σ p kσ − o V ~k . Proof. Let o σ be the lattice in V k σ generated by { ( v s ) : k σ − ≤ s ≤ k σ − } . It is easy to checkthat if p σ ∈ J , then ς ( n p ) p σ sends o σ to p n p σ kσ − σ ς p σ o σ . Let c ∈ C × p be such that ς J o V ~k ⊂ c o V ~k . Weobtain our assertion.Now, we assume d ϕ ∞ is “algebraic” in the sense that(alg) there is a finite extension E of Q p in C p that contains all embeddings of F p ( p | p ),such that d ϕ ∞ lies in M ~k ( U, E ). 30e also make the following “ordinary” assumption.(ord) For each p ∈ J we have | α p | p = | ω p | P σ ∈ Σ p kσ − p , i.e. the p -adic valuation of the U p -eigenvalue α p is P σ ∈ Σ p − k σ .We define the complex period Ω − J,φ byΩ − J,φ : = e Ω − J,φ · | N F/ Q D F | ∞ | N K/ Q D K | ∞ · Q w | c :inert L (1 , τ K w /F w ) · Q w | c : split ζ F w (1) [ O × K ∩ b O × ~n, c : O × F ] N c · Y p ∈ J :inert L (1 , τ K w /F w ) · Y p ∈ J :split ζ F w (1) = 2 L (1 , Ad π ) ζ F (2) h ϕ π ′ , π ′ ( τ n ,B ) ϕ π ′ i G · Y σ ∈ Σ P (1 , v m σ , ν σ ) · | N F/ Q D F | ∞ | N K/ Q D K | ∞ · Y v | n + || ϕ v || v ǫ ( , π v , ψ v ) | d F v | v · Y v ∤ p n + : finite and not split P (1 , φ v , ν v ) · Y v ∤ p n + : finite and split | d F v | v · Q w | c :inert L (1 , τ K w /F w ) · Q w | c : split ζ F w (1) [ O × K ∩ b O × ~n, c : O × F ] N c · Y p ∈ J :inert || ϕ p || p | d K p | p · Y p ∈ J :split || ϕ p || p | d F p | p · Y p | p, p / ∈ J : p is split and r p = 1 || ϕ p || p ǫ ( , π p , ψ p ) | d F p | p · Y p | p, p / ∈ J : p is not split or r p = 0 1 P ( ς p , ϕ p , Theorem 5.9. Under the conditions (alg) and (ord) there exists b Θ J ∈ O E [[Γ − J ]] ⊗ O E E such thatthe following holds:For each continuous p -adic character b χ of Gal( ¯ K/K ) of type ( m , − m ) in the J -branch of b ν suchthat χ · ν − is a character of conductor Q p ∈ J p n p for some ~n = ( n p ) p ∈ J , we have − b Θ J ( b χ b ν − ) = χ ( N + ) · Y v χ v ( ω v ) · Y p ∈ J e p ( π, χ ) · L ( , π K ⊗ χ )Ω − J,φ where v runs over the set ( ) .Proof. By Lemma 3.8 and Lemma 5.8 there exists some integer r independent of ~n such that b Θ ~n, c ∈ ω rE O E [[ G ~n, c ]]. So by Proposition 5.3, the inverse limit lim ←−− ~n b Θ ~n, c exists in ω rE · O E [[ G J ; c ]]. Wedenote it as b Θ J ; c and let b Θ J be its image in O E [[Γ − J ]] ⊗ O E E by the projection G J ; c → Γ − J . Nowour assertion follows from Corollary 4.14, Proposition 5.4 and Lemma 5.5.31 efinition 5.10. The element b Θ ~n, c defines a measure µ J on G ~n, c ∼ = A ∞ , × F \ A ∞ , × K /K × b O × ~n, c . The element b Θ J ; c defines a measure µ J on G J ; c ∼ = A ∞ , × F \ A ∞ , × K /K × b O × J ; c . Here b O J ; c = T ~n b O ~n, c .Put L J ( π K , b χ ) = b Θ J ( b χ b ν − ) , where b χ runs through the J -branch of b ν . Then we define the anticyclotomic p -adic L -function L J ( π K , b χ ) by L J ( π K , b χ ) = L J ( π K , b χ ) . In particular, when b χ b ν − runs over the analytic family { ǫ ~s : ~s = ( s σ ) σ ∈ Σ J , | s σ | ≤ | c | p p − } , we obtain a ♯ (Σ J )-variable analytic function L J ( ~s, π K , b ν ) := L J ( π K , b νǫ ~s )on { ( s σ ) σ ∈ Σ J : | s σ | ≤ | c | p p − } . We also write L J ( ~s, π K , b ν ) := L J ( π K , b νǫ ~s ) . Corollary 5.11. If there exists some p ∈ J such that α p = 1 , and if b χ p b ν − p = 1 , then L J ( π K , b χ ) = 0 . Proof. By continuity we may assume that b χ b ν − is of finite order, i.e. b χ is of ( m , m )-type. Then χ p = ν p − ( b χ b µ − ) = 1 and so ˜ e p ( π, χ ) = 0. Thus our statement follows from Theorem 5.9. In this section we define Harmonic cocycle valued and cohomological valued modular forms. Thenwe construct two maps called the Schneider morphism and the Coleman morphism from Harmoniccocycle valued modular forms to cohomological valued modular forms. Finally we explain L -invariant of Teitelbaum type as the ratio of two cohomological valued modular forms. We alsodefine “multiple-variable Harmonic cocycle” valued modular forms and attach them distributionsneeded in Section 7.The L -invariant of Teitelbaum type was defined by Chida, Mok and Park [9]. They defined L -invariant of this type to be the ratio of two sets of data; those data are evaluations at special32oints of our cohomological valued modular forms. We will fill some details omitted in [9]. Forexample, to define Hecke operators on cohomological valued modular forms, we need corestrictionmaps for cohomological groups, while this is not clearly described in [9].To prove our result in Section 8.2, it is rather convenient to use our language of cohomologicalvalued modular forms than that in [9], since we need evaluations of the cohomological valuedmodular forms at points other than those chosen out in [9]. Let B and G be as in Section 3.3. Then B splits at all primes above p . Let J be the subset of J of elements p ∈ J such that α p = 1. We assume that each p ∈ J splits in K .Let U = Q l U l be a compact open subgroup of G ∞ . We assume that for each p ∈ J , U p = U ( ω p ).Let E be a sufficiently large extension of Q p . In § σ we define two vector spaces V k σ ( E ) and L k σ ( E ) over E with actions ρ k σ ◦ i σ ′ i − σ ′ and ˇ ρ k σ ◦ i σ ′ i − σ ′ of GL ( F p σ ). As we willfrequently use these two notations ρ k σ ◦ i σ ′ i − σ ′ and ˇ ρ k σ ◦ i σ ′ i − σ ′ , to avoid cumbursomeness we use ρ k σ and ˇ ρ k σ to denote them.For each subset Σ of Σ F we form two E -vector spaces V Σ ( E ) = O σ ∈ Σ V k σ ( E )and V Σ ( E ) = O σ / ∈ Σ V k σ ( E )with tensor product actions of G p = Q p | p GL ( F p ). Here we make the convention that, if there isno σ ∈ Σ such that p σ = p , then GL ( F p ) acts trivially on V Σ ; and if there is no σ ∈ Σ F \ Σ suchthat p σ = p then GL ( F p ) acts trivially on V Σ . When V Σ or V Σ is clear, by abuse of notation weuse ρ ~k to denote the action of G p . Similarly, we form the E -vector spaces L Σ ( E ) and L Σ ( E ) withactions ˇ ρ ~k of G p . When Σ = Σ J ( J ⊂ J ), we also write J instead of Σ J in the superscript andthe subscript. First we fix a prime p ∈ J .Consider the Bruhat-Tits T p tree for GL ( F p ). Denote by V p and E p the set of vertices andthe set of oriented edges of T p respectively. The source and the target of an oriented edge e aredenoted by s ( e ) and t ( e ) respectively. A vertex can be represented by a homotopy class of lattices L in F ⊕ p . An edge can be represented by a pair of lattices ( L , L ) with [ L : L ] = | ω p | − p . Let e be the edge ( L , L ) with L = ω − p O F p ⊕ O F p and L = O F p ⊕ O F p . Let ~ p be as in Lemma 3.6.If ~ p = (cid:2) a b c d (cid:3) , we let e ∗ be the edge ( L , L ) with L = { ( a ω − p x + b y, c ω − p x + d y ) : x, y ∈ O F p } and L = { ( a x + b y, c x + d y ) : x, y ∈ O F p } . 33e consider the twisted action of GL ( F p ) on F ⊕ p : if i p i − p ( g ) = (cid:2) a bc d (cid:3) , then g · ( x, y ) =( ax + by, cx + dy ). This action induces an action on T p , and also an action on P ( F p ) = F p ∪ {∞} .The benefit of this twisted action is that i p ( K × p ) acts on P ( F p ) diagonally. Indeed, for any t = ( t P , t ¯ P ) ∈ K × p , we have i p ( t ) · x = t P xt ¯ P for any x ∈ P ( F p ). Hence, the fixed points in P ( F p )of i p ( K × p ) are 0 and ∞ . Remark . For the purpose of defining L -invariants, it is cumbersome and unnecessary to considerthe twisted action. However, it is rather convenience when we apply the result in this section toprove the main result in Section 8.2.The isotropy group of e ∗ is F × p U p , and the isotropy group of e is F × p Ad( ~ p ) − ( U p ). Thus E p isisomorphic to the coset GL ( F p ) /F × p U p . Let G p act on T p by the projection G p → GL ( F p ).Let C ( T p , L ~k ( E )) be the space of L ~k ( E )-valued functions on V p , C ( T p , L ~k ( E )) the spaceof L ~k ( E )-valued functions on E p such that f ( e ) = − f (¯ e ). Let G p act on C i ( T p , L ~k ( E )) by γ ⋆ f = γ ◦ f ◦ γ − p . Then we have a G p -equivariant short exact sequence0 / / L ~k ( E ) / / C ( T p , L ~k ( E )) ∂ / / C ( T p , L ~k ( E )) / / ∂ ( f )( e ) = f ( s ( e )) − f ( t ( e )) . For any subgroup Γ of G p , from (6.1) we get the injective map δ Γ : C ( T p , L ~k ( E )) Γ → H (Γ , L ~k ( E )) . (6.2)Let C ( T p , L ~k ( E )) be the space of harmonic forms f ∈ C ( T p , L ~k ( E )) : f ( e ) = − f (¯ e ) ∀ e ∈ E p , and X t ( e )= v f ( e ) = 0 ∀ v ∈ V p . Observe that C ( T p , L ~k ( E )) is G p -stable. We again use δ Γ to denote the composition C ( T p , L ~k ( E )) Γ ֒ → C ( T p , L ~k ( E )) Γ → H (Γ , L ~k ( E )) . For a subgroup Γ of G p , and h ∈ G p , we have a map r h : H (Γ , L ~k ( E )) → H ( h − Γ h, L ~k ( E )) ,φ ( r h φ )( γ ′ ) = ˇ ρ ~k ( h ) − φ ( hγh − ) . We shall use the following lemma later. It comes from homological theory. Lemma 6.2. Let Γ ⊂ Γ be two subgroups of G p . (a) We have the following commutative diagram C ( T p , L ~k ( E )) Γ δ Γ2 / / (cid:127) _ (cid:15) (cid:15) H (Γ , L ~k ( E )) Res Γ2 / Γ1 (cid:15) (cid:15) C ( T p , L ~k ( E )) Γ δ Γ1 / / H (Γ , L ~k ( E )) . When [Γ : Γ ] is finite, we have the following commutative diagram C ( T p , L ~k ( E )) Γ δ Γ1 / / Cor Γ2 / Γ1 (cid:15) (cid:15) H (Γ , L ~k ( E )) Cor Γ2 / Γ1 (cid:15) (cid:15) C ( T p , L ~k ( E )) Γ δ Γ2 / / H (Γ , L ~k ( E )) . (c) If Γ is a subgroup of G p and h is an element of G p , then we have the following commutativediagram C ( T p , L ~k ( E )) Γ δ Γ / / ˇ ρ ~k ( h − ) (cid:15) (cid:15) H (Γ , L ~k ( E )) r h (cid:15) (cid:15) C ( T p , L ~k ( E )) h − Γ h δ h Γ h − / / H ( h − Γ h, L ~k ( E )) . We extend the above notion of harmonic cocycles to multiple variables.Fix a nonempty subset J of J . We form E J = Q p ∈ J E p . Taking product we obtain actionof G J = Q p ∈ J G p on E J . For each element e ∈ E J we write e p ( e ) for the p -component of e and put s p ( e ) = s ( e p ( e )). For each p ∈ J let e p , ∗ ∈ E p be an oriented edge fixed by U p . Write e J , ∗ = ( e p , ∗ ) p ∈ J .For any edge e ∈ E p let U e ⊂ P ( F p ) denote the set of end points of paths (in T p ) through e . For g ∈ G p we have U ge = g U e . If we identify P ( F p ) with F p ∪ {∞} by [ a, b ] ab , then U e p , = O F p .For e = { e j } j ∈ J ∈ E J we put U e = Q j ∈ J U e j . Definition 6.3. A harmonic cocycle on E J with values in L ~k ( E ) is a function c : E J → L ~k ( E )that satisfies the following two conditions:(a) For each p ∈ J if e p is replaced by ¯ e p , then c ( e ) is changed by a sign.