Triple product p-adic L-functions for balanced weights and arithmetic properties

Transcription

1 Triple product p-adic L-functions for balanced weights and arithmetic properties Marco A. Seveso, joint with Massimo Bertolini, Matthew Greenberg and Rodolfo Venerucci 2013 Workshop on Iwasawa theory and p-adic family of automorphic forms 1 / 43

2 Our first project project: defining p-adic L-functions i = 1, 2, 3 p N i tame levels Ω i X affinoid disks We re given an N i -new Coleman family f i = a n (f i )q n O(Ω i )[[q]]. Thus, if k i Ω i,cl, then n=1 f i,ki = a n (f i, k i )q n S ki +2(Γ 0 (N i p)) N i -new n=1 is a normalized eigenform. 2 / 43

3 f i,ki is p-new for at most one k i Ω i,cl. If f i,ki is p-new, set f i,k i = f i,ki. When f i,ki is p-old, let f i,k i S ki +2(Γ 0 (N i )) be the unique normalized newform such that Explicitly, f i,k i T l = a l (f i, k i )f i,k i for all l N i p. f i,ki (q) = f i,k i (q) pk i +1 a p (f i, k i ) f i,k i (q p ), L p (f i,k i, s) 1 = ( 1 a p (f i, k i )q s) ( 1 pk i +1 a p (f i, k i ) q s ). 3 / 43

4 There are triple product L-functions ( ) L f 1,k 2 f 2,k 2 f 3,k 3, s. If we include nebetype and we require ε 1 ε 2 ε 3 = 1 k 1 + k 2 + k 3 0 mod2, they satisfy a functional equation with ε-factors and central critical point c k := k 1 + k 2 + k We will exclude nebentype for ease of exposition and, consequently, we will focus on triples of even weight. 4 / 43

5 Let M i be the motive attached to f # k i. Then, at least for v N 1 N 2 N 3, ( ) L v f 1,k 2 f 2,k 2 f 3,k 3, s = L v (M 1 M 2 M 3, s). Deligne s critical regions for the motive M 1 M 2 M 3. Let Σ 1 := { (k 1, k 2, k 3 ) Z 3 0 : k 1 k 2 + k } (hence k 1 k 2, k 3 ), Σ 2 := { (k 1, k 2, k 3 ) Z 3 0 : k 2 k 1 + k } (hence k 2 k 1, k 3 ), Σ 3 := { (k 1, k 2, k 3 ) Z 3 0 : k 3 k 1 + k } (hence k 3 k 1, k 2 ), Σ 123 := Z 3 0 (Σ 1 Σ 2 Σ 3 ). 5 / 43

6 The critical regions are the followings k Σ1 j [k 2 + k 3 + 3, k 1 + 1] k Σ2 j [k 1 + k 3 + 3, k 2 + 1] k Σ3 j [k 1 + k 2 + 3, k 3 + 1] k Σ123 j [1, k 1 + k 2 + k 3 + 3] where j is an integer. In particular, when k 1 + k 2 + k 3 is even, ( k ), c is critical. k Then Σ 123, called the balanced region, will be our region of interpolation, i.e. we will interpolate those ( k ), c such that k Σ k 123 and k 1 + k 2 + k 3 is even. 6 / 43

7 Shorthand: f k = f 1,k 2 f 2,k 2 f 3,k 3, Ω = Ω 1 Ω 2 Ω 3 A triple (k 1, k 2, k 3 ) Σ 123 of classical weights is balanced if k 1 + k 2 + k 3 is even. Explicitely k 1 + k 2 + k 3 is even and k 1 k 2 + k 3, k 2 k 1 + k 3, and k 3 k 1 + k 2. Project goal: Construct a p-adic L-function such that L p (f 1 f 2 f 3 ) O(Ω 1 Ω 2 Ω 3 ) L p (f 1 f 2 f 3, k) L for all balanced triples k Ω cl. ( f k, k ) alg 1 + k 2 + k / 43

