Critical p-adic L-functions and applications to CM forms Goa, India. August 16, PDF Free Download

Critical p-adic L-functions and applications to CM forms Goa, India Joël Bellaïche August 16, 2010

Objectives Objectives: 1. To give an analytic construction of the p-adic L-function of a modular form in the missing cases (critical cases) (cf. critical p-adic L-function on my webpage or Arxiv)

Objectives Objectives: 1. To give an analytic construction of the p-adic L-function of a modular form in the missing cases (critical cases) (cf. critical p-adic L-function on my webpage or Arxiv) 2. To compute and study these L-functions, especially in the CM case (work in progress)

A. Previous constructions of p-adic L-functions of modular forms, and their reformulation by Stevens

A. Previous constructions of p-adic L-functions of modular forms, and their reformulation by Stevens We start with a modular form f (z) = a n q n, n=0 q = e 2iπz of weight k + 2 (k 0), level Γ 1 (N), nebentypus ɛ. We assume that f is normalized (a 1 = 1), and an eigenform for the Hecke operators (T l for l N and the diamond operators). Let p be a prime not dividing N. To define a p-adic L-function for f, we need first to refine (or p-stabilize) it. Define α and β the roots of X 2 a p X + p k+1 ɛ(p). One has 0 v p (α), v p (β) k + 1 and v p (α) + v p (β) = k + 1.

There are two forms of level Γ := Γ 1 (N) Γ 0 (p) attached to f that are eigenforms for all Hecke operators (T l for l Np, U p, and the diamond operators): f α (z) = f (z) βf (pz), U p f α = αf α f β (z) = f (z) αf (pz), U p f β = βf β

Theorem (Mazur, Swinnerton-Dyer, Manin, Amice-Vélu, Visik (70 s)) Let f be as above, and assume that a. f is cuspidal.

Theorem (Mazur, Swinnerton-Dyer, Manin, Amice-Vélu, Visik (70 s)) Let f be as above, and assume that a. f is cuspidal. b. v p (α) < k + 1 (non-critical slope)

Theorem (Mazur, Swinnerton-Dyer, Manin, Amice-Vélu, Visik (70 s)) Let f be as above, and assume that a. f is cuspidal. b. v p (α) < k + 1 (non-critical slope) Then there exists a unique analytic p-adic L-function L p (f α, σ), σ W(C p ) = Hom(Z p, C p) such that (INTERPOLATION) For every σ of the form σ(t) = ψ(t)t i, ψ finite image, 0 i k, L p (f α, σ) = e p (f, α, σ) L(f, ψ 1, i + 1). Period (GROWTH) The function L p (f α ) has order at most v p (α), that is L p (f α,.) = O( log vp(α) p ).

I will now explain Stevens beautiful and conceptual proof of this theorem.

I will now explain Stevens beautiful and conceptual proof of this theorem. Classical Modular Symbols:

I will now explain Stevens beautiful and conceptual proof of this theorem. Classical Modular Symbols: Let 0 be the group of divisors of degree 0 on P 1 (Q). GL 2 (Q) acts on it.

I will now explain Stevens beautiful and conceptual proof of this theorem. Classical Modular Symbols: Let 0 be the group of divisors of degree 0 on P 1 (Q). GL 2 (Q) acts on it. Let Γ be a congruence subgroup, and V a Γ-module. We define the abelian group of V -valued modular symbols by Symb Γ (V ) := Hom Γ ( 0, V ). This is a functor in V, obviously left-exact. Suppose that V is acted upon by a subgroup (or submonoid) H of GL 2 (Q) larger than Γ. Then Symb Γ (V ) inherits an action of the Hecke operators [ΓhΓ], h H, since it is a group of Γ-invariants in an H-module.

classical modular symbols Now we begin to feed the modular-symbols machine with interesting Γ-modules V. Let k 0. P k = space of polynomials of degree k or less, with its action of GL 2 (Q). V k = dual of P k. The ( space ) Symb Γ (V k ) has actions of Hecke operators, and if 1 0 normalizes Γ (which we shall assume), an involution ι 0 1 given by this matrix. Theorem (Eichler-Shimura, Manin-Shokurov) Symb Γ (V k ) = S k+2 (Γ) M k+2 (Γ)

classical modular symbols Theorem (Eichler-Shimura, Manin-Shokurov) Symb Γ (V k ) = S k+2 (Γ) S k+2 (Γ) E k+2 (Γ) Symb + Γ (V k) = S k+2 (Γ) E + k+2 (Γ) Symb Γ (V k) = S k+2 (Γ) E k+2 (Γ).

classical modular symbols Theorem (Eichler-Shimura, Manin-Shokurov) Symb Γ (V k ) = S k+2 (Γ) S k+2 (Γ) E k+2 (Γ) Symb + Γ (V k) = S k+2 (Γ) E + k+2 (Γ) Symb Γ (V k) = S k+2 (Γ) E k+2 (Γ). (One idea from the proof: construct S k+2 (Γ) Symb Γ (V k ), f φ f with φ f ({a} {b}) = b a f (z)p(z)dz. The RHS is a linear form of P(z) P k, so is in V k. The modularity of f implies that φ f is a modular symbol. Note: this RHS is sums of special values of L(f, ψ, i + 1) for various finite-order characters ψ and integers 0 i k.)

