MODULAR p-adic L-FUNCTIONS ATTACHED TO REAL QUADRATIC FIELDS AND ARITHMETIC APPLICATIONS

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1 MODULAR p-adic L-FUNCTIONS ATTACHED TO REAL QUADRATIC FIELDS AND ARITHMETIC APPLICATIONS MATTHEW GREENBERG, MARCO ADAMO SEVESO, AND SHAHAB SHAHABI Abstract Let f S 0 +2Γ 0 Np be a normalized N-new eigenform with p N and such that a 2 p p+1 and ord p a p < + 1 By Coleman s theory, there is a p-adic family F of eigenforms whose weight + 2 specialization is f Let K be a real quadratic field and let ψ be an unramified character of GalK/K Under mild hypotheses on the discriminant of K and the factorization of N, we construct a p-adic L- function L F/K,ψ interpolating the central critical values of the Ranin L-functions associated to the base change to K of the specializations of F in classical weight, twisted by ψ When the character ψ is quadratic, L F/K,ψ factors into a product of two Mazur-Kitagawa p-adic L-functions If, in addition, F has p-new specialization in weight + 2, then under natural parity hypotheses we may relate derivatives of each of the Mazur-Kitagawa factors of L F/K,ψ at to Bloch-Kato logarithms of Heegner cycles On the other hand the derivatives of our p-adic L-functions encodes the position of the so called Darmon cycles As an application we prove rationality results about them, generalizing theorems of Bertolini-Darmon, Seveso, and Shahabi Contents 1 Introduction 2 11 Summary 2 12 Setting 2 13 Main results I: interpolation and arithmetic applications 6 14 Main results II: the connection with Darmon cycles Construction of p-adic L-functions 11 Part 1 Real-analytic cycles on Shimura curves and special values of L-functions 12 2 Quaternions Splittings and orders Embeddings and orientations Rational representations of B 15 3 Shimura curves Modular forms and Hece operators Eichler-Shimura cohomology groups Homology classes and values of L-functions 20 Part 2 p-adic L-functions 21 4 Families of cohomology classes 21 5 p-adic L-functions: interpolation properties p-adic L-functions when ɛ K p = p-adic L-functions when ɛ K p = 1 30 Part 3 Derivatives of p-adic L-functions 31 6 p-adic L-functions: relations with Darmon classes The arithmetic p-adic Abel-Jacobi map Darmon classes and conjectures 34 Date: February 24, 2016 MG s research is supported by NSERC of Canada 1

2 63 Darmon classes and derivatives of p-adic L-functions Darmon classes and their rationality 36 7 Proof of Theorem The faux Abel-Jacobi map Faux Abel-Jacobi map and derivatives of p-adic L-functions 41 8 Equality of the arithmetic and the faux Abel-Jacobi maps 42 References 45 1 Introduction 11 Summary Let p be a prime, let N be a squarefree integer with p N, and let E/Q be an elliptic curve of conductor pn Let K be a real quadratic field and let ɛ K be the corresponding quadratic character Under the Star-Heegner hypothesis 1 ɛ K p = 1 and ɛ K l = +1 for all l N, Darmon [12] presented a construction of a Star-Heegner point P K EK p which, he conjectured, actually belongs to EK and governs the arithmetic of E over K in much the same way as a classical Heegner point governs the arithmetic of elliptic curves over imaginary quadratic fields Even though largely conjectural, Darmon s construction provided the beginnings of a coherent approach for studying rational points and associated objects Selmer classes, classical and p-adic L-functions, etc outside of usual framewor of complex multiplication theory In recent joint wor [32] with Rotger, the second author gave a far-reaching generalization of Darmon s original construction and conjectures, replacing elliptic curves by higher weight modular forms, Mordell-Weil groups with Bloch-Kato Selmer groups, and the Star-Heegner hypothesis 1 with a natural condition on signs in functional equations of L-functions The goal of this paper is to provide strong theoretical evidence for the conjectures of [32], thus generalizing results of [5], by proving that the conjectures of loc cit are compatible with analogous theorems from CM-theory in situations where they overlap, as first discovered in [5] We deduce such compatibilities using a class of weight variable p-adic L-functions that we construct in Part 2 of this paper In Part 3, we recall the conjectures of [32], relate them to the p-adic L-functions of Part 1, and prove our main results 12 Setting Fix an algebraic closure Q of Q, an odd prime number p, and embeddings 2 σ : Q C, σ p : Q Q p Let N be a positive integer with p N, let E be a p-adic field, and let Ω be an affinoid dis in the weight space X defined over the p-adic field E A Coleman Ω-family of cuspidal eigenforms of tame level Γ 0 N is a formal q-expansion Fq = n 1 a n q n OΩ[[q]], such that for all Ω cl := {n 2Z : n 0} Ω, there is a normalized eigenform F S +2 Γ 0 Np, Q satisfying F q = n 1 a n q n Since a p is rigid analytic, the function ord p a p may be assumed to be constant, up to shrining in an open affinoid neighbourhood of any Ω We assume this condition satisfied, and call this quantity the slope of F Restricting to a case of particular interest, we assume that F is N-new for all Ω cl It follows that, for each Ω cl, the form F is a p-stabilized newform, ie, either F is p-new, or there is a newform F S +2Γ 0 N such that 3 F q = F p+1 q a p F qp 2

3 If F 0 is p-new, we set F = F There is at most one Ω cl at which F 0 is p-new: F is p-new exactly when a p = ±p 0/2 That this equation can hold for at most one Ω cl follows from the constancy of the slope of F We set Ω p-old cl = { Ω cl : F is p-old} Our first goal in this paper is to study the p-adic variation of the central critical L-values associated to the newforms F as varies over Ω cl By a theorem of Shimura [43, Theorem 1], there are nonzero Shimura periods u ± C such that for every quadratic Dirichlet character χ, 4 L F j 1!τχc χj, χ, j := 2πi j 1 u ± LF, χ, j QF, 1 j + 1, where τχ is the Gauss sum associated to χ, c χ its conductor, QF is the field generated by the Fourier coefficients of F, and the sign of the Shimura period is chosen so that 5 ± 1 j 1 = χ 1 The quantity L F, χ, j is called the algebraic part of the special value Scaled appropriately by p-adic periods and Euler-lie factors, these algebraic parts can be p-adically interpolated: Theorem 11 There exist nonzero p-adic periods λ ± E for Ω cl such that for every quadratic Dirichlet character χ there is a p-adic analytic function L F,χ on Ω Z p satisfying 1 χppj 1 L L F,χ, j F, χ, j if F is p-new, a p λ ± = 1 1 χppj 1 χpp j+1 L F, χ, j a p a p if F is p-old whenever 1 j + 1 and the parity condition 5 holds Remar 12 For each Ω cl, there is a unique normalization of L F,χ such that λ ± = 1 When F has a p-new specialization at we normalize L F,χ such that λ ± = 1 The function L F,χ is called the 2-variable p-adic L-function associated to the Coleman family F and the character χ It s construction is due to Mazur unpublished notes and Kitagawa [24] when the family F has slope zero, and to Stevens see [3], [29] and [30] in the general case Considering the restriction 6 L F,χ := L F,χ, /2 + 1, Ω, of L F,χ to the critical line = 2j 2 we see that 1 χpp/2 L F,χ a p 7 λ ± = 1 χpp/2 a p L F, χ, / L F, χ, /2 + 1 if if F is p-new, F is p-old It is possible that L F,χ is identically zero: Let ω N, be the eigenvalue of the Atin-Lehner involution W N acting on F The completed L-function satisfies the functional equation ΛF, χ, s := 2π s ΓsLF, χ, s ΛF, χ, s = 1/2+1 χ Nω N, fχ 2 N /2+1 s ΛF, χ, + 2 s, implying ΛF, χ, /2 + 1 = 1/2+1 χ Nω N, ΛF, χ, /2 + 1 It can be shown that for, Ω cl, ω N, = 1 + /2 ω N, 3