8 This talk Theorem: (Greenberg-S) Suppose N 1 = N 2 = N 3 = N with N squarefree. Then there is a function L p satisfying (a precise version of) the above interpolation property for balanced triples of even weights. The assumption N 1 = N 2 = N 3 = N is for ease of exposition. We can deal with odd weights, although there are some issues beyond simply admitting forms with compatible nebentype. Cyclotomic variable? 8 / 43

9 ε-factors Since we assume N is squarefree, 1 if v N, ε v (f k, 1) = 2 1 if v = and kis balanced, +1 otherwise If ε(f k, 1 ) = 1, then the interpolation problem is trivial. 2 If ε(f k, 1 2 ) = +1 and k is unbalanced, then the corresponding L p was constructed by Harris & Tilouine (arxiv 1996, Math. Ann. 2001) and Darmon-Rotger. If ε(f k, 1 2 ) = +1 and k is balanced, then ω(n) is odd. 9 / 43

10 The quaternion algebra B Assume ε(f k, 1 2 ) = +1 for balanced k Ω cl. B = quaternion Q-algebra ramification set {v N } π ki : automorpic representation of GL 2 (A) with new vector f i,k i Theorem: (Prasad 1990) B is characterized by: 1 each π ki admits a Jacquet-Langlands lift π B k i to B (A), 2 Hom B v (πk B 1,v πb k 2,v πb k 3,v, C) 0 for all v. Theorem: (Harris & Kudla 1991) With B as above, Hom B (A)(π B k 1 π B k 2 π B k 3, C) 0 L(f k, k 1+k 2 +k ) / 43

11 Trilinear forms and special values Theorem: (Harris & Kudla 1991, Gross & Kudla 1992, Böcherer & Schulze-Pillot 1996, Watson 2002, Ichino 2008) With B as above, there exist local factors C v 0, v N, a quantity T (f k ) Q(f k ) such that L(f k, k 1+k 2 +k ) = f k, f k v N C v T (f k ). 11 / 43

12 Let ϕ i be a Jacquet-Langlands lift of the Coleman family f i to B (more soon) and set ϕ = ϕ 1 ϕ 2 ϕ 3. We produce an analytic function k t 3 (ϕ k ) on Ω such that ( ) t 3 (ϕ k ) 2 2 ϕ k, ϕ k W p = E 2 p, a 1 p k 3 p (f 3,k3 ) T (f k a p (f 1,k1 )a p (f 2,k2 ) k ), for k Ω cl, where k 3 = k 1 + k 2 k 3 2 Symmetrically, we also get functions k t 1 (ϕ k ) and k t 2 (ϕ k ). The t i are trilinear forms on spaces of quaternionic Coleman families, and are the subject of the remainder of the talk. 12 / 43

13 Quaternionic modular forms G = GL 2 (Q p ), K = GL 2 (Z p ), K 0 = { A ( ) } (mod p) K R B maximal order, R 0 R suborder of level p N, splitting B p = B Q p M 2 (Q p ) such that R 0,p R p B p K 0 K G is commutative. 13 / 43

14 ( ) Z Semigroup: Σ = G p Z p pz p Z p Let V be a right Q p [Σ]-module. V -valued quaternionic modular forms of level J = Kor K 0 : { M(V, J) = f : B } /B V : f (kx)k p = f (x)for all k J. M(V, K 0 ) admits an action of T l, l Np, and of U p. Jacquet-Langlands correspondence: There is a Hecke-equivariant, Q p -linear isomorphism S k+2 (Γ 0 (Np), Q p ) N-new M(V k, K 0 )/Eis, where V k is the irred. rep. of G of dimension k / 43

15 Trilinear forms Theorem: (Clebsch & Gordan) ) dim Qp Hom G (V k1 V k2 V k3, Q p (k 1 + k 2 + k 3 ) { 1 if = kis balanced, 0 otherwise. 15 / 43

