TWO-VARIABLE p-adic L-FUNCTIONS

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1 TWO-VARIABE p-adic -FUNCTIONS PAYMAN KASSAEI 1. Introduction This is a write-up of my talk in the Stanford reading group on the work of Bertolini- Darmon. The objective of my talk is to present a construction a two-variable p-adic - function attached to a Hida family, and show that, in some sense, it p-adically interpolates special values of -functions of the specializations of the Hida family. et me be very imprecise in this passage and give a quick introduction to what lies ahead. et f be an ordinary eigenform of weight k 2 living in a Hida family f. To f, one can associate a concrete object called a modular symbol, Ĩf, and one can even divide by an appropriate period, and obtain an algebraically-defined modular symbol, I f. et a, b be elements in P 1 Q. The modular symbol is defined such that to any path from a to b in the completed complex upper half-plane, and any homogenous polynomial of degree k 2 in two variables X, Y, it associates an element in our coefficient field, essentially, by integrating f against P on the path. In particular, given a 0, b, and P X, Y X s 0 1 Y k 1 s 0 for integers 1 s 0 k 1, we obtain an element in our coefficient field denoted I f, k, s 0 1 s 0 1 I f {0 }X s 0 1 Y k 1 s 0. et us call these the special values of the -function of the modular symbol I f. Results of Birch and Manin show that I f, k, s 0 coincides with the classical special value f, s 0 up to a certain factor. Moreover, there is a way to twist I f by a Dirichlet character, so that this procedure produces the special value of the -function of f twisted by the Dirichlet character. It is natural, then, to try to p-adically interpolate the values I f, k, s 0 in both variables k, s. The starting idea is to do so, through p-adically interpolating the modular symbols {I fκ } κ, corresponding to various specializations of f, to a new kind of modular symbol I f associated to f, and applying the same formalism as in The interpolation of the modular symbols is done through replacing homogenous polynomials of degree k 2, which are really functions in one variable, with certain functions in two variable: more precisely, continuous functions on the primitive vectors of Z 2 p. The modular symbols obtained this way are called Λ-adic modular symbols; they specialize in various weights to produce classical modular symbols as above. A Λ-adic modular symbol Φ associates to any a, b P 1 Q, and any continuous function on the primitive part of Z 2 p, an element in C p. In particular, given a 0, b, and the two-variable function 1 fx, y x s 1 y k 1 s 1 Z p Z, we obtain an element p 1 The inclusion of 1Z p Z p is for removing the Euler factor at p. 1

2 2 PAYMAN KASSAEI Φ, k, s 1 s 1 Φ{0 }X s 1 Y k 1 s 1 Z p Z p Thinking of s, k as p-adic variables 2, this is the two-variable p-adic -function attached to Φ. And it is rather formal to prove that, in some sense, it does p-adically interpolate the special values of -functions of the specializations. Given the Birch and Manin s result mentioned above, it is enough to show that a Λ-adic modular symbol can be constructed which p-adically interpolates the modular symbols attached to the specializations of the Hida family f. This is what Greenberg and Stevens do. In the following, I will make the above ideas precise. 2. Notation and background The conventions are, for the most part, in accordance with [BD]. However, the presentation closely follows the article [GS], where the conventions are different from those of [BD]. To keep with the notation of the rest of the talks in the seminar, I will rephrase all the content from [GS] in the conventions of [BD], and hint, at some points, at how the notation relates to that of [BD]. So keep that in mind, when referring to [GS] for more details. et us quickly recall some notations and results from the previous lectures. et p be a prime number, and M an integer prime to p. et N Mp. et k 2 be an integer, and f S k Γ 0 N be a U p -eigenform, U p f a p f, where a p is a p-adic unit. For any field K, et P k K denote the K-vector space of all homogeneous polynomials of degree k 2 in two variables X, Y. It is equipped with a right action of G 2 K, where γ G 2 K acts on P P k K via the formula P γ X, Y P γx, Y T. et V k K be the K-linear dual of P k K. The above action, induces a left action of G 2 K on V k K. For η V k K and P P k K, we have γηp ηp γ. We let P k P k C p, and V k V k C p. In particular, both have left actions of G 2 Q p. A modular symbol with values in an abelian group G is a function φ : P 1 Q P 1 Q G, sending a, b to {a, b}, satisfying φ{a b} + φ{b c} φ{a c}, for all a, b, c P 1 Q. Assume G is equipped with a fixed left action of G 2 Q. If γ G 2 Q, and φ is a modular symbol with values in G, we set γφ{r s} γφ{γ 1 r γ 1 s}. For any subgroup Γ G 2 Q, we let Symb Γ G denote the group of modular symbols which are fixed under the above action of Γ. 2 k, s should be thought of elements in X : Zp Z/p 1Z.