4 Therefore, ω N := 1 /2+1 ω N, is independent of and the sign in the functional equation of ΛF, χ, s is χ Nω N If χ Nω N = 1, then LF, χ, /2 + 1 = 0 for all and L F,χ is identically zero by 7 It is therefore natural, studying L F,χ, to wor under the assumption 8 χ Nω N = +1 Even though we exclude the possibility of 7 forcing L F,χ to be identically zero, it is still possible that the interpolation formula imposes an isolated zero on L F,χ, namely, when the Euler-lie factor 2 1 χpp/2 a p vanishes Suppose there is a weight Ω cl such that 9 χp = a p p 0/2 = ω p, There is at most one such by the constancy of the slope of F By the above discussion together with our assumption that χ is quadratic, this can happen only when F 0 is p-new When this does happen, though, the quantity L F,χ /2 + 1 is related to Heegner cycles This result, which is proved in [4] and generalized to our setting in [38], is recalled in Theorem 14 below We introduce the terminology necessary to state this relationship precisely We need to assume the existence of an auxiliary factorization N = MQ with M and Q coprime and Q squarefree and with an odd number of primes factors For such a decomposition to exist, we need the existence of a prime q N, in which case we may tae Q = q Let B Qp be the unique indefinite quaternion algebra of discriminant Qp and write S 0+2Γ Qp 0 M for the space of weight + 2-modular forms on B Qp of level Γ Qp 0 M See 31 for the notation Let M 0 resp M Qp be the Chow motive associated to the space S 0+2Γ 0 Np resp S 0+2Γ Qp 0 M and let V Np resp V Qp M be its p-adic étale realization, viewed as a continuous, Q p -adic representation of G Q By the Jacquet-Langlands correspondence, the Eichler Shimura relations and the Brauer-Nesbitt principle see for example [22, Lemma 59] there is an identification, both Hece and Galois equivariant, of V Np Qp-new V Qp M It induces an identification V Np new V Qp M new between the associated new parts We fix such an identification once and for all and simply write V for either of these two identified representations The new subspace S 0+2Γ 0 Np new has a natural Q-structure, preserved by the Hece operators, arising be the Q p -algebra of S 0+2Γ 0 Np, Q p new generated by all the Hece operators By Fontaine s theory, we may associate to the restriction to a decomposition group at p of V a filtered Frobenius module with a monodromy operator D := D st V from q-expansion Write S 0+2Γ 0 Np, Q p new for the Q p -space obtained by base change and let T new Q p which is a T new Q p -monodromy module defined over Q p by results of [9] Then D is a free module over T new Q p of ran two Let L be the Fontaine-Mazur L-invariant of D as defined in [26] and let mean Q p -dual in the following theorem Theorem 13 There is an isomorphism of T new Q p -monodromy modules defined over Q p D S 0+2Γ 0 Np, Q p new, S 0+2Γ 0 Np, Q p new, under which the only non-trivial step in the filtration, namely F j D for j = 1,, 1, maps isomorphically onto { Lx, x : x S 0+2Γ 0 Np, Q p new, } Set m = /2 + 1 If E is a finite extension of Q p, then the Bloch-Kato exponential gives an isomorphism D E 10 exp : F m H 1 D E ste, V, 4

5 where Theorem 13 gives an isomorphism H 1 ste, V m = er H 1 E, V m H 1 E, V m B st D E F m S 0+2Γ 0 Np, E new,, D E where now means E-dual Composing this map with exp 1, we obtain a isomorphism log : H 1 ste, V m S 0+2Γ 0 Np, E new, On the global side, we have the p-adic étale Abel-Jacobi map cl := cl 0/2+1 0,L : CH 0/2+1 0 M Qp L H 1 L, V for any number field L Q, where the Chow groups are taen with rational coefficients Here, the i-chow group of a motive M := X, p, m, where X is a smooth scheme, p is an idempotent and m is an integer, is defined to be CH i M := Hom 1, M i; if m = 0, this is simply the p-component of the Chow group CH i X The subscript 0 denotes the subgroup of cycles that are homologically equivalent to zero It is nown that the image of cl is contained in the in the semistable Bloch-Kato Selmer group Sel st L, V See 65 for its definition Let p be the prime of L Q over p determined by the embedding σ p : Q Q p and let L p be the p-adic completion Then we may consider the composite log cl CH 0/2+1 0 M Qp L Sel st L, V m H 1 stl p, V m S 0+2Γ 0 Np, L p p-new,, where means L p -dual Whenever L F,χ vanishes at to order at least two, L F,χ does not depend on the family F through F 0 see Remar 59 This double vanishing occurs when 8 and 9 hold, by results of [4] generalized in [38] to our setting If [F 0 ] is the companion also called Galois conjugacy class of F 0, the [F 0 ]-isotypic component S 0+2Γ 0 Np, C p [F0 ], which is the sum of all σ F 0 -isotypic components for all σ G Q, descends to Q p and, indeed, to Q In other words, there is a space of modular forms S 0+2Γ 0 Np, Q p [F0 ] whose base change to C p is S 0+2Γ 0 Np, C p [F0 ] see [32, 43] We note that 8 and 9 are really conditions on [F 0 ], because ω N, ω p,0 = ω p,f0 Q and χ taes its values in {±1} Q In particular, it maes sense to consider the subspace S 0+2Γ 0 Np, Q p new χ N=ω N,χp= ω p S 0+2Γ 0 Np, Q p new spanned by new forms on which 8 and 9 hold Similarly, we may consider the Q p -adic representation V χ N=ωN,χp= ω p resp V [F0 ] of G Q cut out by the conditions 8 and 9 resp attached to F 0 S 0+2Γ 0 Np, C p new χ N=ω N,χp= ω p Let Q χ be the quadratic extension cut out by the character χ and let Q χ p be its p-adic completion Then F 0 L F,χ may be viewed as a C p -valued function on S 0+2Γ 0 Np, C p new χ N=ω N,χp= ω p that we denote L χ Indeed, it restricts on S 0+2Γ 0 Np, Q χ p new χ N=ω N,χp= ω p to a Q χ p -valued function L χ : S 0+2Γ 0 Np, Q χ p new χ N=ω N,χp= ω p Q χ p and similarly for its restriction to the [F 0 ]-component S 0+2Γ 0 Np, Q χ p [F0 ] Let χ denote the χ- component Projecting onto the new subspaces where 8 and 9 hold resp the [F 0 ]-component, we may consider the composite log cl S new resp log cl : S[F0 χ N=ω N,χp= ωp ] CH 0/2+1 0 M Qp Q χ χ Sel st Q χ, Vχ N=ω new N,χp= ω p m HstQ 1 χ p, Vχ N=ω new N,χp= ω p m S 0+2Γ 0 Np, Q χ p new, χ N=ω N,χp= ω p, CH 0/2+1 0 M Qp Q χ χ Sel st Q χ, V [F0 ]m H 1 stq χ p, V [F0 ]m S 0+2Γ 0 Np, Q χ p [F 0 ] Let T new χ N=ω N,χp= ω p be the semisimple rational Hece algebra of S 0+2Γ 0 Np new χ N=ω N,χp= ω p and let T [F0 ] its projection onto the [F ]-component for any F 0 S 0+2Γ 0 Np, C p new χ N=ω N,χp= ω p Note 5