16 Explicitly: P k = P x,y k = Q p [x, y] k, V k = Hom Qp (P k, Q p ) If k is balanced, then v k = x k 1 y 1 3 k x 1 y 1 2 x 2 y 2 x 2 y 2 x 3 y 3 x 3 y k 1 P k ( k 1 k 2 k 3 ) G, 3 where ki = k i + k i + k i, P k = P x 1,y 1 k 2 1 P x 2,y 2 k 2 P x 3,y 3 k 3. When it is nonzero, ) Hom G (V k1 V k2 V k3, Q p (k 1 + k 2 + k 3 ) = Q p t k, where ˆ t k (µ) = v k dµ. 16 / 43

17 Trilinear forms on coefficients induce trilinear forms on modular forms: M(V k1 ) M(V k2 ) M(V k3 ) M(V k1 V k2 V k3 ) Abusing notation slightly, t k M(Qp (k 1 + k 2 + k 3 )),Eis Q p t k : M(V k1 ) M(V k2 ) M(V k3 ) Q p for the composition of the above maps. If ψ i M(V ki, K)[f i,k i ] is nonzero and ψ = ψ 1 ψ 2 ψ 3, then T (f k ) = t k (ψ) 2 ψ, ψ 17 / 43

18 p-adic families of trilinear forms The p-adic L-function arises from deforming t k over the weight space. Let k i : Z p O i = O(Ω i ) be associated to Ω i X. Let X = Z p Z p and O = O 1 O 2 O 3. We want (but won t quite get) a diagram M(D k1 (X )) M(D k2 (X )) M(D k3 (X )) t k O k k M(V k1 ) M(V k2 ) M(V k3 ) t k Q p k k coming from a trilinear form D k1 (X ) D k2 (X ) D k3 (X ) t k O. 18 / 43

19 Canonical isomorphism: D k1 (X ) D k2 (X ) D k3 (X ) D k1 k 2 k 3 (X 3 ) where, for t Z p, t k 1 k 2 k 3 := t k 1 t k 2 t k 3 O 1 O 2 O 3 = O. First crack at defining t k : ˆ t k (µ) = X 3 x k 1 y 1 3 k x 1 y 1 2 x 2 y 2 x 2 y 2 x 3 y 3 x 3 y 3 k 1 dµ, where k i = k i k i k i. 2 Problem: These determinants need not belong to Z p. 19 / 43

20 Fix: If Y = pz p Z p = X ( ) 0 1, then p 0 x k 1 y 1 2 x 2 y 2 x 3 y 3 x 3 y 3 k 1 makes sense on X 2 Y, while if Z = {( (x 1, y 1 ), (x 2, y 2 ) ) X 2 : x 1 y 2 x 2 y 1 pz p }, then makes sense on X 2 Z. x 1 y 1 x 2 y 2 k 3 20 / 43

21 Therefore, we may define t k : D((X 2 Z) Y ) O by ˆ t k (µ) = (X 2 Z) Y x k 1 y 1 3 k x 1 y 1 2 x 2 y 2 x 2 y 2 x 3 y 3 x 3 y 3 k 1 dµ Dualizing the extension-by-zero map on analytic functions, we get D k1 k 2 k 3 (X 2 Y ) D k1 k 2 k 3 ((X 2 Z) Y ), so we may view t k as a map D k1 k 2 k 3 (X 2 Y ) O. We get an induced map on quaternionic modular forms: t k : M(D k1 (X )) M(D k2 (X )) M(D k3 (Y )) O. 21 / 43

22 The p-adic L-function Let ϕ i M(D ki (X ))[f i ] be the Jacquet-Langlands lift of the Coleman family f i. We (finally) define by L p (f 1 f 2 f 3 ) = L p (f 1 f 2 f 3 ) O(Ω 1 Ω 2 Ω 3 ) t k (ϕ 1 ϕ 2 ϕ 3 W p ) 2 ϕ 1, ϕ 1 W p ϕ 2, ϕ 2 W p ϕ 3, ϕ 3 W p. 22 / 43