3 TWO-VARIABE p-adic -FUNCTIONS 3 et f be a modular form of weight k 2 and level Γ 0 N. To f, one can associate a modular symbol Ĩf Symb Γ0 NV k C, defined as follows Ĩ f {a b}p 2πi b a fzp z, 1dz. et Ĩ± f denote the plus/minus eigencomponent of Symb Γ 0 NV k C p under the involution action of the matrix c. A result of Shimura states that there are two non-canonical complex periods Ω ± f, such that both I ± f : Ĩ± f /Ω± belong to Symb Γ0 NV k K f, where K f is the number field generated by the coefficients of the q-expansion of f. Our convention is to fix a sign, w ±1, and let I f resp., Ω f denote the corresponding choice from the two I ± f resp., Ω± f. From now on, we will consider I f as an element of Symb Γ0 NV k. There is an action of the Hecke algebra on Symb Γ0 NV k, induced by the Hecke correspondences on the modular curve of level Γ 0 N. For any modular form f, and Hecke operator T, we have I T f T I f. The action of U p on φ Symb Γ0 NV k can be explicitly described as p 1 p a U p φ φ. Definition 2.1. For any φ Symb Γ0 NV k. We define φ φ{0 }. et χ be a Dirichlet character of conductor m. We define the twist of φ by χ to be m 1 m a φ χ χa φ. For any 1 s k 1, define More explicity, we have φ, χ, s 1 s 1 φ, χ, s 1 s 1 φ χx s 1 Y k 1 s. m If χ is trivial, we simply write φ, s for φ, χ, s. χaφ{a/m }mx ay s 1 Y k 1 s. Remark 2.2. Our I f, χ, s is the same as Bertolini-Darmon s f, χ, s up to a factor of 1 s m s 1 See [BD, Prop 1.3]. We recall the following theorem of Birch and Manin which relates the above values obtained from the modular symbol attached to a modular form to the spacial values of the -function of that modular form. See [MTT, 8.6], for example.

4 4 PAYMAN KASSAEI Theorem 2.3. et k 2, and 1 s k 1 be an integer. et χ be of conductor p m, and set w 1 s 1 χ 1. Assume f S k Γ 0 N is an eigenform, and let φ I f be the modular symbol associated to it. Then, we have I f, χ, s ms 1 s 1!τχ 2πi s 1 f, χ, s, Ω f where τχ is the Gauss sum, and Ω f is the complex period introduced above. 3. The p-adic -function of an ordinary modular form This chapter is meant as a motivation for Greenberg-Stevens s construction of twovariable p-adic -functions. The construction applies the formalism which emerges in this chapter to Λ-adic modular symbols attached to Hida families. We recall the construction of p-adic -functions attached to modular forms. We continue to consider p to be a prime number, and M an integer prime to p, and N Mp. et k 2 be an integer, and f S k Γ 0 N be a U p -eigenform, U p f a p f, where a p is a p-adic unit. We recall the definition of the weight space W. It is a rigid analytic space over Q p, whose C p points are given by WC p Hom cont Z p, C p A modular form of weight k 2, level Γ 1 Mp m, with p-nebentypus δ, will be assigned a weight character κ WC p given by κx x k 2 δx. In particular, the weight character of f S k Γ 0 N, will be κx x k 2. et Λ Z p [ 1 + pz p ], and Λ Z p [ Z p ]. Then, we have the following equivalences {bounded analytic functions on W} Λ {p adic measures on Z p } For a measure µ on Z p, let µ denote the corresponding analytic function on W. One way to construct the p-adic -function of a modular form f is to first construct a p-adic measure µ f, and then use the analytic function µf to construct the p f, s. We recall the construction of µ f [MS], [MTT]. Definition 3.1. Fix w ±1. The p-adic measure µ f attached to f is defined via for all a Z. µ f 1 a+p n Z p a n p I f {a/p n }Y k 2, In the rest of this section we sketch an alternate formalism to obtain these p-adic measures, which can naturally be interpolated. We first make some definitions. Definition 3.2. et B 1 be the p-adic open unit disc. For any integer k 0, we define A k {rigid analytic functions defined on a neighborhood of Z p in B 1 }, D k Hom cont A k, C p.