(b) For each p ∈ J and ♯ ( J ) elements v p ∈ V p and e p ′ ∈ E p ′ ( p ′ ∈ J , p ′ = p ), we have X s p ( e )= v,e p ′ ( e )= e p ′ c ( e ) = 0 , where the sum runs through all e ∈ E J such that s p ( e ) = v p and e p ′ ( e ) = e p ′ ( p ′ = p ).Let G p act on the space of harmonic cocycles by( g ⋆ c )( e ) = ˇ ρ ~k ( g ) c ( g − J e ) . Definition 6.4. We say that a harmonic cocycle c is bounded , if { ( g ⋆ c )( e J , ∗ ) : g ∈ G J } is bounded for any norm | · | on L ~k ( E ), i.e.sup g ∈ G J | ˇ ρ ~k ( g ) c ( g − e J , ∗ ) | < ∞ . p ∈ J we fix an ι p ∈ Σ p . Write ι J = ( ι p ) p ∈ J .Let LP ι p be the space of of locally polynomials on P ( F p ) of degree ≤ k ι p − 2. Precisely, afunction on P ( F p ) belongs to LP ι p if and only if for each point x of P ( F p ) there exists an openneighborhood U x of x such that f | U x is a polynomial with coefficients in E of degree ≤ k ι p − 2. Wedefine an action of GL(2 , F p ) on LP ι p by(Ad( ~ − p ) (cid:2) a bc d (cid:3) ) − f ( x ) = ι p ( cx + d ) k ι p − ( ad − bc ) kι p − ! f (cid:18) ax + bcx + d (cid:19) . Let LP ι J be the tensor product N p ∈ J LP ι p (of E -vector spaces) with the tensor product action of G J = Q p ∈ J GL ( F p ); LP ι J can be considered as a space of functions on P J := Q p ∈ J P ( F p ).Using the relation (b) in Definition 6.3 we attach to each harmonic cocycle c an L ι J ( E )-valuedlinear functional µ ι J c of LP ι J such that h Z U e Y p ∈ J x j p p µ ι J c , Q i = h c ( e ) , ( N p ∈ J X j p p Y k ι p − − j p p ) ⊗ Q i Q p ∈ J (cid:0) k ι p − j p (cid:1) for e ∈ E J , Q ∈ V ι J ( E ), and j p ∈ { , , · · · , k ι p − } .In the following, for the purpose of simplifying notations, we will write Q p ∈ J x j p p for the element N p ∈ J X j p p Y k ι p − − j p p in V ι J ( E ). Lemma 6.5. For g ∈ G p we have Z U e ( ρ ~k ( g − ) P ) µ ι J c = ˇ ρ ~k ( g − ) Z g U e P µ ι J g⋆c (6.3) for each P ∈ V ι J ( E ) .Proof. If P = Q p ∈ J x j p p ∈ V ι J ( E ) and Q ∈ V ι J ( E ), then h Z U e ρ ~k ( g − ) P µ ι J c , Q i = h c ( e ) , ( ρ ~k ( g − ) P ) ⊗ Q i Q p ∈ J (cid:0) k ι p − j p (cid:1) = h ˇ ρ ~k ( g ) c ( e ) , P ⊗ ρ ~k ( g ) Q i Q p ∈ J (cid:0) k ι p − j p (cid:1) = h ( g ⋆ c )( ge ) , P ⊗ ρ ~k ( g ) Q i Q p ∈ J (cid:0) k ι p − j p (cid:1) = h Z U ge P µ ι J g⋆c , ρ ~k ( g ) Q i = h ˇ ρ ( g − ) Z U ge P µ ι J g⋆c , Q i , as wanted. Remark . If k p = 2 for each p ∈ J , then for each g ∈ G J we have µ ι J c ( g · U e J , ∗ ) = c ( g · e J , ∗ ) . For a point a J ∈ Q p ∈ J F p , and a vector of positive integers m J = ( m p ) p ∈ J , we use U ( a J , m J )to denote the product of closed discs U ( a J , m J ) = Y p ∈ J U ( a p , m p ) . U ( ∞ , m p ) = { x ∈ P ( F p ) : | x | ≥ | ω p | − m p } . For each point a p ∈ P ( F p ), put A ( a p ) = (cid:26) max { , | a p |} k p if a p = ∞ , a p = ∞ . Then we put A ( a J ) = Q p ∈ J A ( a p ). Proposition 6.7. If c is bounded, then there exists a constant A > such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U ( a J ,m J ) Y p ∈ J ( x p − a p ) j p µ ι J c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A · A ( a J ) · Y p ∈ J | ω p | m p ( j p +1 − kι p ) for each a J and m J ≥ . Here m J ≥ means that each component of m J is nonnegative.Proof. Put g p = i p i − p (cid:2) − a p ω m pp (cid:3) if a p = ∞ ,i p i − p (cid:2) ω m pp (cid:3) if a p = ∞ and g = ( g p ) p ∈ G J . Then g p ( U ( a p , m p )) = O F p , and so g U ( a J , m J ) = U (0 J , J ).When none of a p is infinite, as g Y p ∈ J x j p p = ω m p ( k ι p − p ω m p ( kι p − p (cid:18) x p − a p ω m p p (cid:19) j p = Y p ∈ J ω m p ( kι p − − j p ) p ( x p − a p ) j p , by (6.3) we have Z U ( a J ,m J ) Y p ∈ J ( x p − a p ) j p µ ι J c = Y p ∈ J ω m p ( j p +1 − kι p ) p g − Z U (0 J , J ) Y p ∈ J x j p p µ ι J g⋆c . As c is bounded and g − is bounded by A ( a J ), this yields our assertion. When some of a p isinfinite, the argument is similar, and we omit it. Proposition 6.8. If c is bounded, then there is a unique L ι J -valued analytic distribution µ ι J c on P J such that h Q, Z U e Y p ∈ J x j p p µ ι J c i = h Q p ∈ J X j p ι p Y k ι p − − j p ι p ⊗ Q, c ( e ) i Q p ∈ J (cid:0) k ι p − j p (cid:1) , ≤ j p ≤ k ι p − for each Q ∈ V ι J ( E ) . roof. This follows from Proposition 6.7 and a standard Amice-Velu and Vishik’s argument ([1, 27],see also [23, § f on some disc U ( a J , m J ) we write f = X ~j =( j p ): j p ≥ c ~j ( x − a ) ~j . Here, we write ( x − a ) ~j for Q p ∈ J ( x p − a p ) j p ; the coefficients c ~j are in C p ⊗ E L ι J .We write I ( f, a J , m J ) for the lattice of C p ⊗ E L ι J generated by c ~j · Y p ∈ J ι p ( ω p ) m p ( j p − k ι p +1) for all ~j . If U ( a ′ J , m ′ J ) ⊂ U ( a J , m J ), writing f | U ( a ′ J ,m ′ J ) = X ~j c ′ ~j ( x − a ′ ) ~j we have c ′ ~j = X ~n c ~n Y p ∈ J (cid:0) n p j p (cid:1) ( a ′ p − a p ) n p − j p . Since a ′ p − a p ∈ ( ω m p p ), we obtain I ( f, a ′ J , m ′ J ) ⊂ I ( f, a J , m J ). In particular, I ( f, a J , m J ) isindependent of the choice of the center a J .Consider the truncations f a J = X ~j : j p ≤ k ι p − c ~j ( x − a ) ~j and f a ′ J = X ~j : j p ≤ k ι p − c ~j ( x − a ′ ) ~j . Write f a J − f a ′ J = X ~j : j p ≤ k ι p − b ~j ( x − a ) ~j . Then we have b ~j Y p ∈ J ω m p j p p ∈ Y p ∈ J ι p ( ω p ) m p ( k p − · I ( f, a J , m J ) . (6.4)It follows from (6.4) and the estimate in Proposition 6.7 that, for any analytic function f onan open set U and sufficient large m J , writing U as disjoint union of U ( a i,J , m J ), the “ m J -thRiemann sum” X i Z U ( a i,J ,m J ) f a i,J µ ι J c converges when m J → ∞ , yielding our integral R U f µ ι J c .38 orollary 6.9. When k p = 2 for each p ∈ J , if c is bounded, then R ? µ ι J c extends to all compactlysupported continuous functions on F J .Proof. By Proposition 6.7, for any fixed compact open subset U of F J , there exists a constant C U only depending on U such that, for any locally constant function g on F J that is supported in U ,we have | Z gµ ιc | ≤ C U max x ∈ U | g ( x ) | . So, for any continuous function g supported on U , we can take a series of locally constant functions g i supported on U , such that g i → g . Then we put R gµ ι J c = lim i R g i µ ι J c . When g is further locallyanalytic, the integral coincides with that defined in the proof of Proposition 6.8. Indeed, in thiscase we may take g i as in that proof. Proposition 6.10. Fix p ∈ J . For every locally analytic function f p on P J \{ p } := Y p ′ ∈ J : p ′ = p P ( F p ′ ) and each integer j ∈ { , · · · , k p − } we have Z ( f p ⊗ x j p ) µ ι J c = 0 . Proof. We easily reduce to the case when f p is an analytic function on a disc U e ′ ( e ′ ∈ Q p ′ = p E p ′ )and is 0 outside of this disc. As the integral is defined as limit of “Riemann sum”, it suffices toconsider the truncation ( f p ) a ′ (for some a ′ ∈ U e ′ ) instead of f p . But in this case, the assertion isexactly (b) in Definition 6.3. Proposition 6.11. For any g = ( i p i − p ( (cid:2) a p b p c p d p (cid:3) )) p ∈ G p and any locally analytic function f on P J we have ˇ ρ ~k ( g ) Z f ( gx ) · Y p ∈ J ι p ( c p x p + d p ) k ι p − ( a p d p − b p c p ) kι p − ! µ ι J c ( x ) = Z f ( x ) µ ι J g⋆c ( x ) . In particular, if c is Γ -invariant, then for any γ = ( (cid:2) a p b p c p d p (cid:3) ) p ∈ Γ we have ˇ ρ ~k ( γ ) Z f ( γx ) · Y p ∈ J ι p ( c p x p + d p ) k ι p − ( a p d p − b p c p ) kι p − ! µ ι J c ( x ) = Z f ( x ) µ ι J c ( x ) . (6.5) Proof. This follows from (6.3) and a limit argument similar to that in the proof of Proposition6.10. Remark . In the case when k p = 2 for each p ∈ J , (6.5) says that µ ι J c is Γ-invariant.39 .3 Harmonic cocycle valued modular forms Assume that π p = σ ( | · | / , | · | − / ) for each p ∈ J . In this case we have U p ϕ p = ϕ p and w p ϕ p = − ϕ p for each p ∈ J . Definition 6.13. A J -typle “harmonic cocycle” valued modular form of trivial central character,weight ~k and level U = U J Q p ∈ J U ( ω p ) is a function c : G ∞ ,J × E J → L ~k ( E ) that satisfies thefollowing conditions:(a) For each g ∈ G ∞ ,J , c ( g, · ) is a harmonic cocycle.(b) If z ∈ ( F ∞ ,J ) × , then c ( zg, · ) = c ( g, · ).(c) If h ∈ U J , then c ( gh, · ) = ( h − p ⋆ c )( g, · ) . That is c ( gh, · ) = ˇ ρ ~k ( h − p ) c ( g, · ) . (d) If x ∈ G ( Q ), then c ( x g, · ) = ( x ,J ⋆ c )( g, · ) . That is c ( x g, e ) = ˇ ρ ~k ( x ,J ) c ( g, x − ,J e ) . Let C J ( U J , L ~k ( E )) be the space of such “harmonic cocycle” valued modular forms.For each g ∈ G ∞ ,J put e Γ J g = { γ ∈ G ( Q ) : γ l ∈ gU l g − for l / ∈ J } . We embed e Γ J g into G p by ι J g : e Γ J g → G p , γ g − p γ p g p , and put Γ J p = ι J g ( e Γ J g ). Here, we consider g p as an element of G p whose J -components are all 1. Remark . Though e Γ J g only depends on the coset gU J , both ι J g and Γ J g do depend on g p .Indeed, if y ∈ U J , then Γ J gy = y − p Γ J y p . Lemma 6.15. For each g ∈ G ∞ ,J , c ( g, · ) is Γ J g -invariant.Proof. Let γ be in e Γ J g . Then( ι J g ( γ ) ⋆ c )( g, e ) = ˇ ρ ~k ( g − p γ J p g p )ˇ ρ ~k ( γ J ) c ( g, γ − J e ) = ˇ ρ ~k ( g − p γ J p g p ) c ( γg, e ) . The latter equality follows from Definition 6.13 (d). As γ ∈ e Γ J g , we have g − γg ∈ U l for l / ∈ J .Thus by Definition 6.13 (c) we haveˇ ρ ~k ( g − p γ J p g p ) c ( γg, e ) = c ( γg ( g − γg ) − , e ) = c ( g, e ) , as desired. 