23 Interpolation New problem: t k no longer specializes to t k the trilinear form that evaluates special values for k Ω cl. However, if we let t k : M(D k1 (X )) M(D k2 (X )) M(D k3 (Y )) Q p be the weight- k specialization of t k, then we have: Key Proposition: Let ψ i M(D ki (X )), i = 1, 2, 3, be such that ψ i U p = a i ψ i. Then t k (ψ 1 ψ 2 ψ 3 W p ) = ( 1 p k 3 a 3 a 1 a 2 ) t k (ψ 1 ψ 2 ψ 3 ). 23 / 43

24 Elements of the proof: 1. Analyze the difference between (ψ 1 ψ 2 ) U p and ψ 1 U p ψ 2 U p : (ψ 1 ψ 2 ) U p = a 1 a 2 (ψ 1 ψ 2 (ψ 1 ψ 2 ) X 2 Z 2. Consider t k as a pairing,, : M(D k1 k 2 (X 2 )) M(D k3 (Y )) Q p, with respect to which p k 3 U ι p is right adjoint to U p : (ψ 1 ψ 2 ) U p, ψ 3 W p = ψ 1 ψ 2, ψ 3 W p p k 3 U ι p Theorem: If k Ω cl, then t k (ϕ k1 ϕ k2 ϕ k3 W p ) 2 ϕ k, ϕ k W p ), = p k 3 ψ1 ψ 2, ψ 3 U p W p = p k 3 a3 ψ 1 ψ 2, ψ 3 W p ( ) 2 = E 2 p, a 1 p k 3 p (f 3,k3 ) T (f k a p (f 1,k1 )a p (f 2,k2 ) k ). 24 / 43

25 Arithmetic aspects of p-adic L-functions (in progress with Bertolini and Venerucci) Suppose that the Coleman families f i are ordinary and let G be the Galois group of the maximal algebraic extension of Q which is unramified outside Np. We may define a representation V of G with coefficients in O such that, if k : O K with k Ωcl, then K k V V k where V k is the self dual Q p -adic representation of the tensor product of the Deligne s representation attached to the triple f # 1,k 1 f # 2,k 2 f # 3,k / 43

26 The ordinariness implies that we may define a natural filtration by O [G p ]-submodules specializing to a filtration 0 V + V V 0 0 V + k V k V k / 43

27 Following Nekovar we have extended Selmer groups H f 1(Q, V k ), attached to this filtration, sitting in a short exact sequence ( ) 0 H 0 Q p, V H 1 k f (Q, V k ) Sel ( ) Q, V k 0. By means of Poitou-Tate duality and the given deformation V of V k it is possible to define canonical weight pairings, wt k : H 1 f (Q, V k ) H 1 f (Q, V k ) I k /I 2 k, where I k O is the ideal of those functions vanishing at k. The autoduality of the representation V implies that this pairing is alternating. 27 / 43

28 There are three square-root triple product p-adic L-functions. One is L (3) p (f 1 f 2 f 3 ) 1/2 = t k (ϕ 1 ϕ 2 ϕ 3 W p ). and the others are obtained in a similar way. We have, for balanced triples k, L (3) p (f 1 f 2 f 3 ) 1/2 ( k ) = E (3) (f 1 f 2 f 3 )( k )L (3) (f 1 f 2 f 3 ) 1/2 ( k ) where E (3) (f 1 f 2 f 3 ) ( k ) = (1 p k 3 a p (f 3,k3 ) a p (f 1,k1 )a p (f 2,k2 ) ), L (3) (f 1 f 2 f 3 ) 1/2 ( k ) = t k (ψ 1 ψ 2 ψ 3 ). 28 / 43

29 Fix a balanced triple k and define the improving plane { (3) k } k H := : k k 3 = k3. Then E (3) (f 1 f 2 f 3 ) ( k ) is an analytic expression when restricted to H (3) and indeed we may write k L (3) p (f 1 f 2 f 3 ) 1/2 = E (3) (f 1 f 2 f 3 )L (3) (f 1 f 2 f 3 ) 1/2 on H (3) k where all the expressions are analytic. 29 / 43