Stevens Overconvergent Modular Symbols

Stevens Overconvergent Modular Symbols Define A as the Frechet space of analytic functions g(x) = a n x n, a n Q p that converge on a ball of center 0 and some radius > 1 (depending of g).

Stevens Overconvergent Modular Symbols Define A as the Frechet space of analytic functions g(x) = a n x n, a n Q p that converge on a ball of center 0 and some radius > 1 (depending of g). For every k Z, one can define an action of the monoid ( ) a b S 0 (p) = { γ = on A, by setting c d (γ k g)(x) = (a + cx) k g } M 2 (Z p ) GL 2 (Q), p a, p c, ( ) b + dx. a + cx A k = A with this action. D k = dual of A k. For Γ Γ 0 (p) S 0 (p), the space of overconvergent modular symbols is Symb Γ (D k ). This space has an action of the Hecke operators (T l, U p, diamonds, ι). It is very big, but

overconvergent modular symbols Theorem (Stevens (90 s)) For any h 0, is finite dimensional. Symb Γ (D k ) slope h (Idea: compacity of U p ) For k 0, obvious exact sequence of Γ-modules: d k+1 dx 0 P k A k+1 k A 2 k 0. This is even an exact sequence of Σ 0 (p)-module if we twist the action on A 2 k by det k+1.)

overconvergent modular symbols Theorem (Stevens-Pollack) The induced 0 Symb Γ (D 2 k ) Symb Γ (D k ) Symb Γ (V k ) 0. is still exact. This is an Hecke-sequence if we twist the action of T l, U p on Symb Γ (D 2 k ) by (1 + l) k+1, p k+1. Idea of proof: Ash-Stevens, long exact sequence of cohomology, and computation of H 2 c (Γ, D 2 k ), which is 0.

End of Stevens Construction:

End of Stevens Construction: Start with f cuspidal, v p (α) < k + 1. Choose a sign ±.

End of Stevens Construction: Start with f cuspidal, v p (α) < k + 1. Choose a sign ±. Let φ fα Symb ± Γ (V k) be the modular symbol corresponding to f α by Eichler-Shimura.

End of Stevens Construction: Start with f cuspidal, v p (α) < k + 1. Choose a sign ±. Let φ fα Symb ± Γ (V k) be the modular symbol corresponding to f α by Eichler-Shimura. Lift φ fα uniquely to Φ fα Symb ± Γ (D k) by the control theorem (possible since v p (α) < k + 1). Set µ fα = Φ fα ({0} { }) D. Define L ± p (f α, ) as the Mellin transform of the destribition µ fα. One gets this way one-half of the p-adic L-function (the values on characters σ such that σ( 1) = ±1). One uses the other sign for the other half. Proving the interpolation property is a simple computation using the way φ fα is defined. The growth property results easily from Φ fα being an eigenform for U p of slope v p (α).

B. The missing p-adic L-functions

B. The missing p-adic L-functions Recall the map θ : M k (Γ) M k+2 (γ) which is (q d dq )k+1 on q-developements.

B. The missing p-adic L-functions Recall the map θ : M k (Γ) M k+2 (γ) which is (q d dq )k+1 on q-developements. Theorem (Stevens-Pollack (00 s)) If v p (α) = k + 1 but f α is not in the image of θ, one can still lift φ fα to Φ fα uniquely and define the p-adic L-function. Such an f α is called non-critical (a slightly weaker condition than non-critical slope). What happens in the critical case? Notations: if M is a space on which the Hecke operators acts, M[f α ]= eigenspace for the Hecke operators with same eigenvalues as f α. M (fα) = generalized eigenspace for the Hecke operators with same eigenvalues as f α.

Conjecture (Stevens-Pollack) Assume f is cuspidal, new. Then Symb ± Γ (D k)[f α ] has dimension 1.