5 TWO-VARIABE p-adic -FUNCTIONS 5 a b For any γ G c d 2 Z p, and f A k, we define γf A k, defined as γfz f az + b cz + d cz + dk. This gives a left action of G 2 Z p on A k, and induces a left action of G 2 Z p on D k, so that for any δ D k, we have γδγf δf. Remark 3.3. Since every step function on Z p inclusion D k MeasZ p. is an element of A k, we obtain a natural There is a G 2 Z p -equivariant inclusion P k A k given by P X, Y fz P z, 1. This induces a G 2 Z p -equivariant map D k V k, and in turn, a map Symb Γ0 ND k Symb Γ0 NV k. Theorem 3.4. et Symb Γ0 N ord denote the U p -ordinary part of Symb Γ0 N. The map Symb Γ0 ND k ord Symb Γ0 NV k ord is an isomorphism. et ϕ f denote the lifting of I f under the above isomorphism. We claim that the p-adic -measure m f can be obtained as a simple evaluation of ϕ f. Proposition 3.5. For f as above, we have Proof. For any a Z, we write µ f ϕ f {0 }. ϕ f {0 }1 a+p n Z p a n p Up n ϕ f {0 }1 a+p n Z p p n 1 a n p n j p ϕ f {0 }1 a+p n Z p j0 p n 1 a n p ϕ f {j/p n }1 a+p n Z p p n z j j0 a n p ϕ f {p n a/p n }1 a+p n Z p a n p I f {a/p n }Y k 2.

6 6 PAYMAN KASSAEI 4. Λ-adic modular symbols and their p-adic -function 4.1. Λ-adic Modular Symbols. As before, p prime, p, M 1, and N pm. et Z 2 p be the subset of primitive vectors in Z 2 p. First, we define spaces of measures. et ContZ 2 p be the space of continuous C p -valued functions on Z 2 p, and StepZ 2 p be the subspace of locally constant functions. We define D Hom Cp StepZ 2 p, C p to be the space of C p -valued measures on Z 2 p. Every such measure has a unique extension to an element of Hom cont ContZ 2 p, C p, where the continuous functions on Z 2 p are considered with supremum norm. If µ D, h ContZ 2 p, and K Z 2 p is a compact open subset, we define h dµ h1 K dµ µh1 K. K As usual, we set µk 1 K dµ. et D be the subspace of D consisting of measures which are supported on Z 2 p. There is a projection D D defined via the restriction of measures from Z 2 p to Z 2 p, making D a direct summand of D. There is a left action of M 2 Z p on D : for γ M 2 Z p, and h ContZ 2 p, we have h dγµ h γ dµ, Z 2 p where h γ v hγv T. The kernel of the projection D D consists of those measures supported on pz p 2, and, hence, is preserved by the above action. Therefore, the action of M 2 Z p descends to D. Explicitly, If γ M 2 Z p, h : Z 2 p C p a continuous function, and µ D, then this action can be described as Z 2 p h dγµ h γ dµ, Z 2 p Z 2 p where h is the extension by zero of h to Z 2 p, and, hence, { hγx, y h T, if γx, y T Z 2 γ x, y p 0 otherwise. In particular, we have the continuous action of Z p on D and D via diagonal matrices. This shows that both D an D are continuous Z p [[Z P ]] C p-modules. We define two spaces of measure valued modular symbols W Symb Γ0 M D, W Symb Γ0 MD. These spaces carry an action of the Hecke algebra. The natural projection D D induces a Hecke-equivariant surjective map W W.