40here exist Hecke operators T l ( l ∤ p , U l is maximal compact in G l ) on C J ( U J , L ~k ( E )). Forsuch l we fix an isomorphism G l ∼ = GL ( F l ) such that Σ l ∼ = GL ( O F l ) and writeGL ( O F l ) (cid:2) ω l 00 1 (cid:3) GL ( O F l ) = Y t i GL ( O F l ) . Then we define (T l · c )( g, e ) = X i c ( gt i , e ) . Let T pU be the algebra generated by T l ( l ∤ p , U l is maximal compact in G l ).We attach to each “harmonic cocycle” valued modular form a p -adic modular form. For each c ∈ C J ( U J , L ~k ( E )) let b f c be the function on G ∞ defined by b f c ( g ) = ( g − J ⋆ c )( g J , e J , ∗ ) = ˇ ρ ~k ( g − J ) c ( g J , g J e J , ∗ ) . Proposition 6.16. (a) If c ∈ C J ( U J , L ~k ( E )) , then b f c ∈ M ~k ( U, E ) . (b) The map C J ( U J , L ~k ( E )) → M ~k ( U, E ) c b f c is injective and T pU -equivariant. (c) For p ∈ J we have U p b f c = b f c and w p b f c = − b f c . Conversely, if b f ∈ M ~k ( U, E ) satisfies w p b f = − b f , (6.6)and U p b f = b f , (6.7)for each p ∈ J , we associate to it a “harmonic cocyle” valued modular form c b f defined by c b f ( g J , e ) = ˇ ρ ~k ( g e ) b f ( g J g e ) , where g e is an element of G J such that g e ( e J , ∗ ) = e . Note that the value c b f ( g J , e ) does notdepend on the choice of g e . Proposition 6.17. (a) If b f ∈ M ~k ( U, E ) satisfies ( ) and ( ) , then c b f ∈ C J ( U J , L ~k ( E )) . (b) For c ∈ C J ( U J , L ~k ( E )) we have c b f c = c .Proof of Propositions 6.16 and 6.17. It is easy to check that c b f c is a one-to-one correspondencebetween the set of functions c : G ∞ ,J × E J → L ~k ( E ) that satisfy (b), (c) and (d), and the set M ~k ( U, E ). The function c is harmonic cocycle valued, i.e. c ( g, · ) for each g ∈ G ∞ ,J is a harmoniccocycle, if and only if U p b f c = b f c and w p b f c = − b f c for each p ∈ J . That c b f c is T pU -equivariantfollows directly from the definitions of Hecke operators.The following is a direct consequence of Lemma 3.8 and Proposition 6.16.41 orollary 6.18. For any norm | · | on L ~k ( E ) we have sup g =( g J ,g J ) ∈ G ∞ | ( g J ⋆ c )( g J , e J , ∗ ) | < ∞ . In particular, for each g J ∈ G J , the harmonic cocycle c ( g J , · ) is bounded. Assume that ∅ ( J ( J . Assume that b f ∈ M ~k ( U, E ) satisfies (6.6) and (6.7) for each p ∈ J .Let c and c be the J -type “harmonic cocycle” valued modular form and the J -type one attachedto b f respectively.Let g be an element of G J \ J . Proposition 6.19. Suppose k p = 2 for each p above p . For any g ∈ G J \ J , h ∈ G J , and anycompactly supported continuous function α on F J ⊂ P J we have Z ( α ⊗ g U J \ J , ∗ ) µ c ( h, · ) = Z αµ c ( gh, · ) . Proof. We only need to consider the case that α is locally constant, say α is of the form α = 1 x U e , ∗ , x ∈ G J . In this case, both sides of the above equality is b f ( xgh ). We fix a prime p ∈ J and an embedding ι p : F p ֒ → E . In this subsection we define cohomologicalvalued modular form (of level U ).For each g ∈ G ∞ , p we have the groups e Γ p g and Γ p g . As p is clear to us, we denote them by e Γ g andΓ g . As L ~k ( E ) is a Γ g -module, we can form the cohomological group H (Γ g , L ~k ( E )). This groupconsists of equivalence classes of the L ~k ( E )-valued 1-cocycles on Γ g .For a subgroup Γ of G p , and h ∈ G p , we have the map r h : H (Γ , L ~k ( E )) → H ( h − Γ h, L ~k ( E )) ,φ ( r h φ )( γ ′ ) = ˇ ρ ~k ( h ) − φ ( hγ ′ h − ) . If h ∈ U p we have e Γ gh = e Γ g and Γ gh = h − p Γ g h p . So we have the map r h p : H (Γ g , L ~k ( E )) → H (Γ gh , L ~k ( E )) . If g ′ = xg with x ∈ G ( Q ), then e Γ g ′ = x e Γ g x − and Γ g ′ = x p Γ g x − p . So we have the map r x − p : H (Γ g , L ~k ( E )) → H (Γ xg , L ~k ( E )) . Definition 6.20. By a cohomological valued modular form (on G ∞ , p of level U ) we mean a function f on G ∞ , p which satisfies the following conditions:(a) f ( g ) is in H (Γ g , V ).(b) If z ∈ ( F ∞ , p ) × , then f ( zg ) = f ( g ).(c) If h ∈ U p , then f ( gh ) = ( r h p f )( g ). 42d) If x ∈ G ( Q ), then f ( xg ) = ( r x − p f )( g ).Let M H p ( U p , L ~k ( E )) be the space of such cohomological valued modular forms.There exists a T pU -action on M H p ( U p , L ~k ( E )).Let l ∤ p be a prime of F such that U l ∼ = GL ( O F l ). Let b l be in U l (cid:2) ω l 00 1 (cid:3) U l . For any g ∈ G ∞ , p ,let Γ ′ g be the subgroup Γ ′ g = Γ g ∩ Γ gb l of G p . We have the restriction mapRes Γ gb l / Γ ′ g : H (Γ gb l , L ~k ( E )) → H (Γ ′ g , L ~k ( E ))and the corestriction map Cor Γ g / Γ ′ g : H (Γ ′ g , L ~k ( E )) → H (Γ g , L ~k ( E )) . We define the action of T l on M H p ( U p , L ~k ( E )) by(T l f )( g ) = Cor Γ g / Γ ′ g Res Γ gb l / Γ ′ g f ( gb l )for each f ∈ M H p ( U p , L ~k ( E )).Now, we define the Schneider morphism κ sch : C p ( U p , L ~k ( E )) → M H p ( U p , L ~k ( E )) . For each c ∈ C p ( U p , L ~k ( E )) and g ∈ U p , noting that c ( g, · ) is Γ g -invariant, we put κ sch c ( g ) = δ Γ g ( c ( g, · )) ∈ H (Γ g , L ~k ( E )) , where δ Γ g is the map (6.2). Proposition 6.21. (a) If c ∈ C p ( U p , L ~k ( E )) , then κ sch c is in M H p ( U p , L ~k ( E )) . (b) The map κ sch : C p ( U p , L ~k ( E )) → M H p ( U p , L ~k ( E )) c κ sch c is a T pU -equivariant isomorphism.Proof. First we show that κ sch c is really in M H p ( U p , L ~k ( E )). For this we only need to deduce(a)-(d) in Definition 6.20 term by term from Definition 6.13 (with J = { p } there). By definition wealready have (a). If z is in the center of G ∞ , p , then Γ g = Γ zg , so δ Γ g = δ Γ zg . By Definition 6.13 (b)(with J = { p } there) we have c ( zg, · ) = c ( g, · ). Thus κ sch c ( g ) = κ sch c ( zg ), which yields (b). Items(c) and (d) follow from Definition 6.13 (c, d) and Lemma 6.2 (c).Next we prove (b). Put e Γ ′ g = e Γ g ∩ e Γ gb l = { γ ∈ G ( Q ) : γ l ∈ g l ( U l ∩ b l U l b − l ) g − l and γ l ′ ∈ g l ′ U l ′ g − l ′ for l ′ = l , p } . ′ g = ι p g ( e Γ ′ g ). We decompose e Γ g into cosets e Γ g = a i α i e Γ ′ g . (6.8)Via ι p g we obtain Γ g = a i g − p α i,p g p Γ ′ g . Thus Cor Γ g / Γ ′ g c ( gb l , e ) = X i g − p α i,p g p ⋆ c ( gb l , e )= X i ˇ ρ ~k ( g − p α p i,p g p )ˇ ρ ~k ( α i, p ) c ( gb l , α − i, p e )= ˇ ρ ~k ( g − p α p i,p g p ) c ( α i gb l , e ) . By Lemma 6.22 below, from (6.8) the decomposition of e Γ g , we obtain b − l U p b l = a i b − l g − α i gb l · ( b − l U p b l ∩ U p ) . Then T l c ( g, · ) = X i ˇ ρ ~k ( g − p α p i,p g p ) c ( gb l · b − l g − α i gb l , · )= X i ˇ ρ ~k ( g − p α p i,p g p ) c ( α i gb l , · ) = Cor Γ g / Γ ′ g c ( gb l , e ) . By Lemma 6.2 (a, b)(T l κ sch c )( g ) = Cor Γ g / Γ ′ g Res Γ gb l / Γ ′ g ( κ sch c )( gb l )= Cor Γ g / Γ ′ g Res Γ gb l / Γ ′ g δ Γ gb l ( c ( gb l , · ))= δ Γ g (Cor Γ g / Γ ′ g Res Γ gb l / Γ ′ g ( c ( gb l , · ))) = δ Γ g ((T l c )( g, · )) . This proves that κ sch is T pU -equivariant.That κ sch is an isomrphism follows from [9, Proposition 2.9]. Indeed, in loc. cit. it is showedthat δ Γ g : C ( T p , L ~k ( E )) Γ g → H (Γ g , L ~k ( E ))is an isomorphism. Lemma 6.22. gU p g − /g (cid:16) U p , l ( U l ∩ b l U l b − l ) (cid:17) g − ∼ = ˜Γ g / ˜Γ ′ g .Proof. This follows from the strong approximation theorem for SL ( B ), the algebraic group with Q -points B × , = { x ∈ B : N B/F ( x ) = 1 } , N B/F : B × → F × is the reduced norm. By this strong approximation theorem weobtain that, for any compact open subgroup V of G ∞ , p , one has ♯ (cid:16) G ( Q ) \ G ( A Q ) /G ∞ G p V (cid:17) = ♯ (cid:16) F × \ b F × /F × p N B/F ( V ) (cid:17) . So, from the fact N B/F ( gU p g − ) = N B/F (cid:16) g ( U p , l ( U l ∩ b l U l b − l )) g − (cid:17) , we obtain gU p g − = ( gU p g − ∩ G ( Q )) · g (cid:16) U p , l ( U l ∩ b l U l b − l ) (cid:17) g − , or equivalently, the coset gU p g − /g (cid:16) U p , l ( U l ∩ b l U l b − l ) (cid:17) g − has a set of representatives in e Γ g = gU p g − ∩ G ( Q ). L -invariant Let p and ι p be as in Section 6.4. For any u ∈ C × p that is not a unit, let log u : C × p → C p denotethe p -adic logarithm such that log u ( u ) = 0.In Section 6.2 we attach to c ∈ C ( T p , L ~k ( E )) a rigid distribution µ ι p c . For a point z ∈ H ι p = C p − ι p ( F p ), we define an L ~k ( E )-valued function λ z ,c ; u on G p by h λ z ,c ; u ( g ) , P ⊗ Q i = h Z P ( t )log u ( t − g p z t − z ) µ ι p c , Q i (6.9)with P ∈ V ι p ( E ) and Q ∈ V ι p ( E ). Here P P ( t ) is the linear map defined by X j p Y k ι p − − j p t j . Lemma 6.23. (a) For any x ∈ G p we have h Z P ( t )log u ( x − p · t − γ p z x − p · t − z ) µ ι p c , Q i = h Z P ( t )log u ( t − x p γ p z t − x p z ) µ ι p c , Q i . (b) For x ∈ G p we have ˇ ρ ~k ( x ) λ z ,c ; u ( γ ) = λ x p · z ,x⋆c ; u ( xγx − ) . (c) If c is Γ -invariant, then for any γ ∈ Γ and g ∈ G p we have λ z ,c ; u ( γg ) = λ z ,c ; u ( γ ) + ˇ ρ ~k ( γ ) λ z ,c ; u ( g ) . (d) Let z , z be two points of H ι p . If c is Γ -invariant, then there exists ℓ ∈ L ~k ( E ) that onlydepends on z , z and c such that λ z ,c ; u ( γ ) − λ z ,c ; u ( γ ) = ( γ − ℓ for all γ ∈ Γ . roof. Write i p i − p ( x − p ) = (cid:2) a bc d (cid:3) , Then x − p · t − γ p z x − p · t − z = t − x p γ p z t − x p z cγ p z + dcz + d . Thus h Z P ( t )log u ( x − p · t − γ p z x − p · t − z ) µ ι p c , Q i = h Z P ( t )log u ( t − x p γ p z t − x p z ) µ ι p c , Q i + log u ( cγ p z + dcz + d ) h Z P ( t ) µ ι p c , Q i . By Proposition 6.10 (in the case of J = { p } ), the latter term of the right hand side of this equalityis zero. This proves (a).Let x be in G p . We have h ˇ ρ ~k ( x ) λ z ,c ; u ( γ ) , P ⊗ Q i = h ˇ ρ ~k ( x ) Z ( ρ k p ( x − p ) P )( t )log u ( t − γ p z t − z ) µ ι p c , Q i = h Z P ( t )log u ( x p − · t − γ p z x − p · t − z ) µ ι p x⋆c , Q i by Proposition 6 . h Z P ( t )log u ( t − x p γ p z t − x p z ) µ ι p x⋆c , Q i by (a)= h λ x p · z ,x⋆c ; u ( xγx − ) , P ⊗ Q i which shows (b).Using (a) we obtain h λ z ,c ; u ( γg ) − λ z ,c ; u ( γ ) , P ⊗ Q i = h Z P ( t )log u ( t − γ p g p z t − γ p z ) µ ι p c , Q i = h Z P ( t )log u ( γ − p t − g p z γ − p t − z ) µ ι p c , Q i . By Proposition 6.11 (in the case of J = { p } ) and the fact γ ⋆ c = c , we have h Z P ( t )log u ( γ − p t − g p z γ − p t − z ) µ ι p c , Q i = h ˇ ρ ~k ( γ ) Z ( ρ k p ( γ − p ) P )( t )log u ( t − g p z t − z ) µ ι p c , Q i = h ˇ ρ ~k ( γ ) λ z ,c ; u ( g ) , P ⊗ Q i . So we get (c). 