6 that λ F0 : T [F0 ] Q F 0 for any F 0 [F 0 ], where λ F0 is the eigenpacet of F 0 In particular, σ λ F0 = λ σf0 for any σ G Q The following theorem is proved in [4], in the special case = 0, when one restricts to the subspace of S 0+2Γ 0 Np, Q p new χ N=ω N,χp= ω p spanned by modular forms with rational Fourier coefficients It is proved in the general case in [38] Theorem 14 Suppose that there is a prime q with q N and choose a factorization N = MQ as above Then the following facts hold 1 There is a cycle y χ = y χ Qp CH0/2+1 0 M Qp Q χ χ and t = t Qp χ T new, χ N=ω N,χp= ω p 2 If 0 cl y χ [F0 ] Sel stq χ, V [F0 ]m, then L χ = t log 2 cl y χ S new χ N=ω N,χp= ωp Sel st Q χ, V [F0 ]m χ = T [F0 ] Q p cl y χ [F0 ] T [F 0 ] Q p, where cl y χ [F0 ] is the [F ]-component of cl y χ 3 Suppose = 0 Then 0 cl y χ [F0 ] if and only if L F 0, χ, / If t F0 := λ F0 t Q F 0, then for any quadratic Dirichlet character ɛ such that the congruence holds ɛ N = χ N, ɛ p = χ p and L F 0, ɛ, / , t F0 L F 0, ɛ, /2 + 1 in Q F 0 /Q F 0 2 such that Proof We simply remar that an inspection to the proof of [38, Theorem 61] shows that the theorem is true with our slightly more general condition 4 We also note that the case = 0 appears in detail in [4] only for modular forms with rational Fourier coefficients; however a generalization of this result to S 0+2Γ 0 Np, Q p new χ N=ω N,χp= ω p is possible, thans to [4, Introduction, Remar 5] See [20] for more details 13 Main results I: interpolation and arithmetic applications Let K Q be a real quadratic field whose discriminant d K is prime to Np We mae the following convenient but unnatural assumption, which can liely be removed with more effort: Assumption 15 If a prime l divides N and is inert in K, then ord l N = 1 We introduce some notation concerning the field K We write N = N + D as the product of the primes N + that are split in K and those primes D that are inert Let p be the prime ideal of K above p determined by σ p If p splits in K, let p be the other prime of K above p The embedding σ also pics out a real embedding of K which we will also denote by σ Let H K resp H + K be the Hilbert class field resp narrow Hilbert class field of K The extension H + K /H K has degree one or two If the degree is two, let s denote the nontrivial element of GalH + K /H K If H + K = H K, let s denote the identity element of GalH + K /H K Let ψ : GalH + K /K Q be a character In this paper, we prove interpolation results in the spirit of Theorems 11 and 14 for central critical Ranin L-values LF /K, ψ, /2 + 1 Prototypes of these results have been proved by several authors [4, 5, 38, 41] The goal of this paper is to unify the methods of these papers into a cohomological framewor, simultaneously simplifying the treatment and generalizing the results After stating our results, we will point out the precise overlap between these and the results of [4, 5, 38, 41] Let ɛ K be the Dirichlet character associated to K If is in Ω cl, then the completed twisted L-functions of F over K satisfies the functional equation ΛF /K, ψ, + 2 s = ɛ K NΛF /K, ψ, s 6

7 Thus, the central critical values LF /K, ψ, vanish for all Ω cl,, when ɛ K N = 1 Thus, to avoid interpolating the zero function for reasons of signs, we wor under the assumption that ɛ K N = +1 From here, our analysis falls into two cases as ɛ K p = +1 or ɛ K p = 1 Note that in the latter case, ɛ K Np = 1 and LF 0 /K, ψ, /2 + 1 = 0 In either case, we have 11 L F /K, ψ, /2 + 1 := /2! d K 2πi u ± 2 LF /K, ψ, /2 + 1 Qψ, F, The sign of the Shimura period is 1 /2 ψs Our main interpolation result is the following: Theorem 16 There is a unique rigid analytic function L F/K,ψ OΩ such that for all Ω p-old 2 1 p L F /K, ψ, /2 + 1 if ɛ Kp = 1, L F/K,ψ λ ± 2 = a p ψpp/2 ψp p /2 a p a p cl, 2 L F /K, ψ, /2 + 1 if ɛ Kp = +1 Moreover, if ɛ K p = 1 resp ɛ K p = +1 and F 0 is p-new resp F 0 is new at p and ψp = a p /p /2, then L F/K,ψ vanishes at to order at least two Remar 17 Theorem 16 is expected to hold for all ring class characters with conductor prime to d K Np The proof, however, relies on a special value formula of [31] that is proved only in the special case of unramified characters One observes an intesting phenomenon when ψ is a genus character of K, ie, a quadratic character of GalH + K /K The genus character ψ corresponds to a factorization d K = d K1 d K2 of d K into two fundamental discriminants, corresponding to the fields K 1 and K 2, say, and a pair of Dirichlet characters χ i : Z/d Ki Z {±1} satisfying χ 1 χ 2 = ɛ K and χ i Normq = ψq for all degree-one primes of K In addition, one can also easily establish the following factorization formula for central critical values: 12 L F /K, ψ, /2 + 1 = L F, χ 1, /2 + 1L F, χ 2, /2 + 1, Ω cl Combining this factorization formula with 7 and Theorem 16, we obtain: Corollary 18 The following factorization holds on Ω: L F/K,ψ = L F,χ1 L F,χ2 Combining Theorem 14 with Corollary 18 yields interesting arithmetic consequences Suppose that F 0 is p-new and that ε K p = 1 resp ε K p = 1 Under our Assumption 15 and ɛ K N = 1, the complex L-functions L F 0, χ i, s have opposite signs resp the same sign at s = /2 + 1, i = 1, 2, equal to /2+1 ω pn,0 χ i pn = ω N χ i N ω p,0 χ i p Note that, under our running Assumption 15 and ε K N = ε K N = 1, ω N χ i N = ω N ψ n = ω N ψ n +, where n resp n + is a prime of K above N resp N +, and is the class of complex conjugation in G HK /K In particular, ω N χ i N does not depend on i = 1, 2, and 8 holds for χ i It follows from 13 that, in this case, sign L F 0, χ i, s = 1 is negative 9 holds for χ i Let H ψ K be the quadratic extension of K cut out by the character ψ, and let Hψ K,p and ψ have the same meaning as above with L = H ψ K Suppose that W is a G H ψ /Q-module; since IndG Q G K K ψ = χ 1 χ 2, we have W ψ = W χ 1 W χ 2, where the left hand side is viewed as a GH ψ K /K-module, while the right hand side is 7