7 TWO-VARIABE p-adic -FUNCTIONS Two-variable p-adic -functions. et Φ W. We define µ Φ Φ{0 } D. Using µ Φ, we can define an analytic function on W W as follows. et κ, σ WC p Hom cont Z p, C p. We define p Φ, κ, σ κyσ x/y dµ Φ x, y. Z p Z p If σ is arithmetic, i.e., σx x r ɛx for some integer r 0, and some Dirichlet character ɛ : Z/p m C p, we extend σ to Z p, by setting σp p r if ɛ is trivial, and σp 0, otherwise. With this definition at hand, we can define the improved two-variable p-adic -function attached to Φ via pφ, κ, σ κyσ x/y dµ Φ x, y, where κ, σ WC p, and σ is arithmetic. Finally, we define a different form of the two-variable p-adic -function attached to Φ which is presented in two variables in WQ p. We set WQ p Hom cont Z p, Z p Z p Z/p 1Z : X, considered additively. If k k 1, k 2 X, and x Z p, we set x k <x> k 1 x/<x> k 2, where <x> denotes the projection of x onto 1 + pz p. Definition 4.3. et Φ W. et k, s X. et κ, σ WQ p be defined as κx x k 2, and σx x s 1. We set p Φ, k, s p Φ, κ, σ. This is an analytic function on X The specialization maps. et κ WC p be an arithmetic weight, κx x k 2 δx, for some integer k 2. We define the specialization map in weight k, and character δ to be ρ κ : D V k via the formula ρ κ µp The map ρ κ induces a Hecke-equivariant map P x, yδy dµx, y. ρ κ : W Symb Γ0 MD Symb Γ0 NV k. For Φ W, we often denote ρ κ µ with Φ κ. If κx x k 2 for an integer k 2, we write Φ k for Φ κ.

8 8 PAYMAN KASSAEI 4.5. The U p operator. The Z p [[Z p]]-module W is equipped with an action of the Hecke algebra. We recall the explicit definition of U p. Recall that γ G 2 Q acts on Φ W as follows: γφ{r s} γφ{γ 1 r γ 1 s}. We define p 1 U p Φ p a Φ. Using the explicit description of the action of M 2 Z p on D 4.1.1, we have p fx, ydµ UpΦx, y fpx ay, ydµ a x, y, Z 2 p where µ a Φ{a/p }. Z p Z P 4.6. The interpolation property. We can now prove the p-adic interpolation property of p Φ,,. Recall the notions of φ, s and φ, χ, s for a classical modular symbol φ, given in Definition 2.1. Proposition 4.7. et Φ W satisfy U p Φ a p Φ with a p 0. et s, k Z X satisfy 1 s k 1. et κx x k 2. Then p Φ, k, s 1 a p k 1 p s 1 Φ k, s. This follows immediately from the following result. Proposition 4.8. et Φ W. et κ, σ WC p such that σ is arithmetic: σx x s 1 ɛx for an integer s 1, and some Dirichlet character ɛ of conductor p m. Recall that σp p s 1 if ɛ is trivial, and σp 0 otherwise. We have 1 p U p Φ, κ, σ pu p Φ, κ, σ σp pφ, κ, σ. 2 Assume, further, that κ is arithmetic: κx x k 2 δx, for an integer k 2, and some Dirichlet character δ. Then, pu m p Φ, κ, σ Φ κ, ɛ, s. Proof. To prove the first statement, using Equation 4.5.1, we can write p U p Φ, κ, σ κyσ x/y1 Z Z 2 p p Z dµ U p pφ Similarly, we write pu p Φ, κ, σ p 1 κyσ κyσ x/y1 Zp Z dµ U Z 2 p p pφ κyσ p 1 px + ay 1 y Z p Z dµ ax, y. p px + ay 1 y Zp Z dµ ax, y. p