30 We say that k is exceptional if there are periods, i.e. if ( ) H 0 Q p, V 0. k Then we have: k is exceptional E (i) (f 1 f 2 f 3 ) ( k ) = 0 for some i = 1, 2, 3. Up to a permutation we will assume from now on that E (3) (f 1 f 2 f 3 ) ( k ) = 0. Let O O (3) be the natural map corresponding to the inclusion H (3) k X / 43

31 We write I (3) O (3) to denote the ideal of those functions vanishing k at k, so that E (3) (f 1 f 2 f 3 ) I (3) k. Theorem: (Bertolini-S-Venerucci) Suppose that k = (0, 0, 0) is exceptional. ( With the) above notations there exist non-zero periods q 1, q 2 H 0 Q p, V such that k q 1, q 2 wt (3) k H = E (3) (f 1 f 2 f 3 ) in I (3) /I (3)2, k k k where ( ) (3) H : I k /I 2 I (3) /I (3)2 is induced by O O (3). k k k k 31 / 43

32 We say that k is p-new if f # i,k i = f i,ki is p-new for i = 1, 2, 3. The ε factor at p of the complex L function of f k is then given by ε p ( k ) = a p (f 1,k1 )a p (f 2,k2 )a p (f 3,k3 )p k 1 +k 2 +k 3 2. Further aims of the project are the following. Suppose that k is a balanced triple and that ε p ( k ) = 1. Then L (i) (f 1 f 2 f 3 ) 1/2 ( k ) = 0 for i = 1, 2, 3. We expect a relation between a cohomology class k Sel ( ) Q, V k and derivatives of the p-adic L-functions in the same spirit as for the Euler factors. 32 / 43

33 Suppose that k Σ i with i = 1, 2, 3 and that ε p ( k ) = 1. Then the p-adic L-functions need not to vanish and we expect a relation of their value at k with a cohomology class k Sel ( ) Q, V k in the spirit of a theorem of Darmon-Rotger for forms on GL 2 (A). Natural candidates for these cohomology classes arise from analogues of the diagonal cycles appearing in the work of Darmon-Rotger. 33 / 43

34 More motivation: p-adic regulator and Venerucci s p-adic BSD conjecture Let L p (f, k, k/2 + 1) be the restriction to the critical line of the Mazur-Kitagawa p-adic L-function attached to a Coleman family f, that we assume to be ordinary. We focus on the specialization to k = k = 0, suppose that the generic sign ε gen is +1 (hence L p (f, k, k/2 + 1) should not vanish identically) and let A be the elliptic curve (for simplicity) attached to the specialization V k of the selfdual representation V attached to f. 34 / 43

35 Theorem: (Bertolini-Darmon+Venerucci) Suppose that p N E, sign (A, Q) = 1 and that there is an exceptional zero. We have L p (f, k, k/2 + 1) Ik 2 and Q Q A (Q) s.t. L p (f, k, k/2+1) BD = t log A (Q) 2 = V t ( q A, Q wt ) 2 0 in I 2 k /Ik 3. We have that Q 0 if and only if L (E/Q, 1) 0 and, in this case, t Q and P generates Q A (Q). Here, wt k : H 1 f (Q, V k ) H 1 f (Q, V k ) I k /I 2 k is a weight pairing, which is alternating similarly as above. 35 / 43

36 This result has an interpretation in the framework of Venerucci s p-adic BSD conjecture. Let us define r gen := rankã (Q) e gen where e gen = 0 if ε gen = +1 and e gen = 1 if ε gen = 1. Let L gen p (f, k, k/2+1) := L p (f, k, s) (expected 0). (s k/2 + 1) egen s=k/2 36 / 43

37 Define the weight regulator R A,p as follows. Let à (Q) be the kernel of Z p, wt k. Take a basis {P i} of à (Q) Z p à (Q) and complete it to a basis {P i, Q j } of Zp Ã(Q) (this latter being free). ( à (Q) Define R MTT A,p := det P i, P j MTT k (Rmk:, MTT k and R wt A,p := det ( Q i, Q j wt ) k and set ) R A,p := normalization R MTT A,p R wt A,p which does not depend on the choice of the basis. is symmetric) 37 / 43