8 viewed as a G H ψ K /Q-module This remar applies to W = CH0/2+1 0 M Qp H ψ K and W = Sel sth ψ K, V for any Q p -adic representation of G Q, and gives 14 CH 0/2+1 0 M Qp H ψ K ψ = CH 0/2+1 0 M Qp Q χ 1 χ 1 CH 0/2+1 0 M Qp Q χ 2 χ 2, Sel st H ψ K, V = Sel st Q χ 1, V χ 1 Selst Q χ 2, V χ 2 In case ε K p = 1, we may order χ 1, χ 2 in such a way that the sign of L F 0, χ 1, s is 1 The conjectures of Bloch and Beilinson predict that the corresponding Bloch-Kato Selmer groups have positive dimension We expect them to be partially explained by Theorem 14, as it is shown in the subsequent Corollary 19 Indeed we see that conditions 8 and 9 on χ 1 are compatible with the above sign condition on L F 0, χ 1, s The factorization formula of Corollary 18, joint with 8 and 9 on χ 1, implies that L F/K,ψ vanishes at to order at least two Similarly as above, L F/K,ψ does not depend on the lift F of F 0 to an eigenfamily F see Remar 59 In this case, F 0 L F/K,ψ gives rise to a function L ψ : S 0+2Γ 0 Np, H ψ K,p new χ N=ω N,χp= ω p H ψ K,p and we may consider the analogous maps log cl S new and log cl S[F0 for L = H ψ χ 1 N=ω N,χ 1 p= ωp ] K and coefficients in H ψ K,p We write L [F 0 ],ψ for the restriction of L ψ to S [F0 ] = S 0+2Γ 0 Np, H ψ K,p [F 0 ] Corollary 19 Suppose that there is a prime q with q N and choose a factorization N = MQ as above Let K be such that ɛ K p = 1 and ɛ K N = +1 Let ψ be a genus character of K associated to the Dirichlet characters χ 1, χ 2, ordered in such a way that the sign of L F 0, χ 1, /2 + 1 is negative Suppose further that ω N χ i N = 1 for one or equivalently both i {1, 2} Then the following facts hold 1 There is a cycle and a t = t Qp ψ y ψ = y ψ Qp CH0/2+1 0 M Qp Q χ 1 χ 1 CH 0/2+1 0 M Qp H ψ K Tnew, χ 1 N=ω N,χ 1 p= ω p 2 If 0 cl y ψ [F 0 ] Sel sth ψ K, V [F 0 ]m, such that L ψ = t log 2 cl y ψ S new χ 1 N=ω N,χ 1 p= ωp Sel st Q χ 1, V[F0 ]m χ 1 = T[F0 ] Q p cl y ψ [F 0 ] T [F 0 ] Q p, where cl y ψ [F 0 ] is the [F ]-component of cl y ψ If, further, we assume L F 0, χ 2, / equivalently, if L F 0 /K, ψ, / , Sel st H ψ K, V [F 0 ]m ψ = Sel st Q χ 1, V[F0 ]m χ 1 = T[F0 ] Q p cl y ψ [F 0 ] T [F 0 ] Q p 3 Suppose = 0 Then 0 cl y ψ [F 0 ] if and only if L F 0 /K, ψ, / If L F 0, χ 2, / eg, if L F 0 /K, ψ, / , there is such that ỹ ψ CH 0/2+1 0 M 0 Q χ 1 χ 1 CH /2+1 0 M 0 H ψ K L [F 0 ],χ = 2 cl ỹ ψ 2 [F 0 ] and such that, if 0 cl ỹ ψ [F 0 ] Sel sth ψ K, V [F 0 ]m, Sel st H ψ K, V [F 0 ]m ψ = Sel st Q χ 1, V[F0 ]m χ 1 = T[F0 ] Q p cl ỹ ψ [F 0 ] T [F 0 ] Q p If, in addition, = 0, then 0 cl ỹ ψ [F 0 ] if and only if L F 0 /K, ψ, /

9 Proof We may wor with the [F 0 ]-component for all F 0 S 0+2Γ 0 Np, C p new χ 1 N=ω N,χ 1 p= ω p Thans to 10, HstH 1 ψ K,p, V [F 0 ]m HstE, 1 V [F0 ]m for any finite extension E/H ψ K,p In particular, the validity of parts 1 and 4 of the corollary are unaffected by viewing L ψ and L [F 0 ],χ as E-valued functionals Assuming that E contains the field generated by the Fourier coefficients of all f [F 0 ] via σ p, S 0+2Γ 0 Np, E [F0 ] decomposes as the direct sum of its λ-components on which T [F0 ],p := T [F0 ] Q p acts through λ, for all λ Hom Qp-alg T [F0 ],p, E There is no loss of generality in assuming that λ is obtained by means of the identification λ F0 : T [F0 ] Q F 0 followed by σ p, and then passing to the completion In this case, the λ-component of L [F 0 0 is L ],ψ F/K,ψ It follows from Corollary 18, Theorem 11 and Theorem 14 applied to χ 1, that there exist y ψ = y χ 1 CH 0/2+1 0 M Qp Q χ 1 χ 1 and t χ1 T new, χ 1 N=ω N,χ 1 p= ω p such that, setting t χ1,f 0 := λ t χ1, L F/K,ψ = L F,χ 1 L F,χ2 = t χ1,f 0 σ p 2L F 0, χ 2, /2 + 1 log cl y ψ 2 F0 Since the quantity t F0 := σ 1 p tχ1,f 0 2L F 0, χ 2, /2 + 1 Q F 0 satisfies t σf0 = σ t F0 for all σ G Q, there is t [F0 ] T [F0 ] inducing t σf0 whenever λ = σ p λ σf0, where λ σf 0 : T [F 0 ] Q σ F 0 is the homomorphism determined by the eigenpacet λ σf0 of σ F Claim 1 follows If L F 0 /K, χ 2, / , then χ 2 satisfies the assumptions on ɛ appearing in Theorem 14 4 It follows that t F0 /2 Q F 0 2 Since the Hece action on the λ-component is through λ = σ p λ F0, our claimed equation in 4 follows setting ỹ ψ := t F0 /2y ψ, where t F0 /2 is any lift of t F0 /2 Q F 0 to a Hece operator acting on the Chow groups The second part of 4 follows from 2, 3, the definition of ỹ ψ and the fact that L F 0 /K, ψ, / if and only if L F 0, χ 1, /2 + 1 when L F 0, χ 2, / The first assertion of 2 follows from Theorem 14 2 The second assertion follows from the implication L F 0 /K, χ 2, / = Sel st Q χ 2, V[F0 ]m χ 2 = 0, proved in [23, Theorem 142 2], and 14 Part 3 is a restatement of Theorem 14 3 We now assume ε K p = 1 If L F 0, χ i, s for i = 1, 2 have negative sign, L F 0 /K, ψ, /2 + 1 = 0 and we expect to have larger ran, again to be partially explained by Theorem 14, joint with the factorization 14, which is the cohomological version of the factorization of the complex L-functions We already remared that condition 8 on χ i does not depend on i = 1, 2; as we assume ε K p = 1, we see that the same is true for 9 relative to χ i In particular, 8 and 9 that are required for the application of Theorem 14 are simultaneously satisfied and S 0+2Γ 0 Np, H ψ K,p new χ i N=ω N,χ i p= ω p does not depend on i = 1, 2 It then follows from Corollary 18 that L F/K,ψ vanishes at to order at least two Similarly as above, L 4 F/K,ψ does not depend on the lift F of F 0 to an eigenfamily F see Remar 59 and we may consider L 4 ψ : S 0+2Γ 0 Np, H ψ K,p new, χ N=ω N,χp= ω p H ψ K,p Corollary 110 Suppose that there is a prime q with q N and choose a factorization N = MQ as above Let K be such that ɛ K p = +1 and ɛ K N = +1 Let ψ be a genus character of K associated to the Dirichlet characters χ 1, χ 2 and suppose that the sign of L F 0, χ i, /2 + 1 is negative for one or equivalently both i {1, 2} and that ω N χ i N = 1 for one or equivalently both i {1, 2} 1 There exist cycles y ψ i = y ψ Qp,i CH0/2+1 0 M Qp Q χ i χ i CH 0/2+1 0 M 0 H ψ K and there exist t i = t Qp i,χ i T new, L 4 ψ = 6t 1 t 2 log cl χ i N=ω N,χ i p= ω p, i = 1, 2, such that 2 2 y ψ 1 log cl y ψ 2 S new χ 1 N=ω N,χ 1 p= ωp 9 S new χ 1 N=ω N,χ 1 p= ωp