9 TWO-VARIABE p-adic -FUNCTIONS 9 It is easy to check that all the corresponding terms are equal, except for a 0. For a 0, the term in p U p Φ, κ, σ is zero. It follows that pφ, κ, σ p Φ, κ, σ σp κyσ px y dµ 0x, y σp p Φ, κ, σ, κyσ x y dµ Φx, y as µ 0 Φ{0 } µ Φ. For the second statement, setting µ a,m Φ{a/p m }, we can write pup m Φ, κ, σ κyσ x/y1 Zp Z dµ U Z 2 p p p mφ p m 1 p m 1 p m 1 ɛa κyσ pm x + ay dµ a,m x, y y κyɛ pm x + ay y pm x + ay s 1 dµ a,m x, y y κy pm x + ay s 1 dµ a,m x, y. y On the other hand, we have Φ κ, ɛ, s 1 s 1 Φ κ ɛ{0 }X s 1 Y k 1 s pm 1 1 s 1 p ɛa m a Φ κ {0 }X s 1 Y k 1 s pm 1 1 s 1 pm 1 1 s 1 ɛaφ κ {a/p m }p m X ay s 1 Y k 1 s ɛa δyp m x ay s 1 y k 1 s dµ a,m x, y. To finish the proof, we just need to compare the two sides.

10 10 PAYMAN KASSAEI 5. The two-variable p-adic -function of a Hida family 5.1. The Λ-adic modular symbol attached to an ordinary eigenform. et f S k0 Γ 1 N be an ordinary eigenform. et f be a Hida family f q a n q n, n1 where a n s are analytic functions on some open k 0 U W satisfying 1 The weight k specialization f k q n1 a nkq n is an eigenform of weight k and level Γ 0 N, for all k U Z 2. 2 f k0 q fq. We will use f to construct a Λ-adic modular symbol which specializes to a nonzero multiple of I fk at weight k U Z 2. et R be the local ring of k 0 U. Then R is a Z p [[Z p ]] C p -algebra, and we can form D R, : D Zp[[Z p ]] C p R. Set W R : Symb Γ0 MD R,. An element Φ W R will be of the form Φ 1 λ Φ r λ r, where, for all i, Φ i W, and λ i is defined on an open set around k 0. In particular, there is an open U U around k 0 on which all the λ i s are defined. Therefore, for every κ U, one can define the specialization ρ κ Φ. Theorem 5.2. There is Φ f W R defined over U, such that 1 ρ k0 Φ f I f ; 2 For all k U Z 2, there is a scalar λk C p such that ρ k Φ f λki fk. Proof. See [GS, 6] for a complete proof. Here, we will only give a brief sketch. et Γ r Γ 0 M Γ 1 p r. First, we claim that H 1 parγ 0 M, D lim r H 1 parγ r, Z p. For any r 1, let D r denote the space of Z p -valued functions on Z/p r Z 2, the set of primitive vectors of Z/p r Z 2. The Z p -module D r has a natural action of Γ 0 M. Since Γ r \Γ 0 N Z/p r Z 2 via the map a b c, d mod p c d r, it follows that D r Γ Ind 0 M Γ r Z p. By Shapiro s lemma, for all r 1, we have H 1 parγ 0 M, D r H 1 parγ r, Z P. The Z p [Γ 0 M]-modules {D r : r 1} form a projective system via the maps D r+1 D r, sending µ r+1 µ r, where µ r x y p r x µ r+1y. It follows that and, hence, we have Hecke equivariant maps D lim r D r, HparΓ 1 0 M, D lim H 1 r parγ 0 M, D r lim H 1 r parγ r, Z p