46or (d) we have h λ z ,c ; u ( γ ) − λ z ,c ; u ( γ ) , P ⊗ Q i = h Z P ( t )[log u ( t − γ p z t − z ) − log u ( t − γ p z t − z )] µ ι p c , Q i = h Z P ( t )[log u ( t − γ p z t − γ p z ) − log u ( t − z t − z )] µ ι p c , Q i = h Z P ( t )[log u ( γ − p t − z γ − p t − z ) − log u ( t − z t − z )] µ ι p c , Q i . Let ℓ be the element of L ~k ( E ) defined by h ℓ, P ⊗ Q i = h Z P ( t )log u ( t − z t − z ) µ ι p c , Q i . As γ ⋆ c = c , by Proposition 6.11 we have h Z P ( t )log u ( γ − p t − z γ − p t − z ) µ ι p c , Q i = h ˇ ρ ~k ( γ ) Z ( ρ k p ( γ − p ) P )( t )log u ( t − z t − z ) µ ι p c , Q i = h ˇ ρ ~k ( γ ) ℓ, P ⊗ Q i , as expected.By Lemma 6.23 (c), if c is Γ-invariant for a subgroup Γ of G p , then the restriction of λ z ,c ; u toΓ is an L ~k ( E )-valued 1-cocycle on Γ. By Lemma 6.23 (d) its class [ λ z ,c ; u ] in H (Γ , L ~k ( E )) doesnot depend on the choice of z . In this way we obtain Coleman’s morphism λ Γ; u : C ( T p , L ~k ( E )) Γ → H (Γ , L ~k ( E )) , c [ λ z ,c ; u ] . Lemma 6.24. Let Γ ⊂ Γ be two subgroups of G p . (a) We have the following commutative diagram C ( T p , L ~k ( E )) Γ λ Γ2; u / / (cid:127) _ (cid:15) (cid:15) H (Γ , L ~k ( E )) Res Γ2 / Γ1 (cid:15) (cid:15) C ( T p , L ~k ( E )) Γ λ Γ1; u / / H (Γ , L ~k ( E )) . (b) When [Γ : Γ ] is finite, we have the following commutative diagram C ( T p , L ~k ( E )) Γ λ Γ1; u / / Cor Γ2 / Γ1 (cid:15) (cid:15) H (Γ , L ~k ( E )) Cor Γ2 / Γ1 (cid:15) (cid:15) C ( T p , L ~k ( E )) Γ λ Γ2; u / / H (Γ , L ~k ( E )) . roof. Assertion (a) follows from the definition. We prove (b). Let c be in C ( T p , L ~k ( E )) Γ .Write Γ = ⊔ x j Γ . Then Cor Γ / Γ ( c ) = P j x j ⋆ c .As u in the subscript is clear, we will omit it in the following.For any γ ∈ Γ and any i , there exists j = j γ,i (depending on i and γ ) such that γx i = x j h γ,j with h γ,j ∈ Γ . Then γ X j ˇ ρ ~k ( x j ) λ z ,c ( h γ,j )is an L ~k ( E )-valued 1-cocycle on Γ , and its class in H (Γ , L ~k ( E )) is exactly Cor Γ / Γ [ λ c ]. We haveˇ ρ ~k ( x j ) λ z ,c ( h γ,j ) = λ x j · z ,x j ⋆c ( x j h γ,j x − j ) by Lemma 6 . 23 (b)= λ x j · z ,x j ⋆c ( γx i x − j )= λ z ,x j ⋆c ( γx i ) − λ z ,x j ⋆c ( x j )= λ z ,x j ⋆c ( γx i x − j ) + ˇ ρ ~k ( γx i x − j ) λ z ,x j ⋆c ( x j ) − λ z ,x j ⋆c ( x j ) by Lemma 6 . 23 (c)= λ z ,x j ⋆c ( γ ) − ˇ ρ ~k ( γx i x − j ) λ z ,x j ⋆c ( x j x − i )+ ˇ ρ ~k ( γx i x − j ) λ z ,x j ⋆c ( x j ) − λ z ,x j ⋆c ( x j ) by Lemma 6 . 23 (c)= λ z ,x j ⋆c ( γ ) − ˇ ρ ~k ( γ ) λ x i x − j z ,x i ⋆c ( x j x − i )+ ˇ ρ ~k ( γ ) λ x i x − j z ,x i ⋆c ( x i x j x − i ) − λ z ,x j ⋆c ( x j ) by Lemma 6 . 23 (b)= λ z ,x j ⋆c ( γ ) − ˇ ρ ~k ( γ )( λ z ,x i ⋆c (1) − λ z ,x i ⋆c ( x i x − j ))+ ˇ ρ ~k ( γ )( λ z ,x i ⋆c ( x i ) − λ z ,x i ⋆c ( x i x − j )) − λ z ,x j ⋆c ( x j )= λ z ,x j ⋆c ( γ ) + ˇ ρ ~k ( γ ) λ z ,x i ⋆c ( x i ) − λ z ,x j ⋆c ( x j ) . Here, we have used λ z ,x i ⋆c (1) = 0. Taking sum we obtain X j ˇ ρ ~k ( x j ) λ z ,c ( h γ,j ) = λ z , P j x j ⋆c ( γ ) + ( γ − X j λ z ,x j ⋆c ( x j ) . This shows (b).We define the Coleman morphism κ col ,ι p u : C p ( U p , L ~k ( E )) → M H p ( U p , L ~k ( E )) c κ col ,ι p c ; u ( g ) = λ Γ g ; u ( c ( g, · )) . Proposition 6.25. (a) If c ∈ C p ( U p , L ~k ( E )) , then κ col ,ι p c ; u is in M H p ( U p , L ~k ( E )) . (b) The map κ col ,ι p u : C p ( U p , L ~k ( E )) → M H p ( U p , L ~k ( E )) c κ col ,ι p c ; u is T pU -equivariant. roof. For (a) we need to show that κ col ,ι p c ; u satisfies (a)-(d) in Definition 6.20. Conditions (a) and(b) follows directly form the definition; (c) and (d) follows form Lemma 6.23 (b).The proof of (b) is similar to the assertion that κ sch is T pU -equivariant in the proof of Proposition6.21 . We only need to use Lemma 6.24 instead of Lemma 6.2 (a, b) used there. We omit thedetails.To end this section we define L -invariants of Teitelbaum type. See [9, Definition 3.4].Now, let f be a newform in M ~k ( U, E ). Assume that α p ( f ) = 1. Then f comes from a harmonicmodular form c ∈ C p ( U p , L ~k ( E )). We attached to f two cohomological valued modular forms κ sch c and κ col ,ι p c ; u . By Proposition 6.21 and Proposition 6.25 both κ sch c and κ col ,ι p c ; u lie in the f -componentof M H ( U p , L ~k ( E )) with respect to T pU -action. As f is new, the multiplicity one theorem for GL tells us the f -component in M ~k ( U, E ) and that in ∈ C p ( U p , L ~k ( E )) is 1-dimensional over E . ThenProposition 6.21 implies that f -component M H ( U p , L ~k ( E )) is 1-dimensional over E and generatedby κ sch c . Therefore there exists a unique L Tei ι p ; u ( f ) ∈ E such that κ col ,ι p c ; u = L Tei ι p ; u ( f ) κ sch c . Now, let u vary.Let ord p be the function on C × p such that ord p ( ω p ) = 1, that ord p ( xy ) = ord p ( x ) + ord p ( y ) andthat ord p ( x ) = 0 if x is a unit. Here ω p is a uniformizing element of F p . If u , u ∈ C × p are notunit, then log u − log u = (cid:0) log u ( ω p ) − log u ( ω p ) (cid:1) ord p . Proposition 6.26. We have L Tei ι p ; u ( f ) − L Tei ι p ; u ( f ) = log u ( ω p ) − log u ( ω p ) . Proof. It is easy to check that Lemma 6.23 again holds when the function log u in (6.9) is replacedby ord p . In particular, it defines an element of M H p ( U p , L ~k ( E )). As is showed in [10, § κ sch c , yielding our assertion. Definition 6.27. We call L Tei ι p ; u ( f ) the L-invariant of Teitelbaum type of f at p for the embedding ι p and the logarithm log u . Fix a set J of primes of F above p . Let J be the set of primes p ∈ J such that α p = 1. We assumethat each p ∈ J is split in K and that k p = 2 for each p ∈ J . e µ J In this subsection we write J ′ = J \ J .If ~n is a J ′ -tuple of nonnegative integers, we put b O J ; ~n, c := \ ~m b O ( ~m,~n ) , c e X J ; ~n, c := A ∞ , × F \ A ∞ , × K / b O × J ; ~n, c where ~m runs through all J -tuples of nonnegative integers. Similarly, we put b O J ; c := \ ~m b O ~m, c and e X J ; c := A ∞ , × F \ A ∞ , × K / b O × J ; c , where ~m runs through all J -tuples of nonnegative integers.Let F × \ K × act on e X J ; ~n, c and e X J ; c by multiplication. The quotient ( F × \ K × ) \ e X J ; c is isomor-phic to the group G J ; c defined in Section 2.2.We write e X J ; ~n, c in the form e X J ; ~n, c = ( F × J \ K × J ) × ( A ∞ ,J , × F \ A ∞ ,J , × K / b O × J ; ~n, c ) . For each p ∈ J we consider the twisted action of i p ( K × p ) on P ( F p ) = F p ∪{∞} given in Section6.2. For this action i p ( K × p ) has two fixed points 0 , ∞ . So, for any point ⋆ ∈ P ( F p ) \{ , ∞} , theorbit of ⋆ by i p ( K × p ) is exactly P ( F p ) \{ , ∞} . Taking ⋆ = − β we obtain an isomorphism η p : F × p \ K × p ∼ −→ P ( F p ) \{ , ∞} , ( x, y ) 7→ − βxy . Note that η p ( F × p \ K × p ) = F × p and η p ( O × F p \O × K p ) = O × F p . In this way, we identify F × J \ K × J = Q p ∈ J F × p \ K × p with the subset F × J = Q p ∈ J F × p of P J .Thus e X J ; ~n, c is isomorphic to F × J × ( A ∞ ,J , × F \ A ∞ ,J , × K / b O × J ; ~n, c ) . Let X J ; ~n, c be the larger set X J ; ~n, c := F J × ( A ∞ ,J , × F \ A ∞ ,J , × K / b O × J ; ~n, c ) ⊂ P J × ( A ∞ ,J , × F \ A ∞ ,J , × K / b O × J ; ~n, c ) . We pull back the measure µ J on G J ; c (see Definition 5.10) to a measure e µ J on e X J ; c , so thatfor any compactly supported p -adically continuous function α on e X J ; c , we can define R α e µ J . Thefunction α ( x ) = X a ∈ K × /F × α ( ax )is invariant by F × \ K × and thus can be viewed as a function on G J ; c ; we have R α e µ J = R αµ J .Here, for µ being a measure on a p -adically topological space X , we mean that for each compactopen subset U of X , there exists a positive constant C U such that, if g is a p -adically continuousfunction on X supported on U , then | Z gµ | ≤ C U || g || Gauss , || g || Gauss := sup x ∈ U | g ( x ) | .For any a ∈ A × K and J -tuple of positive integers ~m we have Z a b O × ( ~m,~n ) , c e µ J = 1 ( Q p ∈ J ′ α n p p ) b ν ( a ) h ρ ~k ( i p ( ς ( ~m,~n ) J )) − b v m , \ ˜ φ ∞ , † J ( aς ( ~n ) J ) i . (7.1)We use C ♭J to denote the space of functions of the form g + h , where g is a compactly supported p -adically continuous function on e X J ; c , and h is a compactly supported p -adically continuous functionon X J ; ~n, c for some ~n ( ~n allowed to vary).Below, we will extend the integral R g e µ J to C ♭J .Let φ and φ † J be as in Section 4.3, c the J -typle “harmonic cocycle” valued modular form(Definition 6.13) attached to \ ˜ φ ∞ , † J .Let µ c ( aς ( ~n ) J ′ ,Jp \ J , · ) be the (vector valued) measure on F J × ( A ∞ ,J , × F a b O × J ; ~n, c ) ⊂ P J × ( A ∞ ,J , × F a b O × J ; ~n, c ) ∼ = P J . See Remark 6.6 and Corollary 6.9. Lemma 7.1. For every a ∈ A ∞ ,J , × K the integral g ( Q p ∈ J ′ α n p p ) b ν ( a ) h i p ( ς ( ~n ) J ′ ) − b v m , Z g µ c ( aς ( ~n ) J ′ ,Jp \ J , · ) i is independent of representatives a in the double coset A ∞ ,J , × F a b O × J ; ~n, c .Proof. For any x ∈ A ∞ ,J , × F and y ∈ b O × J ; ~n, c , since ( ς ( ~n ) J ′ ,J p \ J ) − yς ( ~n ) J ′ ,J p \ J ∈ U J , by Definition 6.13(b, c) we have µ c ( xayς ( ~n ) J ′ ,Jp \ J , · ) = ˇ ρ ~k (cid:18)(cid:16) ( ς ( ~n ) J ′ ,J p \ J ) − yς ( ~n ) J ′ ,J p \ J (cid:17) − p (cid:19) µ c ( aς ( ~n ) J ′ ,Jp \ J , · ) . Hence b ν ( xay ) h i p ( ς ( ~n ) J ′ ,J p \ J ) − b v m , Z g µ c ( xayς ( ~n ) J ′ ,Jp \ J , · ) i = b ν ( xay ) h i p (( ς ( ~n ) J ′ ,J p \ J ) − yς − ∅ ,J ) b v m , Z g µ c ( aς ( ~n ) J ′ ,Jp \ J , · ) i = b ν ( xay ) b ν ( y ) − h i p ( ς ( ~n ) J ′ ,J p \ J ) − b v m , Z g µ c ( aς ( ~n ) J ′ ,Jp \ J , · ) i = b ν ( a ) h i p ( ς ( ~n ) J ′ ) − b v m , Z g µ c ( aς ( ~n ) J ′ ,Jp \ J , · ) i . Here, we use i p ( ς − ∅ ,J ) b v m = b v m , i p ( y ) b v m = b ν ( y ) − b v m and b ν ( x ) = 1.Let µ ν,c be the measure on X J ; ~n, c whose restriction to F J × ( A ∞ ,J , × F a b O × J ; ~n, c ) is that givenby Lemma 7.1. 