10 2 If 0 cl y ψ i Sel st H ψ K, V [F 0 ]m, [F 0 ] Sel st Q χ i, V[F0 ]m χ i = T[F0 ] Q p cl y ψ i where cl y ψ i is the [F 0 ]-component of cl y ψ i If 0 cl [F 0 ] Sel st H ψ K, V [F 0 ]m ψ = T [F0 ] Q p cl [F 0 ] T [F0 ] Q p, y ψ i y ψ 1 T [F 0 ] [F0 ] Q p cl y ψ 2 [F 0 ] for i = 1, 2 [F 0 ] 2 T [F0 ] Q p 3 Suppose = 0 Then 0 cl y ψ i [F 0 ] if and only if L F 0, χ i, / Furthermore, they are both non-trivial if and only if L F 0 /K, ψ, / If t i,f0 := λ F0 t i Q F 0, then for any quadratic Dirichlet character ɛ such that the congruence holds ɛ N = χ i N, ɛ p = χ i p and L F 0, ɛ, / , t i,f0 L F 0, ɛ, /2 + 1 in Q F 0 /Q F 0 2 Proof As in the proof of Corollary 19, we fix F 0 and a sufficiently large p-adic field E It follows from Corollary 18 and Theorem 14 applied to χ i, that there exist y ψ i = y χ i CH 0/2+1 0 M Qp Q χ i χ i and t i T new, χ i N=ω N,χ i p= ω p such that, setting t i,f0 := λ F0 t i, 2 2 L 4 F/K,ψ = 6L F,χ 1 L F,χ 2 = 6t 1,F0 t 2,F0 log cl y ψ 1 F0 log cl y ψ 2 F0 The claim is now a restatement of Theorem 14 Remars When = 0, the image of the cycles appearing in Theorem 14, Corollary 19 and Corollary 110 factors through the appropriate Mordell-Weil group A L Q see [20] for more details After extending coefficients to Q p, their local restrictions factor through A L p Q p and the Bloch-Kato logarithm is compatible, up to the Kummer map, with the usual p-adic logarithm Hence, in this case, our p-adic L-functions really control elements coming from A L Q A L p Q p 2 When ε K p = 1 and D = 1, Theorem 16, Corollary 18 and Corollary 19 were proved in [4], when = 0 for modular forms with rational Fourier coefficients, and in [38] for arbitrary even 0 When ε K p = 1 and D = 1, Theorem 16, Corollary 18 and Corollary 110 were proved in [41], when = 0 for modular forms with rational Fourier coefficients The novelty of our methods, which allows us to wor simultaneously in the case where D may be different from 1 and may be greater than 0, resides in two resources The use of purely cohomological methods and, respectively, the use of Ash and Stevens results on slope decompositions 14 Main results II: the connection with Darmon cycles Suppose that p is inert in K, so that K p is the unramified quadratic extension of Q p Consider the Dp-new quotient V pn Dp-new of V pn When L = H ψ K, σ p induces H ψ K K and the restriction map taes the form res p : Sel st H ψ K, V pndp-new m H 1 stk p, V pn Dp-new m With ψ as above, the conjectures of Beilinson and Bloch, in conjunction with the conjectures of Bloch-Kato see [8, Conjecture 515], predict that dim T[F0 ] SelHψ K, V [F 0 ]m ψ = ord s=0/2+1 LF 0 /K, ψ, s In particular, if LF 0 /K, ψ, s has sign 1, as ensured by Assumption 15 and the conditions ɛ K N = 1 and ɛ K p = 1, then one expects Sel st H ψ K, V pndp-new m to be nonzero In this situation, methods 10