11 TWO-VARIABE p-adic -FUNCTIONS 11 where, one can check that the transition maps on the right side are corestrictions. By Hida s results, the Hecke eigensystem of f appears in the ordinary part of lim H 1 r parγ r, Z p, and, hence, in HparΓ 1 0 M, D ord. On the other hand, we have a Hecke equivariant isomorphism Symb Γ0 MD H 1 c Γ 0 M, D. Consider the composite surjective map Symb Γ0 MD ord H 1 c Γ 0 M, D ord H 1 parγ 0 M, D ord where, we have taken ordinary parts, and tensored over Λ with the fraction field of Λ. The Kernel of this map is Eisenstein: for any prime l p, such that l N 1, the operator η l : T l l <l > 1 kills the kernel of the above map. On the other hand, using Weil bounds for cusp forms, we see that η l acts invertibly on HparΓ 1 0 M, D ord. We claim that any Hecke eigenclass in HparΓ 1 0 M, D ord lifts to a Hecke eigenclass in Symb Γ0 MD ord. et v HparΓ 1 0 M, D ord. et w Symb Γ 0 MD ord be any element in the preimage of η 1 l v. Then w : η l w maps to v, and it is independent of the choice of w, since η l annihilates the kernel of the above map. The commutativity of the Hecke algebra implies that w has the same Hecke eigensystem as v. et Φ f be an element of Symb Γ0 MD ord with the same Hecke system as f. Then, for any integer k 2, at which Φ f is defined, ρ k Φ f has the same Hecke eigensystem as f k, and, hence, as I fk. Since the space of modular symbols of the same Hecke eigensystem as I fk is one-dimensional, it follows that ρ k Φ f λki fk, for some λk C p. By shrinking the domain of regularity of Φ f, and after normalizing, we can ensure the properties required by Theorem 5.2 are satisfied The two-variable p-adic -function of f. et χ be a Dirichlet character of conductor m. We define the twist of a Λ-adic modular symbol Φ by χ to be Φ χ m 1 m a χa Φ. emma 5.4. et Φ W satisfy U p Φ a p Φ. et χ be a Dirichlet character of conductor m prime to p. Then, U p Φ χ a p χp 1 Φ χ.

12 12 PAYMAN KASSAEI Proof. Since m, p 1, we can write p 1 p bm U p Φ χ b0 m 1 m 1 p 1 χa b0 p 1 χa b0 m 1 χp 1 m 1 p bm m ap χap a p χp 1 Φ χ. m a χa m ap m a p b p 1 b0 Φ Φ Φ p b Definition 5.5. et f S k0 Γ 0 N be an ordinary eigenform, f a Hida family through f as in the previous section, and Φ f W R a modular symbol as in Theorem 5.2. et χ be a Dirichlet character with conductor m prime to p. We define the two-variable p-adic -function associated to f to be p f, χ, k, s : p Φ f χ, k, s, where k U, s X, and the right side is given in Definition 4.3. Remark 5.6. The definition of p f, χ, k, s here differs from that of [BD] by a factor of 1 s m s 1. We finally prove the interpolation property of the two-variable p-adic -function of f. Theorem 5.7. et k U Z 2, and s Z satisfy 1 s k 1. Set w 1 s 1 χ 1. Then, p f, χ, k, s λk1 χpa p k 1 p s 1 I fk, χ, s. Proof. This follows immediately from Proposition 4.7, Theorem 5.2, and emma 5.4. Remark 5.8. In light of Remarks 2.2 and 5.6, Theorem 5.7 is the same as Theorem 1.12 of [BD]. References [BD] Bertolini, M.; Darmon, H.: Hida families and rational points on elliptic curves. Invent. Math. 168, 2007, no. 2, pp [GS] Greenberg, R.; Stevens, G.: p-adic -functions and p-adic periods of modular forms, Invent. Math. 111, 1993, no. 2, pp [MS] Mazur, B.; Swinnerton-Dyer, P.: Arithmetic of Weil curves, Invent. Math. 25, 1974, pp [MTT] Mazur, B., Tate, J., Teitelbaum, J.: On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84, 1986, pp Φ