Conjecture (Stevens-Pollack) Assume f is cuspidal, new. Then Symb ± Γ (D k)[f α ] has dimension 1. Theorem This is true for f cuspidal, and also for f Eisenstein if v p (α) > 0. More precisely, if e is the degree of the eigencurve at f α over W, the algebra H fα generated by the Hecke operators in End(Symb ± Γ (D k) (fα)) is isomorphic to Q p [T ]/T e, and Symb ± Γ (D k) (fα) is free of rank 1 over H fα This result offers a natural definition of the p-adic L-function of f α in all cases, including the critical ones: start with a generator Φ fα of Symb ± Γ (D k)[f α ], take the distribution µ fα := Φ fα ({0} { }), and define L(f α, σ) as the Mellin transform of µ fα.

This definition coincide with the classical one in the cuspidal, non-critical slope case, and with the definition of Stevens-Pollack in the cuspidal, non-critical case.

This definition coincide with the classical one in the cuspidal, non-critical slope case, and with the definition of Stevens-Pollack in the cuspidal, non-critical case. This p-adic L-function has all expected properties: it has order of growth at most k + 1, it satisfies a functional equation, etc. Also one can construct a two-variables p-adic L-function, the first variable being a point of the eigencurve (in a neighborhood of f α ), the second being σ, that interpolates the one-variable p-adic L-functions, critical or not.

We now give a sketch of the proof of this theorem.

We now give a sketch of the proof of this theorem. Construction of the eigencurve through modular symbols (developing unwritten ideas of Stevens):

We now give a sketch of the proof of this theorem. Construction of the eigencurve through modular symbols (developing unwritten ideas of Stevens): Fix W an affinoid subset of W. I construct an adequate module D W of distributions over W, with action of S 0 (p) such that for all h 0 1. Symb Γ (D W ) slope h is a finite projective W -module. 2. If k Z, k W, the fiber at k of that module is Symb Γ (D k ) slope h, except when k = 0, h 1. We can now construct the eigencurve C Symb. We need only construct the open part of the eigencuve that lies over W and on which the slope v p (U p ) h, which we call C h Symb,W. We define it as the rigid spectrum of the affinoid algebra generated by the Hecke operators in End W (Symb Γ (D W ) slope h ).

Using a theorem of Chenevier, one can prove: Theorem There is a canonical isomorphism between the usual eigengurve C and C Symb.

Using a theorem of Chenevier, one can prove: Theorem There is a canonical isomorphism between the usual eigengurve C and C Symb. Recall from Fabrizio s talk that he has constructed with Iovita and Stevens a natural map Symb Γ (D k ) slope h (M k+2 (Γ) S k+2 (Γ))slope h, and that they hope/conjecture that for every k Z it is surjective, or even an isomorphism. I know nothing of this map, but with the ideas above, I can prove: Theorem If k Z, k 0, the source and target of this map have the same semi-simplication as H-modules, and in particular have the same dimension. If k = 0, the semi-simplification of the source is a line plus the semi-simplification of the target. Remark: I have no idea why this should be true without semi-simplification in general (especially when k = 1).

Geometry of the eigencurve:

Geometry of the eigencurve: Theorem (Bellaïche-Chenevier (2003), Chenevier (2004)) Let f be as in theorem. The eigencurve is smooth at f α.

Geometry of the eigencurve: Theorem (Bellaïche-Chenevier (2003), Chenevier (2004)) Let f be as in theorem. The eigencurve is smooth at f α. Ingredients of proof: Galois deformations, Galois cohomology, pseudocharacters.

End of the proof

End of the proof We study the module Symb ± Γ (D W ) as a module over the eigencurve C slope h Symb,W in a neighborhood of f α (we choose h > v p (α))

End of the proof We study the module Symb ± Γ (D W ) as a module over the eigencurve C slope h Symb,W in a neighborhood of f α (we choose h > v p (α)) Lemma This module is free of rank one in a neighborhood of f α. Ingredients : commutative algebra using the smoothness; theory of newforms and new components over the eigencurve. Corollary The fiber over the weight space of the eigencurve at f α is the spectrum of the algebra H f,α. Results from the definitions and a commutation of End with taking the fiber which is not tautological really needs the lemma. Also, needs to deal with the exceptional case when k = 0.

Lemma This module is free of rank one in a neighborhood of f α. Corollary The fiber over the weight space of the eigencurve at f α is the spectrum of the algebra H f,α. Lemma (Riemann) One can find a uniformizer T on the eigencurve at f α, and a uniformiser Y of W at k, such that locally near f α, the eigencurve has equation T e Y

C. Study of critical p-adic L-functions

C. Study of critical p-adic L-functions Note that for a refined form f α to be of critical slope, we need v p (β) = 0 since v p (α) + v p (β) = k + 1, so we need f ordinary at p. Below we shall assume that v p (α) v p (β). Classification of forms of critical slope: Case 1: f Eisenstein. Such a form is always ordinary. Then v p (α) = k + 1, and f α is of critical slope, and actually is critical (that is not in the image of θ). Case 2: f cuspidal CM by a imaginary quadratic field K. The ordinarity of f forces p to be split in K. In this case again, f α is critical.