11 have been devised to construct local classes in H 1 stk p, V pn Dp-new m that are conjectured to lie in the image of Sel st H ψ K, V pndp-new m under res p see [12], [18] and [32] Since their construction is based on techniques of Darmon, we follow [32] in calling these elements of H 1 stk p, V pn Dp-new m Darmon classes In 64, we give evidence that the Darmon classes are indeed images of global Galois cohomology classes in Sel st H ψ K, V pndp-new m, up to restricting to the new part, in the case where ψ is a genus character of K Thans to the Atin-Lehner theory, there is no loss of generality in our restriction to the new part see for example [40, 62], where the proof of the Teitelbaum conjecture is reduced its proof for the new part As the proof will show, these global cohomology classes come from an appropriate rational Chow group In some sense, this fact presents a slight strengthening of the conjectures formulated in [32], where only the more abstract Sel st group is involved In particular, when = 0, the above Remar 111 applies and, as noticed after Theorem 611, we really get the rationality result at the level of Mordell-Weil groups This generalizes wor of Bertolini and Darmon [5] under the hypotheses = 0, D = 1, and Qf = Q, and of Seveso [38] in the case D = 1 Results of a similar nature have been obtained simultaneously and independently by Longo and Vigni under the hypotheses = 0 and Qf = Q see [25] 15 Construction of p-adic L-functions Modular symbols are the main tool used in the construction of cyclotomic p-adic L-functions associated to modular forms [27] In unpublished wor, Stevens developed a theory of p-adic families of modular symbols that he applied to the construction of cyclotomic 2-variable p-adic L-functions We describe some aspects of Stevens theory since analogous ideas will be employed below Remar 112 Stevens techniques can be readily adapted to define p-adic families of automorphic forms on definite quaternion algebras These families, together with Chenevier s p-adic Jacquet-Langlands correspondence for definite quaternion algebras, can be used to construct 2-variable anticyclotomic p-adic L-functions This is carried out in [4, 38] Let Y be the Q p -manifold Z p Z p see [34, 9] and let DY be the space of E-locally analytic distributions on Y as in [34, 11] The diagonal action of Z p on Y endows DY with the structure of a DZ p -Fréchet module By the theorem of Amice and Vélu, the convolution algebra DZ p is isomorphic to OX, the coordinate ring of the weight space X If Ω is a subspace of X, DY Ω := OΩ OX DY is a Fréchet OΩ-module We define the space Symb Γ0Np DY Ω := Hom Γ0Np Div 0 P 1 Q, DY Ω to be the space of Ω-families of modular symbols The family of norms defining the Fréchet OΩ-module structure on DY Ω are Γ 0 Np-invariant Since Div 0 P 1 Q is a finitely generated Γ 0 Np-module, Symb Γ0Np DY Ω becomes an OΩ-Fréchet module as well A ey fact is that the operator U p acts completely continuously on this space Therefore, Coleman s theory of slope decompositions applies It follows that there is an open affinoid Ω neighbourhood of in X and a Hece-eigenvector Ψ Symb Γ0Np DY Ω whose weight + 2 specialization φ := Ψ 0 Symb Γ0Np V 0 has the same system of Hece eigenvalues as f The eigenvector Ψ is the cohomological version of the Coleman family F The p-adic families which we need to consider in this paper are Hece eigenvectors Φ in H 1 Γ 0, DY Ω whose weight + 2 specializations ϕ := Φ 0 have the same system of Hece eigenvalues as f Here, Γ 0 is a group of quaternionic units see 2 The technical difficulty which arises in this situation is that H 1 Γ 0, DY Ω seems not to have a natural Fréchet module structure, maing Coleman s theory of slope decompositions inapplicable This difficulty has been resolved by Ash and Stevens in [1] By applying Coleman s theory on the level of cochains with respect to a resolution consisting of finitely generated Γ 0 - modules, they prove that the cohomology groups of Γ 0 possess canonical slope decompositions This is the ey step which allows for the construction of p-adic deformations of cohomology classes Based on the very general machinery developed in [1], these issues are discussed in more detail in [19] With an eigenvector Φ H 1 Γ 0, DY Ω in hand, we construct p-adic L-functions L Φ/K,ψ using a combination of the methods of [4, 5, 38, 41] and group-cohomological techniques of the sort employed in [15, 18] Note, however, that we have subscripted these p-adic L-functions with a Φ instead of with an F In order to justify changing this subscript, we must show that the specializations of Φ in classical weights correspond to those of F under the Jacquet-Langlands correspondence This result, interesting in its own right, is proved 11

12 in [19] With this compatibility in hand, the interpolation property of L F/K,ψ = L Φ/K,ψ see the end of 51 d κ/2 K ν for the definition of ν is deduced in 51 and 52 from Popa s formula relating the central critical values LF, χ, /2 + 1 to integrals of certain real-analytic cycles on Shimura curves Part 1 Real-analytic cycles on Shimura curves and special values of L-functions 2 Quaternions 21 Splittings and orders We recall the following assumption on the real quadratic field K Let D = l : l N, ɛ K l = 1 By Assumption 15 and our running hypothesis that ɛ K N = +1, D has an even number of prime factors Therefore, there is a unique indefinite quaternion Q-algebra B of discriminant D Let : B B be the involution and let nrd : B Q denote the reduced norm: nrdx = xx When B = M 2 Q, we have a b d b a b a b =, nrd = det c d c a c d c d By construction, K is a splitting field of B, ie, there is an isomorphism 15 ι : B Q K M 2 K If D = 1, we choose ι to be the canonical isomorphism ι : M 2 Q Q K M 2 K Let p denote the prime ideal of O K corresponding to the valuation on K induced by the embedding σ p of 2 If ɛ K p = 1, we write p for the other prime ideal of O K above p Write ι p for the composite { M 2 Q p if ɛ K p = +1, B M 2 K M 2 K p = M 2 Q p 2 if ɛ K p = 1 Here, Q p 2 denotes the quadratic, unramified extension of Q p In either case, the image of B is contained in M 2 Q p, and ι p induces an isomorphism For an ideal I of a ring A, let ι p : B Q Q p M 2 Q p { a b R 0 I = c d } M 2 A : c I If m is an ideal of O K, we set R0 D m = {x B : ιx R 0 m} The embedding ι p induces an isomorphism ι p : R D 0 m Z p R 0 p ordpm Z p Set Γ D 0 m = er nrd : R0 D m {±1} Let N + = N/D Since ɛ K l = +1 for all l N +, we may choose an ideal n + of O K of norm N + Of particular interest are the rings R and R 0 They are Eichler orders in B of levels N + and N + p, respectively We will use the following shorthand: 16 R = R D 0 n +, R 0 = R D 0 n + p, Γ = Γ D 0 n +, Γ 0 = Γ D 0 n + p The order R 0 contains a unique bilateral ideal of norm p, and this ideal is principal Let w p R 0 be a generator It is characterized by the properties that 0 17 nrdw p = p, ι p w p mod p

13 22 Embeddings and orientations Let O K be an order in K, of conductor prime to md and the discriminant of K We say that an embedding j : K B is an optimal embedding of O of level M if j 1 R D 0 m = O, where M is a positive generator of m Z, and let EO, R D 0 M be the set of such oriented embeddings A necessary and sufficient condition for such embeddings to exist is that the primes dividing M are split in K together with our running assumption on the primes dividing D We assume that this is the case Note that, replacing m with an ideal m M such that O/m O/ m, the order R D 0 M := R D 0 m = R D 0 m is unchanged, thus justifying our notation for the set of optimal embeddings We say that j EO, R D 0 M is m-oriented if the diagram 18 O j R D 0 m ι R 0 m O/m commutes, where the vertical arrow is the map a b c d a and the diagonal arrow is the natural projection For a prime l D, let B l be the l-adic completion of B, and fix once and for all an identification B l /m Bl = F l 2, where m Bl is the unique } maximal bilateral ideal of the unique maximal order O Bl of B l Note that is the set with two elements Any j EO, R D 0 M induces d l j : O F l 2 Hom Zl -alg O, F l 2 = { δ ± l by means of the identification B l /m Bl = F l 2 If d = δ ε l l l D with ε l {±1} is a choice of homomorphisms one for every l D, we say that j EO, R0 D M is d-oriented if d l j = d l for every l, where d l := δ ε l l is the l-component of d Let E md O, R D 0 M be the subset of such optimal md-oriented embeddings Clearly, R D 0 M acts on E md O, R D 0 M by conjugation on the target Let Cl + O be the narrow ideal class group of O There is a faithful, transitive action of Cl + O on Γ D 0 M\E md O, R0 D M Let b be an ideal of O relatively prime to Dm and the conductor of O Then R0 D mjb is an invertible left ideal of an indefinite quaternion order, and thus is principal, say R0 D mjb = R0 D mb By the norm theorem, R0 D m contains elements of norm 1, so we may assume that nrdb > 0 Since jojbr0 D m jbr0 D m, the image jo is contained in the left order of jbr0 D m But this left order is just br0 D mb 1, so we conclude that jo b 1 R0 D mb Therefore, we may define [b] j : O R 0 m, [b] jx = bjxb 1 One may show that [b] j E md O, R0 D M and that the class of [b] j in Γ D 0 M\E md O, R0 D M depends only on [b] For a prime l MD, let w l R0 D M be an element of norm l; it normalizes Γ D 0 M := Γ D 0 m = Γ D 0 m In particular, it induces by conjugation a well defined action on Γ D 0 M \EO, R0 D M If t MD is a squarefree integer, we may uniquely write m = tt c, where t and t c are coprime, t Z = tz, and D = D t D t,c, where D t and D t,c are coprime and D t = gcd D, t Define w t := l t w l and let W t be the action induced by w t on Γ D 0 M \EO, R0 D M Set W t m := tt c, where t is the conjugate of t by the non-trivial dl automorphism τ of K/Q, and W t d =, d l Dt l l Dt,c, where d l := d l τ Let w R0 D M be an element of reduced norm 1 Again, w normalizes Γ D 0 M and we let T be the corresponding operator on Γ D 0 M-equivalence classes The element τ induces an action by the rule τ j = j := j τ We let W be the group generated by the involutions W t and T If j E md O, R D 0 M, there exists a unique extension j p of j such that the following diagram commutes: 19 O K Z p j p O K j R D 0 m ι p R 0 p ordpm Z p 13