51 roposition 7.2. For any compactly supported continuous function g on e X J ; ~n, c we have Z gµ ν,c = Z g e µ J . Proof. Since both µ ν,c and e µ J are measures on e X J ; ~n, c , we only need to verify the formula when g is locally constant. Without loss of generality we may assume that g = 1 U × ( A J , × F b b O × J ~n, c ) where U is a compact open subset of F × J ∼ = F × J \ K × J . Since U is a disjoint union of open subsets of the form a U (1 , ~m ), we may further assume that U = − aβ U (1 , ~m ) for some a ∈ F × J . Note that U = η J ( a, Y p ∈ J (1 + p m p O K p ) · F × J /F × J . (7.2)By a simple computation we have ζ ( ~m ) J ,J · U e J , ∗ = − β U (1 , ~m ) . Thus i J ( a, ζ ( ~m ) J ,J · U e J , ∗ = aβ U (1 , ~m ) = U. Now we have Z gµ ν,c = 1 ( Q p ∈ J ′ α n p p ) b ν ( b ) h i p ( ς ( ~n ) J ′ ) − b v m , c ( bς ( ~n ) J ′ ,J p \ J , i J ( a, ς ( ~m ) J ,J e J , ∗ ) i = 1 ( Q p ∈ J ′ α n p p · Q p ∈ J α m p p ) b ν ( b ) h i p ( ς ( ~m,~n ) J ) − b v m , \ ˜ φ ∞ , † J ( i J ( a, bς ( ~m,~n ) J ) i where the last equality follows from Remark 6.6. Here, we use the fact that ˇ ρ ~k | G J is trivial. By(7.1) the last term of the above formula is exactly R g e µ J .Now, we extend e µ J to C ♭J by putting Z ( g + h ) e µ J = Z g e µ J + Z hµ ν,c . This definition is compatible when ~n varies. Indeed, this follows from an argument similar to theproof of Proposition 5.3.The action of F × \ K × on X J ; ~n, c and e X J ; c induces an action on C ♭J : γ ∗ ( f )( x ) := f ( γx )for γ ∈ F × \ K × and f ∈ C ♭J . It induces an action of F × \ K × on the dual of C ♭J . Lemma 7.3. e µ J is F × \ K × -invariant.Proof. This follows from Definition 6.13 (d) and the fact that ˇ ρ ~k | G J is trivial.Let J be a subset of J . With J \ J instead of J , we have a distribution e µ J \ J on e X J \ J ; c and X J \ J ; ~n, c . We compare e µ J and e µ J \ J . 52 roposition 7.4. If V is a compact open subset of A ∞ ,J , × F \ A ∞ ,J , × K / b O × J ; c , then e µ J ( Y v ∈ J O F v × V ) = e µ J \ J ([1 J ] × V ) , where [1 J ] denotes the point F × J · · O × K J in F × J \ K × J / O × K J .Proof. By the same argument as in the proof of Proposition 7.2, we may take V to be of the form i J \ J ( a, ζ ( ~m ) J \ J ,J \ J · U e J \ J , ∗ × A J , × F b b O × J ; ~n, c where ~m and ~n are J \ J -tuple and J ′ -tuple of positive integers respectively. Note that Y v ∈ J O F v = U e J , = ζ ∅ ,J · U e J , ∗ . Thus we have e µ J ( Y v ∈ J O F v × V )= µ ν,c ( i J \ J ( a, ζ ( ~m ) J \ J ,J · U e J , ∗ × A J , × F b b O × J ; ~n, c )= 1 ( Q p ∈ J ′ α n p p ) b ν ( b ) h i p ( ς ( ~n ) J ′ ) − b v m , \ ˜ φ ∞ , † J ( i J \ J ( a, ζ ( ~m ) J \ J ,J bς ( ~n ) J ′ ,J p \ J ) i = 1 ( Q p ∈ J ′ α n p p · Q p ∈ J \ J α m p p ) b ν ( b ) h i p ( ς ( ~n ) J \ J ) − b v m , \ ˜ φ ∞ , † J \ J ( i J \ J ( a, bς ( ~m,~n ) J \ J ) i = e µ J \ J ([1 J ] × V ) . Here, the first equality is just Proposition 7.2, the second equality follows from the definition of µ ν,c ,the third equality follows from the fact that \ ˜ φ ∞ , † J = \ ˜ φ ∞ , † J \ J and that the action of i p ( ς ( ~m ) J \ J ,J )on L ~k ( C p ) is trivial, and the last equality follows from the definition of e µ J \ J . For each p ∈ J ( p O K = PP ) we choose β p ∈ K × such that ( β p ) = P − h p with h p > 0. We mayeven choose β p such that h p is the smallest with this property. For any subset J of J , let ∆ J bethe subgroup of K × /F × generated by β p ( p ∈ J ).The p -component of β p , denoted by β p , p , acts on K × p /F × p ∼ = F × p , ( x, y ) F × p 7→ − βxy ;the fundamental domain of this action is D p := { x ∈ F × p : 0 ≤ v p ( x ) < h p } . 53e put D p := O × F p and B p := { x ∈ F × p : v p ( x ) ≥ } . Then (1 − β ∗ p , p )1 B p = 1 D p . For each finite place v / ∈ J we set D v = O × K v / O × F v . For any element a ∈ A ∞ , × K with a p = 1, and two disjoint subsets J and J of J , we put Z J ,J ,a = ( Y p ∈ J B p ) × ( Y p ∈ J D p ) × a ( Y v / ∈ J ∪ J D v ) . We form the indexed set I = LOG J × { finite places of F } . If i = ( l, v ), we put pr ( i ) = l and pr ( i ) = v .For each pair i = ( l, v ) let v l denote the function on A ∞ , × K / A ∞ , × F such that v l ( x ) = l ( x v ) , where x v denotes the element of A ∞ , × K / A ∞ , × F whose v -component is x v and other components areall 1. Then l = P v v l . We also write ℓ i for v l . Lemma 7.5. If pr ( i ) / ∈ Σ J , then ℓ i ( β p ) = 0 for each p ∈ J .Proof. Write i = ( l, v ). As v = p , β p ,v ∈ O × K v . For each σ ∈ Σ J we have v log σ | O × Kv / O × Fv = 0. Inparticular v log σ ( β p ) = 0. Thus v l ( β p ) = 0.For a vector ~t = ( t i ) i ∈ I of nonnegative integers (with t i = 0 for almost all i ∈ I ) and two disjointsubsets J ⊂ J and J ⊂ J , we write ℓ ~t := Y i ∈ I ℓ t i i , λ ( ~t, J , J , a ) := 1 Z J ,J ,a · ℓ ~t . If ( ~t, J , J ) satisfies the condition( ♭ ) t i = 0 when pr ( i ) ∈ Σ J ,then λ ( ~t, J , J , a ) is in C ♭J . When J = ∅ , ( ♭ ) automatically holds.Put | ~t | = X i ∈ I t i and | ~t | J = X pr ( i ) ∈ Σ J t i . If t ′ i ≤ t i for each i ∈ I , and if | ~t ′ | J < | ~t | J , we write ~t ′ < J ~t . Lemma 7.6. Let p be in J . For any ~t with t i = 0 for almost all i ∈ I , (1 − β ∗ p ) ℓ ~t is a linearcombination of ℓ ~t ′ with ~t ′ < J ~t . Writing (1 − β ∗ p ) ℓ ~t = X ~t ′ < J ~t c ~t ′ ℓ ~t ′ ; if | ~t ′ | = | ~t | − with t ′ i = t i − , then c ~t ′ = − t i ℓ i ( β p ) . roof. We use the relation(1 − β ∗ p )( f g ) = (1 − β ∗ p ) f · g + f · (1 − β ∗ p ) g − (1 − β ∗ p ) f · (1 − β ∗ p ) g (7.3)and the fact that (1 − β ∗ p ) ℓ i = − ℓ i ( β p ) is a constant. Lemma 7.7. If ( ~t, J , J ) satisfies ( ♭ ) , if | ~t | J + ♯ ( J ) + ♯ ( J ) < ♯ ( J ) , and if b χ is a ∆ J \ ( J ∪ J ) -invariant element of C ♭J , then Z b χ · λ ( ~t, J , J , a ) e µ J = 0 . Proof. We prove the assertion by induction on | ~t | J . Since | ~t | J + ♯ ( J ) + ♯ ( J ) < ♯ ( J ), there exists p ∈ J \ ( J ∪ J ) such that t i = 0 when pr ( i ) = p . Then ( ~t, J ∪ { p } , J ) satisfies ( ♭ ) and thus λ ( ~t, J ∪ { p } , J , a ) is in C ♭J .By Lemma 7.5, when | ~t | J = 0, ℓ ~t is β ∗ p -invariant. Then λ ( ~t, J , J , a ) = (1 − β ∗ p ) λ ( ~t, J ∪ { p } , J , a ) . As e µ J and b χ are β p -invariant, we have Z b χ · λ ( ~t, J , J , a ) e µ J = 0 . Now we assume that | ~t | J > 0. As e µ J is β p -invariant, we have Z (1 − β ∗ p ) (cid:16)b χ · λ ( ~t, J ∪ { p } , J , a ) (cid:17)e µ J = 0 . By (7.3) we obtain(1 − β ∗ p ) (cid:16) b χ · λ ( ~t, J ∪ { p } , J , a ) (cid:17) = b χ · (1 − β ∗ p )(1 Z J ∪{ p } ,J ,a ℓ ~t )= b χ · [((1 − β ∗ p ) ℓ ~t ) · Z J ∪{ p } ,J ,a + ℓ ~t · (1 − β ∗ p )1 Z J ∪{ p } ,J ,a − (1 − β ∗ p ) ℓ ~t · (1 − β ∗ p )1 Z J ∪{ p } ,J ,a ]= b χ · ((1 − β ∗ p ) ℓ ~t ) · Z J ∪{ p } ,J ,a + b χ · ℓ ~t · Z J ,J ,a + b χ · ((1 − β ∗ p ) ℓ ~t ) · Z J ,J ,a By Lemma 7.6, (1 − β ∗ p ) ℓ ~t is a linear combination of ℓ ~t ′ with ~t ′ < J ~t . Since ( ~t, J ∪ { p } , J )satisfies ( ♭ ) and since ~t ′ < J ~t , both ( ~t ′ , J ∪ { p } , J ) and ( ~t ′ , J , J ) satisfy ( ♭ ). On the other hand | ~t ′ | J + ♯ ( J ) + ♯ ( J ) < | ~t ′ | J + ♯ ( J ∪ { p } ) + ♯ ( J ) < | ~t | J + ♯ ( J ∪ { p } ) + ♯ ( J ) ≤ ♯ ( J ) . Thus by the inductive assumption we have Z b χ · ((1 − β ∗ p ) ℓ ~t ) · Z J ∪{ p } ,J ,a e µ J = Z b χ · ((1 − β ∗ p ) ℓ ~t ) · Z J ,J ,a e µ J = 0 . It follows that R b χ · ℓ ~t · Z J ,J ,a e µ J = 0. 