The case of CM forms Take f CM by K, and assume that p splits in K. Let g M k (Γ) be the companion form of f α : θg = f α. It is an ordinary overconvergent modular form, whose p-adic L-function is defined since the nineties by work of Greenberg-Stevens and Kitagawa. Since g is CM, one can also use Katz construction. To describe our formula, we need to introduce the analytic function log p [k] on the weight space W(C p ) = Hom(Z p, C p). Let us choose a generator γ of 1 + pz p. Set s = σ(γ), so s is in the open ball of center 1 and radius 1 in C p. Set log [k] p (σ) = k i=0 log p(γ j s) (log p γ) k+1

Theorem L p (f α, σ) = log [k] (σ)l p (g, σ/t k ). Corollary L p (f α, σ) vanishes on every special character. It is a function of order exactly k + 1.

The case of an Eisenstein series If f is an Eisenstein series, and v p (α) = k + 1, one expects a somewhat similar formula for L p (f α, ), namely a product of log [k] p and two p-adic Dirichlet L-functions. This was conjectured in a special case back in the nineties by Stevens and Pasol (for a critical p-adic L-function that whose existence was only conjectural). This is much harder that in the CM case, probably because of the poles that may have the p-adic Dirichlet L-functions. What I can prove is just Theorem If f is an Eisenstein series, and v p (α) = k + 1, then L p (f α, σ) is non zero. This is in contrast with the case of Eisenstein series with their ordinary refinements.

Other constructions of p-adic L-function What we constructed was the analytic p-adic L-function of f α. One can also attach to f α an algebraic p-adic L-function:

Other constructions of p-adic L-function What we constructed was the analytic p-adic L-function of f α. One can also attach to f α an algebraic p-adic L-function: We consider the Galois representation ρ f of G Q attached to f, together with its refinement attached to α, that is the line L α in D crys ((ρ f ) GQp ) on which the crystalline Frobenius acts by α. Note: in the case of an Eisenstein f, the natural representation ρ f to consider here, and in many other situations, is not the obvious semi-simple sum of two characters, but the only indecomposable, crystalline at p representation with the same trace.

To (ρ f, L α ) one can attach an algebraic p-adic L-function. Greenberg was the first to define it in a big generality, but this generality is not sufficient here as for our critical f α, (ρ f, L) does not satisfy Panchiskin s condition. Instead we can use either Perrin-Riou s very abstract construction, or Pottharst s more recent (2010) construction using (φ, Γ)-modules. It is not yet known if the two algebraic p-adic L-functions are the same. It is natural to make an Iwasawa main conjecture, stating the equality, up to a function with no zeros and no poles, of the analytic p-adic L-function of f α and the algebraic p-adic L-function of (ρ f, L α ). Theorem Assuming the conjectural computation of the p-adic L-function of an Eisenstein series, the Iwasawa s conjecture (version Pottharst) is true for Eisenstein series. Ingredients of the proof: use the Mazur-Wiles main conjecture for the Dirichlet characters, and account for the extra zeros of log [k] using Colmez comutations on the cohomology of (φ, Γ)-module.

This theorem confirms that Potthart s definition of the algebraic L-function is correct. (or perhaps confirms that my definition of the analytic L-function is, that s a matter of point of view)

This theorem confirms that Potthart s definition of the algebraic L-function is correct. (or perhaps confirms that my definition of the analytic L-function is, that s a matter of point of view) I ignore if the same results is true with Perrin-Riou s definition of the p-adic L-function. I actually even ignore if this definition make sense for a reducible Galois representation like ρ f. There is also a construction of a p-adic L-function of the third type, which is in some sense intermediate between the analytical and algebraic p-adic L-function, which is often easier to compare to both. I mean the construction with an Euler system like Rubin s in the CM case an Kato s in general. This has led to Kato s proof of one-half of the main conjecture in the ordinary case. In the non-critical slope case, one knows that Kato s p-adic L-function is the same as the classical analytic p-adic L-function of Mazur et al. This results would be easily to generalized to any refined form f α if we know that Kato s construction can be put in family, and that Kato s p-adic L-function of f α is non zero. I believe the first point is a work in progress of a student of Colmez. The second point seems harder, especially in the critical case.

Critical p-adic L-functions and applications to CM forms Goa, India. August 16, 2010