14 When R 0 m = R 0 n + resp R 0 n + p we write E O, R resp E O, R 0 for the set of such embeddings, where may be empty or md In 3 we will need to consider a slightly different type of embeddings Suppose that p is inert and set Õ := O [1/p], R := R [1/p] = R 0 [1/p] and Γ := R 1, where R 1 is the subgroup of norm one elements in R We define the set of optimal embeddings of Õ of level M, that we write EÕ, R, by the requirement j 1 R = Õ, while the subset of optimal n+ d-oriented embeddings is defined in exactly the same way We recall how to define an orientation at p Let T be the Bruhat-Tits tree attached to GL 2 Q p, whose set of vertices we denote by V Set L := Z 2 p and v := [L ] V and write V + resp V for the set of those v V that lie at an even resp odd distance from v We let GL 2 Q p act from the left on V by viewing the elements of Q 2 p as column vectors If j EÕ, R, the group K acts on V by means of j p := j p with a unique fixed point v j V Let E n+ d ± Õ, R be the set of optimal n + d-oriented embeddings such that v j V ± Again there is a faithful, transitive action of Cl + O on Γ\E n+ d ± Õ, R, as well as involutions W t for every squarefree integer t pn + D, an involution T and an action of τ, all defined in the same way as above We define signs W t ± = if p t and W t ± = ± otherwise We record in the following lemma some basic facts about embeddings and orientations, whose proof we leave to the reader Lemma 21 1 The Cl + O-actions commute with the W-actions We have W t [j] = W t [j], T [j] = T [j] and [b] j = [b] j The involution W t induces bijections W t : Γ D 0 M\E md O, R0 D M Γ D 0 M\E Wtmd O, R0 D M, n W t : Γ\E + d ± Õ, R Γ\E d Wtn+ Õ, R, W t± T preserves orientation and τ induces bijections τ : Γ D 0 M\E md O, R0 D M Γ D 0 M\E WMDmd O, R0 D M, n τ : Γ\E + d ± Õ, R Γ\E W MDn + d Õ, R W MD ± 2 For every [j] Γ D 0 M\EO, R0 D M there is a unique σ [j] Cl + O such that W MD [j] = σ [j] [j] and σ [b] [j] = σ [j] [b] 2 In particular the image σ of σ [j] in Cl + O / Cl + O 2 is a well defined element If O = O K, then σ = md in Cl + O / Cl + O 2, where is the class of complex conjugation in Cl + O and d is the unique squarefree ideal of O dividing D Hence, for a genus character ψ attached to the Dirichlet characters χ 1, χ 2, the value ψ σ [j] = ψ md = χi MD does not depend on [j] 3 Suppose that p splits resp is inert in K Then the natural inclusion induces a bijection, commuting with all the actions described above, Γ 0 \E n+ dp O, R 0 Γ\E n+d O, R resp Γ\E n+d O, R E n+ d + Õ, R Suppose p splits in K and that j EO K, R 0 It follows from 18 that { } 0 20 jp α R : ι p α mod p 0 The set on the right is a left R-ideal of norm p, generated by P, say Since R contains elements of norm 1, we may assume that nrd P = p Finally, it s worth noting that the ideal Rjp does not depend on j so long as the image of j is contained in R 0 14

15 Again assume that p splits in K and that j EO K, R 0 Let P j be a generator of R 0 jp as a left R 0 -ideal Unlie Rjp, the ideal R 0 jp does depend on j By reasoning similar to that of the last paragraph, R 0 jp = R 0 P j { α R 0 : ι p α } 0 0 Although the set on the right is a left R 0 -ideal, it is not invertible as its localization at p is not principal Thus, R 0 P j is an invertible subideal of this set Since the orientation of j implies R 0 P j R 0 w p, it follows that there is an integer t j, unique modulo p, such that 21 R 0 pz p j p p 1 tj = R 0 pz p 0 p 23 Rational representations of B 231 The split case Consider the matrix algebra B := M 2 E over a field E and let 22 B 0 = {α B : trdα = 0} Define a left action of B on B 0 by 23 α b = αbα This action induces a map : B GLB 0 GL 3 E, the so-called symmetric-square lift Explicitly, a 2 2ac c 2 a b = ab ad + bc cd c d b 2 2bd d 2 The matrices form a basis of B 0 Thus, X = 0 1, Y = , and Z = 1 0 Sym r B 0 = E[X, Y, Z] r, where E[X, Y, Z] r is the space of degree r homogeneous polynomials in X, Y, and Z E[X, Y, Z] by linear change of variable: A P X, Y, Z = P X, Y, ZA GL 3 E acts on A routine computation shows the actions of B and GL 3 E on Sym r B 0 and E[X, Y, Z] r are compatible with respect to the symmetric-square lift: α P = α P, P Sym r B 0 = E[X, Y, Z] r For a representation V and an integer m, let V m be the representation whose underlying space is V, but with the twisted action defined by the rule α v := Nrα m αv, where Nr is the reduced norm We also write E for the trivial representation with undelying vector space the base field E With these notations we may consider the trace form on B, regarded as a B -representation by 23, as a B -equivariant morphism of representations, : B E B E2, α, β = trdαβ = Trαβ, where Tr denotes the usual trace of a matrix The trace form induces an orthogonal decomposition B = E1 B 0 where E1 is identified with the space generated by the identity matrix and the restriction of this pairing to B 0 is nondegenerate The dual basis of {X, Y, Z} with respect to this pairing is {L X, L Y, L Z } = {Y, X, 1 2 Z} The trace form induces pairings, r on Sym r B 0 defined by 24 α 1, α r, β 1, β r r = σ S r α 1, β σ1 α n, β σn 15