55hen | ~t | = 0 and ♯ ( J ) + ♯ ( J ) = ♯ ( J ), we have the following useful average vanishing result. Lemma 7.8. Let { a k } be a set of representatives in A ∞ , × K of the coset A ∞ , × F K × \ A ∞ , × K / b O × K .Assume that J F J = J and that J is nonempty. Let b χ be a character of Γ − J that factors through Γ − J \ J , and assume that b χ p = 1 for some p ∈ J . Then X k Z b χ · Z J ,J ,ak e µ J = 0 . Proof. Note that the map [ k Z ∅ ,J ,a k → K × A ∞ , × F \ A ∞ , × K is surjective and sends [ O × K : O × F ] elements to one element. Thus, it follows from Proposition 7.4that X a k Z b χ · Z J ,J ,ak e µ J = X a k Z b χ · Z ∅ ,J ,ak e µ J \ J = [ O × K : O × F ] Z b χ µ J \ J = [ O × K : O × F ] L J \ J ( π, b ν b χ ) . Applying Theorem 5.9 to J \ J instead of J , we obtain L J \ J ( π, b ν b χ ) = 0. Note that b ν p = 1 since k p = 2. In this subsection we fix an element l of LOG J , and a subset J of J .If Ξ is a subset of J \ J , we use ~t l, Ξ to denote the vector with ~t l, Ξ ,i = (cid:26) i ∈ { l } × Ξ , a ∈ A ∞ , × K with a p = 1, we putΛ l, Ξ ,J ,a = λ ( ~t l, Ξ , ( J \ J ) \ Ξ , J , a ) . When Ξ is the empty set ∅ , Λ l, ∅ ,J ,a is independent of l , so we write it by Λ ∅ ,J ,a .For every J ′ ⊆ J \ J we denote by M ( J ′ ) the set of maps f : J ′ → J \ J such that f ( S ) * S for all S ⊆ J ′ , S = ∅ .Let ~t l,f be the vector with ~t l,f,i = (cid:26) ♯ ( f − ( v )) if i = ( l, v ) , ( i ) = l. Lemma 7.9. If ( ~t, J , J ) satisfies that t i = 0 unless i ∈ { l } × ( J \ ( J ∪ J )) , if | ~t | + ♯ ( J ) + ♯ ( J ) = ♯ ( J ) , and if b χ is a ∆ J \ ( J ∪ J ) -invariant element of C ♭J , then Z b χ · λ ( ~t, J , J , a ) e µ J = ~t ! X (Ξ ,f ) Y v ∈ J \ ( J ∪ J ∪ Ξ) f ( v ) l ( β v ) Z b χ · Λ l, Ξ ,J ,a e µ J , where the sum runs through all pairs (Ξ , f ) with Ξ ⊆ J \ ( J ∪ J ) and f ∈ M (( J \ ( J ∪ J ∪ Ξ)) such that ~t l,f + ~t l, Ξ = ~t . roof. In the case t i > i ∈ { l } × ( J \ ( J ∪ J )), both sides of the equality are equal to Z b χ · Λ l,J \ ( J ∪ J ) ,J ,a e µ J . Indeed, so is the left hand side because we must have ~t = ~t l,J \ ( J ∪ J ) . For (Ξ , f ) appeared in theright hand side, we have ~t l,f = ~t l,J \ ( J ∪ J ) − ~t l, Ξ = ~t l,J \ ( J ∪ J ∪ Ξ) . This implies that f ( J \ ( J ∪ J ∪ Ξ)) = J \ ( J ∪ J ∪ Ξ). So the condition f ∈ M (( J \ ( J ∪ J ∪ Ξ))forces J \ ( J ∪ J ∪ Ξ) = ∅ or equivalently Ξ = J \ ( J ∪ J ).Suppose that t i = 0 for some i = ( ℓ, p ), p ∈ J \ ( J ∪ J ). By Lemma 7.6 and Lemma 7.7 wehave Z b χ · λ ( ~t, J , J , a ) e µ J = − Z b χ · (cid:16) (1 − β ∗ p ) Y v ∈ J ℓ ~t (cid:17) · Z J ∪{ p } ,J ,a e µ J = Z b χ · X ( f,~t ′ ) t l,f ( p ) f ( p ) l ( β p ) · λ ( ~t ′ , J ∪ { p } , J , a ) e µ J , where ( f, ~t ′ ) runs over the pairs with f ∈ M ( { p } ) and t l,f + ~t ′ = ~t . Now our assertion follows byinduction.We apply Lemma 7.9 to the case that l satisfies X v ∈ J \ J v l ( β p ) = 0 (7.4)for each p ∈ J \ J . Put h = | J | − | J | , and write J \ J = { p , · · · , p h } . Proposition 7.10. If b χ is a ∆ J \ J -invariant element of C ♭J , then Z b χ X v ∈ J \ J vl h · Z ∅ ,J ,a e µJ = h ! Z b χ det Λ l, p ,J ,a − p l ( β p ∅ ,J ,a − p l ( β p ∅ ,J ,a · · · − p l ( β p h )Λ ∅ ,J ,a − p l ( β p ∅ ,J ,a Λ l, p ,J ,a − p l ( β p ∅ ,J ,a · · · − p l ( β p h )Λ ∅ ,J ,a ... ... ... ... − p hl ( β p ∅ ,J ,a − p hl ( β p ∅ ,J ,a · · · Λ l, p h,J ,a − p h l ( β p h )Λ ∅ ,J ,a e µJ Here we denote Λ l, { p i } ,J ,a by Λ l, p i ,J ,a . Proof. Appliying Lemma 7.9 to the case of J = ∅ we obtain Z b χ · X v ∈ J \ J v l h · Z ∅ ,J ,a e µ J = h ! X (Ξ ,f ) Y v ∈ ( J \ J ) \ Ξ f ( v ) l ( β v ) Z b χ · Λ l, Ξ ,J ,a e µ J , where the sum runs through all pairs (Ξ , f ) with Ξ ⊆ J \ J and f ∈ M ( J \ ( J ∪ Ξ)).57n the other hand, det Λ l, p ,J ,a − p l ( β p )Λ ∅ ,J ,a − p l ( β p )Λ ∅ ,J ,a · · · − p l ( β p h )Λ ∅ ,J ,a − p l ( β p )Λ ∅ ,J ,a Λ l, p ,J ,a − p l ( β p )Λ ∅ ,J ,a · · · − p l ( β p h )Λ ∅ ,J ,a ... ... . . . ... − p h l ( β p )Λ ∅ ,J ,a − p h l ( β p )Λ ∅ ,J ,a · · · Λ l, p h ,J ,a − p h l ( β p h )Λ ∅ ,J ,a = X Ξ ⊆ J \ J d Ξ Λ l, Ξ ,J ,a where d Ξ = det( − p j ℓ ( β p i )) p i , p j ∈ J \ ( J ∪ Ξ) . As (7.4) holds for each p ∈ J \ J , our assertion follows from the following lemma. Lemma 7.11. [25, Lemma 4.9] Let k ≤ m be positive integers, and let ( c ij ) ≤ i ≤ k, ≤ j ≤ m be a k × m -matrix with entries in a commutative ring such that P mj =1 c ij = 0 for all i = 1 , · · · , k . Then det( − c ij ) ≤ i,j ≤ k = X f k Y i =1 c if ( i ) where the sum runs through all maps f : { , · · · , k } → { , · · · , m } with f ( S ) * S for all S ⊆{ , · · · , k } , S = ∅ . L -invariants and group cohomology For each p ∈ J let β p be the element in K × /F × provided in Section 7.2. Let ∆ = ∆ J be thesubgroup of K × /F × generated by β p ( p ∈ J ). Then ∆ is a free abelian group with generators β p ( p ∈ J ).We fix a J \ J -tuple ~n .For a subset J ⊂ J we consider the spaces e X J J ; ~n, c = ( F × J \ J \ K × J \ J ) × ( A ∞ ,J , × F \ A ∞ ,J , × K / b O × J ; ~n, c )and X J J ; ~n, c := F J \ J × ( A ∞ ,J , × F \ A ∞ ,J , × K / b O × J ; ~n, c ) . Via η J we identify e X J J ; ~n, c with a subset of X J J ; ~n, c . If J \ J = J , F J , is a disjoint union, wehave a set X J J ; ~n, c ( J , , J , ) := ( F J , × F × J , ) × ( A ∞ ,J , × F \ A ∞ ,J , × K / b O × J ; ~n, c ) . Let ¯ C J ( J , , J , ) be the completion of the space of compact supported functions on X J J ; ~n, c ( J , , J , )for the Gauss norm. When J , = J \ J and J , = ∅ , we denote it by ¯ C J . Then ¯ C J ( J , , J , ) isclosed in ¯ C J for each pair ( J , , J , ).Let ∆ act on the spaces X J J ; ~n, c and X J J ; ~n, c ( J , , J , ). These actions induce actions of ∆ on¯ C J and ¯ C J ( J , , J , ). 58et I J , be the subset of I consisting of i = (log ι , p ) with p ∈ J , and ι ∈ Σ p . Let C J , ≤ N denote the subspace M J , ⊂ J \ J M ~m : IJ , ,~m > , | ~m | ≤ N (cid:0) ¯ C J ( J , , J , ) (cid:1) ∆ J · ℓ ~m endowed with the product topology, where ~m > m ι, p of ~m is positive.There exists a natural action of ∆ on C J , ≤ N .Similarly, let ¯ C ( F J ) denote the completion of the space of compactly supported functions on F J = Q p ∈ J F p . Then ∆ acts on ¯ C ( F J ). Let C ≤ NJ be the space M J ′ ⊂ J M ~m : IJ ′ -tuple ,~m > , | ~m | ≤ N ¯ C ( F J \ J ′ × F × J ′ ) ∆ J \ J · ℓ ~m equipped with the product topology. L -invariants revisited Let p be in J . Write p O K = P ¯ P . Then K p = K P ⊕ K ¯ P . Write β p , p = ( u ω − h p p , u ) ∈ K p = K P ⊕ K ¯ P ∼ = F p ⊕ F p for u , u ∈ O × F p . Put u p = u u ω h pp . Proposition 8.1. If ξ ∈ ( ¯ C p ) ∆ p , then for each ι ∈ Σ p we have Z p log ι · D p · ξ e µ J = L Tei ι,u ι p ( f ) Z h p X i =1 ω i p B p · ξ e µ J . (8.1)Note that p log ι is exactlylog ιu ι p : K × p → C p , ( a , a ) log u ι p (( a /a ) ι ) . Indeed, both of them are additive characters that coincide with each other on O × K p and vanish at β p , p .Since e µ J is a measure, to prove (8.1) we may restrict to the case that ξ is locally constant. Put J ′ = J \{ p } . We write ξ = X k d k U ( a k ,m k ) × A ∞ ,J , × F t k b O × J ~n, c , where U ( a k , m k ) are open discs of F J ′ , and d k are constants in C p . Let h k be an element ofGL ( F J ′ ) such that U ( a k , m k ) = h k U e J ′ , ∗ .Write ˜ φ = P j b j π ′ ( g j ) ϕ with b j ∈ E and ( g j ) ∞ ,p = 1. Then ˜ φ ∞ , † J = P j b j π ′ ( g j ) ϕ ∞ , † J . Let c ( · , · ) be the p -type “harmonic cocycle” valued modular form attached to \ ϕ ∞ , † J (Section 6.3).Let V be a compact open subgroup of A ∞ , p , × K such that i p p ( V ) ⊂ ς ( ~n ) J \ J h k g j U p g − j h − k ( ς ( ~n ) J \ J ) − j, k . Let r be a positive integer such that β r p is contained in V . Enlarging r if necessary wemay assume that the out- p part ( β r p ) p of β r p is in b O × c .For ℓ = t k ς ( ~n ) J \ J h k g j let γ ℓ be the element ι p ℓ ( β r p ) in Γ p ℓ . Note that the p -component of γ ℓ is β r p , p ,independent of j and k . We write γ p for it. The out- J -part of γ ℓ is also independent of j and k ;we write γ J for it.Fix a point z ∈ H p . Put I ℓ := Z P ( F p ) log u ι p ( ι ( x ) − γ p z ι ( x ) − z ) µ c ( ℓ, · ) . Lemma 8.2. We have I ℓ = Z F (log u ι p ( z ) − log u ι p ι ( x )) µ c ( ℓ, · ) . Proof. Let F = F γ p be a fundament domain in P ( F p ) \{ , ∞} for the action of γ p . We may take F = r − S i =0 β − i p , p D p .By Remark 6.12, µ c ( ℓ, · ) is Γ p ℓ -invariant. So I ℓ = Z P ( F p ) log u ι p ( ι ( x ) − γ p z ι ( x ) − z ) µ c ( ℓ, · ) = Z F ∞ X j = −∞ log u ι p ( ι ( γ j p x ) − γ p z ι ( γ j p x ) − z ) µ c ( ℓ, · ) As i p i − p ( γ p ) = (cid:2) u r ω − rh pp u r (cid:3) , we have ∞ X j = −∞ log u ι p ( ι ( γ j p x ) − γ p z ι ( γ j p x ) − z ) = ∞ X j = −∞ log u ι p ( ι ( u rj p x ) − ι ( u r p ) z ι ( u rj p x ) − z )= ∞ X j = −∞ log u ι p ( ι ( u r ( j − p x ) − z ι ( u rj p x ) − z ) = lim N → + ∞ log u ι p ( ι ( u − r ( N +1) p x ) − z ι ( u rN p x ) − z )= lim N → + ∞ log u ι p ( ι ( u − r ( N +1) p x ) − z ι ( x ) − ι ( u − rN p ) z ) = log u ι p z − log u ι p ι ( x ) . Thus we obtain the equality in the lemma. Lemma 8.3. i p ( ς ( ~n ) J \ J ) − b v m is fixed by i p ( γ ℓ ) .Proof. Since k q = 2 for each q ∈ J , J -part of i p ( γ ℓ ) acts trivially on V ~k ( E ). So, i p ( γ ℓ ) i p ( ς ( ~n ) J \ J ) − b v m = i p ( γ J ) i p ( ς ( ~n ) J \ J ) − b v m = i p ( ς ( ~n ) J \ J ) − i p ( β J p ,p ) r b v m = b ν ( β r p ,p ) i p ( ς ( ~n ) J \ J ) − b v m . Here β J p ,p denotes the J p \ J -part of β . But b ν ( β r p ,p ) = b ν (( β r p ) p ) − = 1 since ( β r p ) p is in b O × c and since b ν is trivial on b O × c . The reader should be cautious that ( β r p ) p denotes the out- p part of β r p , not the p -th power of β r p . 