38 Let now A have good ordinary or multiplicative reduction at p. Conjecture: (Venerucci, weak version) We have L p (f, k, k/2 + 1) I rgen and L gen p (f, k, k/2+1) = R A,p in I rgen /I rgen+1 up to (precisely defined) local Euler type factors and Shafarevich-Tate/Tamagawa factors. 38 / 43

39 In the above setting we have e gen = 0, hence L p = L gen p. Since we need a basis to compute the regulator, suppose L (E/Q, 1) 0. Then r gen := rankã (Q) = 2 and, wt k is expected to be non degenerate. Thanks to the fact that the weight pairing is alternating, L gen p (f, k, k/2+1) = t ( ( q, Q wt ) 2 qa, q k = t det A wt k q A, Q wt k = R wt expect A,p = R A,p in Ik 2 /Ik 3 q A, Q wt k Q, Q wt k Going back to our setting, assuming that k is exceptional, we have: { ( ) dim Qp H 0 Q p, V 2 if k is not p-new case 1 = k 3 if k is p-new case 2 ) 39 / 43

40 Naively applying the above terminology, we are in case e gen = 0 (we should define e gen = 1 if we interpolate in a central critical region including a cyclotomic variable, but the restriction of the resulting function to the weight space is the zero function: we can not define L gen p, since we do not have a cyclotomic variable) and we assume that k belongs to the balanced region (relative to V k ). We will always assume to be in the lower rank setting as much as it is possible, both analytic and arithmetic. Case 1 Then e gen = 0 and, since at least one f ki does not ) have multiplicative reduction (suppose this happens for i = 3), ε (f k # = +1. Hence we have r gen := rankã k (Q) := rank H 1 f (Q, V k ) = 2 ( and à (Q) is purely generated by two periods q 1, q 2 H 0 Q p, V ). k 40 / 43

41 Recalling that E (3) (f 1 f 2 f 3 ) comes from a square root p-adic L-function and that we assume ( k ) t := L (3) (f 1 f 2 f 3 ) 0 we see that, since the weight pairing is alternating, L (3) p (f 1 f 2 f 3 ) = 2t 2 E (3) (f 1 f 2 f 3 ) 2 ( ) 2 = 2t 2 q 1, q 2 wt (3) k H = 2t 2 det q 1, q 1 wt k q 2, q 1 wt (3) k H k (3) k H k q 1, q 2 wt (3) k H k q 2, q 2 wt (3) k H k = 2t 2 R wt = (3) k H,p 2t2 R (this line q (3) k H,p 1, q 2 wt (3) k H 0), k k k the equalities in I (3)2 k /I (3)3 k. 41 / 43

42 Note that q 1, q 2 wt (3) k H = 0 if and only if, wt (3) k H = 0, thus k k justifying R wt = (3) k H,p R and, by a direct computation of k H (3),p k k the Euler factors, this happens if and only if L an 1 = L an 2 = L an 3, where L an i are, formally, the analytic L-invariants of f ki. In the spirit of the above conjecture we shoud expect that L (3) p (f 1 f 2 f 3 ) = I (3)3 k may happen only if rank H 1 f (Q, V k ) 2 forces r gen > 2, a case we where excluding. Then we should have to define a MTT p-adic cyclotomic height pairing, that we can not see working only in the weight space. One can show that, in this case, f k3 is the unique form having multiplicative reduction and then recover the equality L alg 3 = L an / 43

43 Case 2 ( k is p-new) Then r gen := rankã (Q) := rank H 1 f (Q, V k ) = ) 3 if ε (f k # ) (f k # 4 if ε = +1, = 1. In the first case, R wt (3) = 0 (being the determinant of a k H,p k skew-symmetric matrix on an odd rank space) and we should have again to define a MTT p-adic cyclotomic height pairing. Second case: more soon! 43 / 43