16 Using the evaluations 25 X, X = Y, Y = X, Z = Y, Z = 0, X, Y = 1, and Z, Z = 2 one deduces that X i Y j Z, X i Y j Z r = { 2 i!j!!, if i = j, j = i, and =, 0, otherwise In particular, we see that the pairing, r on Sym r B 0 is nondegenerate Define the hyperbolic Laplacian or Casimir operator i + j + = i + j + = r : E[X, Y, Z] r E[X, Y, Z] r 2 2 by P = X Y 2 Z 2 That is B -equivariant follows from standard properties of Casimir operators, or from a direct calculation The hyperbolic Laplacian admits a coordinate free description, that we will exploit in an essential way in the nonsplit case see 232 Lemma 22 Viewing as a map from Sym r B 0 into Sym r 2 B 0 2, we have 26 P = α i, α j α 1 α i α j α r, where P = α 1 α r Sym r B 0 1 i<j r Proof Since {X, Y, Z} is a basis of B 0, we may assume P has the form P = } X {{ X } Y } {{ Y } Z } {{ Z }, a + b + c = r a b c By 25, the right hand side of 26 equals c ab X, Y X } {{ X } } Y {{ Y } Z } {{ Z } + Z, Z X 2 } {{ X } Y } {{ Y } a 1 b 1 c a b Define Lemma 23 is adjoint to, ie, Z Z }{{} = abx a 1 Y b 1 Z c cc 1X a Y b Z c 2 = : E[X, Y, Z] r 2 2 E[X, Y, Z] r by Q = Z 2 4XY Q P, Q r 2 = P, Q r c 2 X Y 2 Z 2 P Proof Just compute both sides with P = X i Y j Z, and Q = X i Y j Z Set H r = er : E[X, Y, Z] r E[X, Y, Z] r 2 2 We call H r the space of hyperbolic harmonic polynomials Proposition 24 We have the following B -invariant orthogonal decompositions: { Sym r B 0 H r H r 2 2 H 2 r 2 Er, if r is even, = H r H r 2 2 H 3 r 3 H 1 r 1, if r is odd This decomposition is natural with respect to change of base field Proof By Lemma 23, Sym r B 0 = H r Sym r 2 B 0 2 By dimension counting, Sym r 2 B 0 2 = Sym r 2 B 0 2 The desired decomposition now follows by induction The naturality is obvious 16

17 We wish to connect the spaces H r with more standard models of B -representations For a row vector v = x, y E 2, we set v y = x If A B, then 27 va = A v Let P r = P r E = E[x, y] r, equipped with the left action of GL 2 E defined by a b AP x, y = P x, ya = P ax + cy, bx + dy, A = c d It is well-nown that P r = Sym r E 2 is an irreducible representation of B Define P : B 0 P 2 by P A x, y = x y A x y Thus, if A = a b c a then P A x, y = bx 2 2axy cy 2 It is clear from this formula that A P A is a bijection If follows from 27 that A P A is in fact an isomorphism of left B = GL 2 E-modules Define π : Sym r B 0 = Sym r P 2 P 2r by πp 1 P r = P 1 P r This map is B -equivariant Since π is evidently nonzero and has irreducible codomain, it s surjective Proposition 25 The map π restricts to an isomorphism of H r onto P 2r Proof It is easy to see that H r and P 2r both have dimension 2r + 1 Therefore, we need only establish the surjectivity of π Hr By the irreducibility of P 2r, it suffices to show that π Hr is nonzero One verifies directly that X r er = H r On the other hand, P X x, y = x 2, so πx r = 1 r x 2r The nonsplit case We now revert to the notation of 21 Define B 0, the left action of B on B 0, and the trace form on Sym r B 0 as in 22, 23, and 24, respectively Define : Sym r B 0 Sym r 2 B 0 2 as in 26 It is B -equivariant and respects the trace forms on the domain and codomain Let Hr B be the ernel of It is easy to chec that is natural with respect to change of base field The following results follow easily from this fact, together with the results of 231 just choose a splitting field of B Proposition 26 We have the following B -invariant orthogonal decompositions: { Sym r B 0 H B = r Hr 22 B H2 B r 2 Er, if r is even, H r Hr 22 B H3 B r 3 H1 B r 1, if r is odd H B r is an irreducible representation of B Let j : K B be a Q-algebra embedding We obtain an induced map j : Sym r K 0 Sym r B 0, where K 0 is the set of trace-zero elements of K Let δ O K be such that δ 2 = d K and σ δ > 0 Let pr : Sym r B 0 H B r be the orthogonal projection arising from the decomposition of Proposition 26 We mae the following ey definition: 28 Q r j := pr j δ r H B r Since j splits B, it induces an identification H B r Q K = P 2r K In what follows, we can unambiguously identify Q r j with its image in P 2rK Our notation is justified by the facts that, in P 2r K and over any other splitting field, Q r j = Q j r It is easily checed that the polynomial 17

18 Q r j may be characterized, up to sign, by the property of being a generator for the one dimensional subspace of P 2r K on which B acts via nrd r such that Q r j, Qr j = 2r r!d K ; the choice σ δ > 0 fixes the sign This characterization descend to Q 3 Shimura curves 31 Modular forms and Hece operators Let ι to be the composite ι : B ι M 2 K σ M 2 R Via ι, we may view the groups Γ D 0 m as a subgroups of SL 2 R As such, they act on the complex upper half-plane h The quotients X D 0 mc := Γ D 0 m\h are Riemann surfaces, compact if and only if D > 1 As the notation suggests, there are algebraic Shimura curves X D 0 m defined over Q whose loci of complex points are identified with Γ D 0 m\h Let 0 be an integer and let S D +2 m be the space of cusp forms for ΓD 0 m of weight + 2 When D = 1, we will drop the superscript D and write S +2 M, where M = nrdm The space S+2 D m is endowed with an action of a commutative algebra of Hece operators which we now describe Define the semigroup S D 0 m = { α R D 0 m : nrd α 0 and, for all v D, M c v α and a v α, M = 1 } Note that Γ D 0 m is the subgroup of invertible elements of S0 D M with positive reduced norm Let l > 0 be a prime with l D By the Strong Approximation Theorem, we may find an element λ of S0 D M such that nrd λ = l The quotient Γ D 0 m\γ D 0 mλγ D 0 m is finite and we may choose representatives λ i S0 D M such that Γ D 0 mλγ D 0 m = Γ D 0 mλ i i For an element g S+2 D m, define g T where l = f +2 λ i, i g +2 ατ = nrd α +1 cτ + d +2 gι ατ, α B +, a b ι α = c d A standard argument shows that g T l is independent of the choices made above and is an element of S+2 D m Thus, T l is a well-defined endomorphism of S+2 D m We define TS D +2m = C [ {T l : l > 0 prime and l D} ] End C S D +2m The spaces S+2 D m admit rational structures Since XD 0 m admits a canonical model over Q and we assume to be even, we may identify S+2 D m with H 0 X D 0 m Q, Ω 1 X D 0 m/q +2/2 Q C = H 0 X D 0 mc, Ω 1 X D 0 mc +2/2 by associating to g S+2 D m the -fold differential form gτ2πi dτ +2/2 If F is any number field, we define S+2 D m F to be the image of H 0 X0 D m F, Ω 1 X0 Dm/F +2/2 in S+2 D m If D = 1, then the q-expansion principle states that S +2 M F is simply the subspace of S +2 M consisting of forms whose Fourier coefficients belong to F We have: 29 S D +2m F F C = S D +2m The Jacquet-Langlands correspondence identifies the systems of Hece-eigenvalues occuring in S D +2 m with those occuring in spaces of classical cusp forms: 18