60or a choice z ′ different from z , by Lemma 6.23 (d) there exists v ∈ L ~k ( E ) such that( γ − v = Z P ( F p ) log u ι p ( ι ( x ) − γz ι ( x ) − z ) µ c ( ℓ, · ) − Z P ( F p ) log u ι p ( ι ( x ) − γz ′ ι ( x ) − z ′ ) µ c ( ℓ, · ) = (log u ι p z − log u ι p ( z ′ )) Z F µ c ( ℓ, · ) . Thus by Lemma 8.3 we have h i p ( ς ( ~n ) J \ J ) − b v m , Z F µ c ( ℓ, · ) i = 0 . Applying Proposition 6.19 we obtain Z F · p ℓ ι · ξ e µ J = − ( Q q ∈ J \ J α n q q ) X j,k b j d k b ν ( t k ) h i p ( ς ( ~n ) J \ J ) − b v m , I t k ς ( ~n ) J \ J h k g j i . Lemma 8.4. For each ℓ = t k ς ( ~n ) J \ J h k g j , h i p ( ς ( ~n ) J \ J )) − b v m , I ℓ i = L Tei ι,u ι p ( f ) h i p ( ς ( ~n ) J \ J )) − b v m , δ Γ p ℓ c ( ℓ, · )( γ ℓ ) i . Proof. In Section 6.5 we proved that there exists y ∈ L ι ( E ) such that I ℓ = L ι,u p ( f ) δ Γ p ℓ c ( ℓ, · )( γ ℓ ) + ( i p ( γ ℓ ) − y. By Lemma 8.3 we have h i p ( ς ( ~n ) J ′ ) − b v m , ( i p ( γ ℓ ) − y i = h ( i p ( γ ℓ ) − i p ( ς ( ~n ) J ′ ) − b v m , y i = 0. Proof of Proposition 8.1. Write ℓ = t k ς ( ~n ) J \ J h k g j . For any vertex v in the tree T p , δ Γ ℓ c ( ℓ, · )( γ ℓ ) = X e ∈ v → γ p v c ( ℓ, e ) . We may take v = ω − rh p p O F p ⊕ O F p . Then the edges in the chain v → γ p v are i p (( ω i p , e p , (1 ≤ i ≤ rh p ) . Thus when ℓ runs over the set { t k ς ( ~n ) J \ J h k g j } j,k,t , the sum1 ( Q q ∈ J \ J α n q q ) X j,k b j d k b ν ( t k ) h i p ( ς ( ~n ) J ′ )) − b v m , δ Γ p tkς ( ~n ) J \ J hkgj c ( t k ς ( ~n ) J \ J h k g j , · )( γ t k ς ( ~n ) J \ J h k g j ) i is exactly − R ( rh p P i =1 ω i p B p ) ξ e µ J . As β ∗ p ( ξ ) = ξ , we have β ∗ p ( sh p X i =( s − h p +1 ω i p B p · ξ ) = ( s +1) h p X i = sh p +1 ω i p B p · ξ Z ( rh p X i =1 ω i p B p ) ξ e µ J = r Z ( h p X i =1 ω i p B p ) ξ e µ J . Similarly, Z F · p log ι · ξ e µ J = r Z D p · p log ι · ξ e µ J . Now Proposition 8.1 follows from Lemma 8.4. ∆ Lemma 8.5. (a) For each γ ∈ ∆ J , γ ∗ − is strict on C J , ≤ N for the given topology. (b) γ ∗ − C J , ≤ N +1 → C J , ≤ N is surjective.Proof. We consider the big set e C J , ≤ N := M ~m : IJ \ J , | ~m | ≤ N ( ¯ C J ) ∆ J · ℓ ~m endowed with the product topology. The map γ ∗ − e C J , ≤ N → e C J , ≤ N is a homomorphism of finite rank ( ¯ C J ) ∆ J -module, and the matrix of γ ∗ − { ℓ ~m : ~m is an I J \ J -tuple , | ~m | ≤ N } has entries in C p . Thus γ ∗ − J ′ ⊃ J , , ¯ C J ( J , , J , ) ∆ J is closed in ¯ C J (( J \ J ) \ J ′ , J ′ ) ∆ J . Thus C J , ≤ N is closedin e C J , ≤ N . It follows that γ ∗ − C J , ≤ N → C J , ≤ N is strict. This proves (a).The proof of (b) is easy and omitted. Remark . e C J , ≤ N is not a subset of ¯ C J .Let D J , ≤ N and Dist J be the dual spaces of C J , ≤ N and ¯ C J respectively. The natural map C J , ≤ N → ¯ C J is continuous and ∆-equivariant. So by restricting we obtain a ∆-equivariantcontinuous map Dist J → D J , ≤ N .If J is a subset of J disjoint from J , then we have a ∆-equivariant map C ≤ MJ × D J , ≤ M + N → D J ∪ J , ≤ N . We derive from it a pairing H i (∆ , C ≤ MJ ) × H j (∆ , D J , ≤ M + N ) → H i + j (∆ , D J ∪ J , ≤ N ) . Similarly, we have a ∆-equivariant map C ≤ MJ × C ≤ NJ → C ≤ M + NJ ∪ J . 62e derive from it a pairing H i (∆ , C ≤ MJ ) × H j (∆ , C ≤ NJ ) → H i + j (∆ , C ≤ M + NJ ∪ J ) . For each p ∈ J let ord p be the function on F × p such that ord p ( uω n p ) = n for every u ∈ O × F p . Weattach to ord p the 1-cocycle on ∆ with values in C ≤ p : For each γ ∈ ∆ we put c p , ord ( γ ) = ( γ ∗ − − ord p · O p ) . For l ∈ LOG J we define c p ,l by c p ,l ( γ ) = ( γ ∗ − − p l · O p ) . When l = log σ ( σ ∈ Σ J ), we also write c p ,σ, log for c p , log σ . If l = P σ ∈ Σ J s σ log σ , then c p ,l = X σ ∈ Σ J s σ c p ,σ, log . We have c p , ord ( β q ) = h p P i =1 ω i p B p if q = p q = p . When σ ∈ Σ p , c p ,σ, log ( β q ) = (cid:26) p log ι · D p if q = p − p log ι ( β q ) · B p if q = p . When σ / ∈ Σ p , c p ,σ, log = p log σ ( ω p ) c p , ord . (8.2)Hence [ c p , ord ] and [ c p ,ι, log ] ( l ∈ LOG J ) are all in H (∆ , C ≤ p ).The ∆-invariant distribution e µ J induces an element [ e µ J ] in H (∆ , Dist ∅ ). Consider [ c p , ord ]and [ c p ,l ] as elements in H (∆ , C ≤ N p ). Then we obtain elements [ c p , ord ] ∪ [ e µ J ] and [ c p ,l ] ∪ [ e µ J ] in H (∆ , D p , ≤ N ). Proposition 8.7. If p ∈ J and if σ ∈ Σ p , then for each N ≥ we have [ c p ,σ, log ] ∪ [ e µ J ] = L Tei σ,u σ p ( f )[ c p , ord ] ∪ [ e µ J ] in H (∆ , D p , ≤ N ) .Proof. Let µ denote the distribution (cid:16) c p ,σ, log ( β p ) − L Tei σ,u σ p ( f ) c p , ord ( β p ) (cid:17) e µ J ∈ D p , ≤ N +1 . By Proposition 8.1, µ vanishes on ( C p , ≤ N +1 ) ∆ p = ( ¯ C p ) ∆ p . As β ∗ p − µ ∈ D p , ≤ N +1 such that h µ , ( β ∗ p − α i = h µ , α i for each α ∈ C p , ≤ N +1 . Equivalently ( β p ∗ − µ = µ .63ow, for any τ ∈ ∆ and α ∈ C p , ≤ N +1 we have h ( τ ∗ − µ , ( β ∗ p − α i = h ( β p ∗ − τ ∗ − µ , α i = h ( τ ∗ − β p ∗ − µ , α i = h ( τ ∗ − c p ,σ, log ( β p ) − L Tei σ,u σ p ( f ) c p , ord ( β p )) e µ J , α i = h ( β p ∗ − c p ,σ, log ( τ ) − L Tei σ,u σ p ( f ) c p , ord ( τ )) e µ J , α i = h ( c p ,σ, log ( τ ) − L Tei σ,u σ p ( f ) c p , ord ( τ )) e µ J , ( β ∗ p − α i . As ( β ∗ p − C p , ≤ N +1 = C p , ≤ N , we obtain( τ ∗ − µ = ( c p ,σ, log ( τ ) − L Tei σ,u σ p ( f ) c p , ord ( τ )) e µ J in D p , ≤ N . Definition 8.8. For σ ∈ Σ J and p ∈ J , we put L p ,σ ( f ) = (cid:26) L Tei σ,u σ p ( f ) if σ ∈ Σ p , p log σ ( ω p ) if σ / ∈ Σ p . For l = P σ ∈ Σ J s σ log σ , we set L p ,l ( f ) = X σ ∈ Σ J s σ L p ,σ ( f ) . The following is a direct consequence of Proposition 8.7 and (8.2). Corollary 8.9. For each p ∈ J , each l ∈ LOG J and each N ≥ we have [ c p ,l ] ∪ [ e µ J ] = L p ,l ( f )[ c p , ord ] ∪ [ e µ J ] in H (∆ , D p , ≤ N ) . Theorem 8.10. Write J = { p , · · · p r } . Let l i ( i = 1 , · · · , r ) be in LOG J . Then for each N ≥ we have [ c p ,l ] ∪ · · · ∪ [ c p r ,l r ] ∪ [ e µ J ] = r Y i =1 L p i ,l i ( f ) ! [ c p , ord ] ∪ · · · ∪ [ c p r , ord ] ∪ [ e µ J ] in H r (∆ , D J , ≤ N ) .Proof. For j = 1 , · · · , r we write J ( j ) = { p , · · · p j } . We prove[ c p ,l ] ∪ · · · ∪ [ c p j ,l j ] ∪ [ e µ J ] = j Y i =1 L p i ,l i ( f ) ! [ c p , ord ] ∪ · · · ∪ [ c p j , ord ] ∪ [ e µ J ]in H j (∆ , D J ( j ) , ≤ N ) by induction on j . 64hen j = 1, this is just Proposition 8.7. Assume j ≥ j isreplaced by j − 1. Then we have[ c p ,l ] ∪ · · · ∪ [ c p j − ,l j − ] ∪ [ e µ J ] = j − Y i =1 L p i ,l i ( f ) ! [ c p , ord ] ∪ · · · ∪ [ c p j − , ord ] ∪ [ e µ J ]in H j − (∆ , D J ( j − , ≤ N +1 ). So from the pairing H (∆ , C ≤ p j ) × H j − (∆ , D J ( j − , ≤ N +1 ) → H j (∆ , D J ( j ) , ≤ N )we obtain [ c p ,l ] ∪ · · · ∪ [ c p j ,l j ] ∪ [ e µ J ]= ( − j − [ c p j ,l j ] ∪ (cid:0) [ c p ,l ] ∪ · · · ∪ [ c p j − ,l j − ] ∪ [ e µ J ] (cid:1) = j − Y i =1 L p i ,l i ( f ) ! ( − j − [ c p j ,l j ] ∪ (cid:0) [ c p , ord ] ∪ · · · ∪ [ c p j − , ord ] ∪ [ e µ J ] (cid:1) = j − Y i =1 L p i ,l i ( f ) ! [ c p , ord ] ∪ · · · ∪ [ c p j − , ord ] ∪ [ c p j ,l j ] ∪ [ e µ J ]in H j (∆ , D J ( j ) , ≤ N ).Since [ c p j ,l j ] ∪ [ e µ J ] = L p j ,l j ( f )[ c p j , ord ] ∪ [ e µ J ]in H (∆ , D p j , ≤ N + j − ) which is ensured by Proposition 8.7, from the pairing H j − (∆ , C ≤ j − J ( j − ) × H (∆ , D p j , ≤ N + j − ) → H j (∆ , D J ( j ) , ≤ N )we obtain the relation[ c p , ord ] ∪ · · · ∪ [ c p j − , ord ] ∪ [ c p j ,l j ] ∪ [ e µ J ] = L p j ,l j ( f )[ c p , ord ] ∪ · · · ∪ [ c p j − , ord ] ∪ [ c p j , ord ] ∪ [ e µ J ]in H j (∆ , D J ( j ) , ≤ N ). This finishes the inductive proof. We fix a disjoint decomposition J = J ′ F J ′′ of J ; J ′′ may be empty. Put J = J ′ ∩ J and J = J ′′ ∩ J . Assume that r := ♯ ( J ) > σ ∈ Σ J ′ we take a constant s σ . We take l in Proposition 7.10 to be l = X σ ∈ Σ J ′ s σ log σ . Then for any p ∈ J ′ we have X v ∈ J \ J ′′ v l ( β p ) = X v ∈ J \ J v l ( β p ) = 0 . We take a set of representatives { a : a p = 1 } in A ∞ , × K of the coset A ∞ , × F K × \ A ∞ , × K / b O × K .65 roposition 9.1. Let b χ be a character of Γ − J such that b χ p = 1 for every p ∈ J . Then Z b χ · X a X v ∈ J \ J v l r · Z ∅ ,J ,a e µ J = r ! Y p ∈ J ( h p L p ,l ) · L J \ J ( π K , b ν b χ ) , where L p ,l = L p ,l ( f ) is defined in Definition . . When b χ p = 1, b χ factors through Γ − J \ J , and b ν b χ lies in the J \ J -branch of b ν . So, the notation L J \ J ( π K , b ν b χ ) makes sense. Proof. By continuity we may suppose that b χ factors through A ∞ , × F K × \ A ∞ , × K / b O × J ; ~n, c for some J \ J -tuple ~n . Indeed, among the characters satisfying our condition, those of finite order aredense.Write J = { p , · · · , p r } and J ′ = { p , · · · , p d } . For our convenience we identify J ′ with theindex set { , · · · , d } . For each i = 1 , · · · , d , put ℓ i = P σ ∈ Σ p i s σ log σ . Then l = d P i =1 ℓ i .Since J is fixed, we will write Λ ? , p ,a for Λ ? , p ,J ,a ( p ∈ J ), and write Λ ∅ ,a for Λ ∅ ,J ,a . For themeanings of these notations see Section 7.3.Let ¯Ω a be the matrix Λ l, p ,a − p l ( β p )Λ ∅ ,a − p l ( β p )Λ ∅ ,a · · · − p l ( β p r )Λ ∅ ,a − p l ( β p )Λ ∅ ,a Λ l, p ,a − p l ( β p )Λ ∅ ,a · · · − p l ( β p r )Λ ∅ ,a ... ... . . . ... − p r l ( β p )Λ ∅ ,a − p r l ( β p )Λ ∅ ,a · · · Λ l, p r ,a − p r l ( β p r )Λ ∅ ,a . If i = j ∈ { , · · · , r } , then¯Ω a,ij = − p i l ( β p j )Λ ∅ ,a = − X σ ∈ Σ p i s σ p i log σ ( β p j )Λ ∅ ,a . For i ∈ { , · · · , r } we have¯Ω a,ii = Λ ℓ i , p i ,a + X ≤ i ′ ≤ d,i ′ = i Λ ℓ i ′ , p i ,a − p i l ( β p i )Λ ∅ ,a . Let Ω a be the matrix with Ω a,ii = Λ ℓ i , p i ,a and Ω a,ij = ¯Ω a,ij if i = j . Let M a be the diagonal matrix P ≤ i ′ ≤ d,i ′ =1 Λ ℓ i ′ , p ,a − p l ( β p )Λ ∅ ,a . . . P ≤ i ′ ≤ d,i ′ = r Λ ℓ i ′ , p r ,a − p r l ( β p r )Λ ∅ ,a . a = Ω a + M a .For arbitrary two disjoint subsets T and S of J we set Z TS,J ,a = Y p ∈ S B p × Y p ∈ J D p × a ( Y v / ∈ S ∪ T ∪ J D v )and B T = Y p ∈ T B p . Put f Ta : = Y i ∈ J \ T (cid:16) X i ′ :1 ≤ i ′ ≤ d,i ′ = i p i ℓ i ′ · Z TJ \ ( T ∪{ p i } ) ,J ,a − p i l ( β p i ) · Z TJ \ T,J ,a (cid:17) = X S ⊆ J \ T Y i ∈ J \ ( T ∪ S ) X i ′ :1 ≤ i ′ ≤ d,i ′ = i p i ℓ i ′ · Y i ∈ S p i l ( β p i ) · Z TS,J ,a . Then f Ta is ∆ T -invariant, and1 B T ⊗ f Ta = Y i ∈ J \ T X ≤ i ′ ≤ d,i ′ = i Λ ℓ i ′ , p i ,a − p i l ( β p i )Λ ∅ ,a By our condition on b χ , considered as a function on A ∞ , × F \ A ∞ , × K , b χ is the pull-back of a functionon A ∞ ,J , × F \ A ∞ ,J , × K . We write b χ J for the latter function. For any T ⊂ J let b χ T be the pull-backof b χ J to A ∞ ,T, × F \ A ∞ ,T, × K .We use Ω a,T to denote the submatrix of Ω a consisting of the ( i, j )-entries ( i, j ∈ T ). Thendet( ¯Ω a ) = X T det(Ω a,T ) · Y i ∈ J \ T (cid:16) X ≤ i ′ ≤ d,i ′ = i Λ ℓ i ′ , p i ,a − p i l ( β p i )Λ ∅ ,a (cid:17) = X T det(Ω a,T ) · (1 B T ⊗ f Ta )with the convention that det Ω a, ∅ = 1 when T = ∅ .When T = { i , · · · , i k } with i < i < · · · < i k , we define c log ,T ( g , · · · , g k ) = X P ( − P c p i ,ℓ i ( g p ) · · · c p ik ,ℓ ik ( g p k ) , g , · · · , g k ∈ ∆ T , where in the sum P runs over all permutations ( p , · · · , p k ) of { , · · · , k } , and( − P = (cid:26) P is an even permutation − P is an odd permutation . Then c log ,T is a k -cocycle on ∆ T , and[ c log ,T ] = [ c p i ,ℓ i | T ] ∪ · · · ∪ [ c p ik ,ℓ i | T ] . c on ∆ J , c | T means the restriction of c to ∆ T . Note thatdet(Ω a,T ) = c log ,T ( β p i , · · · , β p ik ) ⊗ Z TJ \ T,J ,a . As b χ T f Ta Z TJ \ T,J ,a is ∆ T -invariant, we have Z b χ · det(Ω a,T ) · (1 B T ⊗ f Ta ) e µ J = ([ b χ T f Ta Z TJ \ T,J ,a ] ∪ [ c log ,T ] ∪ [ e µ J ])( β p i , · · · , β p ik ) . By Theorem 8.10 and its proof we obtain Z b χ · det(Ω a,T ) · (1 B T ⊗ f Ta ) e µ J = (cid:0) Y i ∈ T L p i ,ℓ i (cid:1) ([ b χ T f Ta Z TJ \ T,J ,a ] ∪ [ c p i , ord ] ∪ · · · ∪ [ c p ik , ord ] ∪ [ e µ J ])( β i , · · · , β i k )= (cid:0) Y i ∈ T L p i ,ℓ i (cid:1) · Z ( h i X s =1 · · · h ik X s k =1 ω s p i B p i ⊗ · · · ⊗ ω sk p ik B p ik ) ⊗ X S ⊆ J \ T Y i ∈ J \ ( T ∪ S ) X i ′ :1 ≤ i ′ ≤ d,i ′ = i p i ℓ i ′ · Y i ∈ S ( − p i l ( β p i )) · b χ T Z TS,J ,a e µ J . Here, we write h i for h p i (1 ≤ i ≤ d ).When S ( J \ T , writing U := J \ ( S ∪ T ) = { j , · · · , j u } we have Y i ∈ J \ ( T ∪ S ) X i ′ :1 ≤ i ′ ≤ d,i ′ = i p i ℓ i ′ = h j − X t =0 · · · h ju − X t u =0 c t , ··· ,t u ω t p j O × p j ⊗ · · · ⊗ ω tu p ju O × p ju , where c t , ··· ,t u are constants in C p that can be precisely determined but not important. Take b ∈ K × such that( b ) = s P i + · · · + s k P i k + t P j + · · · + t u P j u + sum of primes not above p. Let b ( p ) be the out- p -part of b . Then X a Z b χ (cid:16) (1 ω s p i B p i ⊗ · · · ⊗ ω sk p ik B p ik ) ⊗ (1 ω t p j O × p j ⊗ · · · ⊗ ω tu p ju O × p ju ) ⊗ Z T ∪ US,J ,a (cid:17)e µ J = X a Z b ∗ (cid:18)b χ (cid:16) (1 ω s p i B p i ⊗ · · · ⊗ ω sk p ik B p ik ) ⊗ (1 ω t p j O × p j ⊗ · · · ⊗ ω tu p ju O × p ju ) ⊗ Z T ∪ US,J ,a (cid:17)(cid:19) e µ J = X a Z b χ (cid:16) (1 B p i ⊗ · · · ⊗ B p ik ) ⊗ (1 O × p j ⊗ · · · ⊗ O × p ju ) ⊗ Z T ∪ US,J ,a/b ( p ) (cid:17)e µ J = X a Z b χ · S ∪ T,J ∪ U,a/b ( p ) e µ J = L J \ ( S ∪ T ) ( b ν b χ ) . 68y our condition on b χ we have b χ p = 1 for any p ∈ J \ ( S ∪ T ). So, by Corollary 5.11 we have L J \ ( S ∪ T ) ( b ν b χ ) = 0. Hence, (cid:0) Y i ∈ T L p i ,ℓ i (cid:1) · Z ( h i X s =1 · · · h ik X s k =1 ω s p i B p i ⊗ · · · ⊗ ω sk p ik B p ik ) ⊗ X S ( J \ T Y i ∈ J \ ( T ∪ S ) X i ′ :1 ≤ i ′ ≤ d,i ′ = i p i ℓ i ′ · Y i ∈ S ( − p i l ( β p i )) · b χ T Z TS,J ,a e µ J = 0 . When S = J \ T , we have X a Z (1 ω s p i B p i ⊗ · · · ω sk p ik B p ik ) ⊗ b χ T Z TJ \ T,J ,a e µ J = X a Z b χ Z J ,J ,a e µ J = L J \ J ( π, b ν b χ ) . Therefore X a Z b χ · det(Ω a,T ) · (1 B T ⊗ f Ta ) e µ J = ( r Y i =1 h p i ) · (cid:0) Y i ∈ T L p i ,ℓ i (cid:1) · (cid:16) Y i ∈ J \ T X i ′ :1 ≤ i ′ ≤ d,i ′ = i p i ℓ i ′ ( ω p i ) (cid:17) · L J \ J ( π, b ν b χ ) . Now, applying Proposition 7.10 we obtain X a Z b χ · X v ∈ J \ J v l r · Z ∅ ,J ,a e µ J = r ! X a Z b χ · det( ¯Ω a ) e µ J = r ! X T ⊂ J X a Z b χ · det(Ω a,T ) · (1 B T ⊗ f Ta ) e µ J = r ! · ( r Y i =1 h p i ) · X T ⊂ J (cid:0) Y i ∈ T L p i ,ℓ i (cid:1) · (cid:16) Y i ∈ J \ T X i ′ :1 ≤ i ′ ≤ d,i ′ = i p i ℓ i ′ ( ω p i ) (cid:17) · L J \ J ( π, b ν, b χ )= r ! · ( r Y i =1 h p i ) · r Y i =1 L p i ,ℓ i + X i ′ :1 ≤ i ′ ≤ d,i ′ = i p i ℓ i ′ ( ω p i ) · L J \ J ( π, b ν b χ ) , as expected.With ( s σ ) σ ∈ Σ J fixed, we put L J ( t ; b ν b χ, ( s σ ) σ ) = L J (( ts σ ) σ , π K , b ν b χ ).69 heorem 9.2. We have d k dt k L J ( t ; b ν b χ, ( s σ ) σ ) | t =0 = if ≤ k < r,r ! Q p ∈ J P σ ∈ Σ J \ Σ p s σ p ℓ σ ( ω p ) + P σ ∈ Σ p s σ L Tei σ,u σ p ! · L J \ J ( π K , b ν b χ ) if k = r. Proof. We have d k dt k L J ( t ; b ν b χ, ( s σ ) σ ) | t =0 = Z b χ · l k µ J = 1 Q p ∈ J h p Z X a b χ · X v v l ! k · Z ∅ ,J ,a e µ J , where v runs through all finite places of F . When k < r , by Lemma 7.7 this is equal to 0. When k = r , again using Lemma 7.7 we obtain1 Q p ∈ J h p Z X a b χ · X v v l ! r · Z ∅ ,J ,a e µ J = 1 Q p ∈ J h p Z X a b χ · X v ∈ J v l ! r · Z ∅ ,J ,a e µ J . But X v ∈ J v l ! r · Z ∅ ,J ,a = X v ∈ J \ J v l r · Z ∅ ,J ,a . By Proposition 9.1 we obtain the assertion for k = r .Allowing s σ varying, we obtain the following Theorem 9.3. Let b χ be a character of Γ − J such that b χ p = 1 for every p ∈ J . If ~σ = ( σ , · · · , σ r ) is a tuple of elements in Σ J ′ , then ∂ r ∂s σ · · · ∂s σ r L J ( ~s, π K , b ν b χ ) (cid:12)(cid:12)(cid:12)(cid:12) ~s = ~ = (cid:16) X P =( j , ··· ,j r ) r Y i =1 L p ji ,σ i (cid:17) · L J \ J ( π K , b ν b χ ) , where in the sum P runs over all permutations ( j , · · · , j r ) of { , · · · , r } .Proof. By Theorem 9.2 we have L J ( ~s, π K , b ν b χ ) = L J \ J ( π K , b ν b χ ) Y p ∈ J X σ ∈ Σ J s σ L p ,σ ! + higher order terms . (9.1)For every vector ~n = ( n τ ) τ ∈ Σ J ′ of nonnegative integers with P τ n τ = r , the coefficient of Q τ ∈ Σ J ′ s n τ τ is L J \ J ( π K , b ν b χ ) X f r Y i =1 L p i ,f ( i ) , f runs over all maps f : { , · · · , r } → Σ J ′ such that ♯f − ( τ ) = n τ for each τ ∈ Σ J ′ . Let f be such a map, and write σ i = f ( i ). Then L J \ J ( π K , b ν b χ ) X f r Y i =1 L p i ,f ( i ) = 1 ~n ! L J \ J ( π K , b ν b χ ) X P =( j , ··· ,j r ) r Y i =1 L p i ,σ ji , where P runs over all permutations ( j , · · · , j r ) of { , · · · , r } . Hence, ∂ r ∂s σ · · · ∂s σ r L J ( ~s, π K , b ν b χ ) (cid:12)(cid:12)(cid:12) ~s = ~ = ∂ r Q τ ∈ Σ J ′ ∂s n τ τ L J ( π K , b ν b χ, ~s )= ~n ! · ~n ! · L J \ J ( π K , b ν b χ ) X P =( j , ··· ,j r ) r Y i =1 L p i ,σ ji = L J \ J ( π K , b ν b χ ) X P =( j , ··· ,j r ) r Y i =1 L p i ,σ ji = L J \ J ( π K , b ν b χ ) X P =( j , ··· ,j r ) r Y i =1 L p ji ,σ i , as desired.As last, we prove Theorems 1.1, 1.2, 1.3 and 1.4. Theorem 1.4 follows immediately from (9.1).We fix notations. Take J to be J p , and let J and J be as in the introduction. We take the CMtype Σ K to be set of the places above the infinite place of K corresponding to a fixed embedding K ֒ → C . Write K ∩ − Ω p = P and p O K = P ¯ P . Then in the decomposition p O K = P ¯ P for eachprime p above p , P is above P , and ¯ P is above ¯ P . We take c in Section 2.2 to be O F . Take m tobe the zero vector , and take ν to be the trivial character so that b ν is also trivial.Let π be the automorphic representation of GL ( A F ) associated to f . The local conditions on f ensure that there exists a definite quaternion algebra B over F whose discriminant n − b is theproduct of the primes that divide n − and are inert in K , and an automorphic representation on B attached to π via Jacquet-Langlands correspondence. So we can carry out the constructions inSections 4 and 5. We should note that π , ν = 1 and B satisfy the local assumptions made afterProposition 4.11.Let L J p ( π K , b χ ) be the p -adic L -function constructed in Section 5. Put Ω − f = Ω − J p ,φ and L p ( π K , b χ ) = b χ ( N + ) − · ( Y v b χ v ( ω v )) − · L J p ( π K , b χ )where v runs over the set (4 . L p ( π K , b χ ) to (1.6) we obtain the p -adic L -functiondemanded in Theorem 1.1.We apply Theorem 9.2 to prove Theorem 1.2 and Theorem 1.3.Put L Tei p ( f ) = X ι ∈ Σ p L Tei ι,u ι p ( f ) + X σ ∈ Σ Jp \ Σ p p ℓ σ ( ω p ) . Let β K be an element of K × such that β K is a unit at all finite places outside of P . Let u K be the element β K , P β K , ¯ P of Q p . Consider the infinitesimal of h · i s anti as an character of K × \ A ×K . Its71estriction to K × p = K × P × K × ¯ P becomes ( x, y ) log u K ( xy ). Note that ǫ ( s, ··· ,s ) is the restriction of h · i s anti . So P σ ∈ Σ Jp ℓ σ is the infinitesimal of h · i s anti . Hence, X σ ∈ Σ p \ Σ p p ℓ σ ( ω p ) = 1ord p ( u p ) X σ ∈ Σ Jp p ℓ σ ( u p ) = 1ord p ( u p ) log u K ( N F/ Q ( u p )) = 1ord p ( u p ) X ι ∈ Σ p log u K ( u ι p ) . Observe that 1ord p ( u p ) log u K ( u ι p ) = log u K ( ω p ) − log u ι p ( ω p ) . By Proposition 6.26 we have L Tei p ( f ) = X ι ∈ Σ p L Tei ι,u K ( f ) . Theorem 9.2 tells us d n L p ( s, f /K ) ds n (cid:12)(cid:12)(cid:12) s =0 = n < r, ( r !) Q p ∈ J L Tei p ( f ) ! · L J ((0 , · · · , , π K ) if n = 2 r, and L J p (( s σ ) σ ∈ Σ Jp , π K ) = L J ((0 , · · · , π K ) · Y p ∈ J X σ ∈ Σ Jp s σ L p ,σ + higher order terms . By the interpolation formula (i.e. Theorem 5.9) for L J ( · , π K ) we have L J ((0 , · · · , , π K ) = Y p ∈ J e p (1) · L ( , π K )Ω − J ,φ = Y p ∈ J e p (1) · L (1 , f /K )Ω − J ,φ We show Ω − J ,φ = Ω − J p ,φ to complete our proof. By the formula on Ω − J,φ given in Section 5 we obtainΩ − J ,φ = Ω − J p ,φ · Y p ∈ J ǫ ( 12 , π p , ψ p ) . The root number ǫ ( , π p , ψ p ) is computed by [20, Proposition 3.6] which says that ǫ ( 12 , π p , ψ p ) = ǫ ( 12 , µ p , ψ p ) ǫ ( 12 , µ p | · | − p , ψ p )if π p = σ ( µ p , µ p | · | − p ). Since µ p | · | − p = 1, it follows from Tate’s local functional equation (see [6,Proposition 3.1.5, 3.1.9]) that ǫ ( , µ p , ψ p ) ǫ ( , µ p | · | − p , ψ p ) = 1.72 eferences [1] Y. 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