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

Transcription

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

2 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)

3 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)

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

5 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).

6 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.

7 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

8 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 To refine f is to choose one of those forms. The forms f α and f β are called refinements of f.

9 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 To refine f is to choose one of those forms. The forms f α and f β are called refinements of f.

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

11 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)

12 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

13 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 ).

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

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

16 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.

17 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.

18 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.

19 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. Theorem (Ash-Stevens (80 s)) For every Γ-module V, Symb Γ (V ) H 1 c (Γ, V ) functorially in V and as Hecke-modules.

20 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. Theorem (Ash-Stevens (80 s)) For every Γ-module V, Symb Γ (V ) H 1 c (Γ, V ) functorially in V and as Hecke-modules. Idea of proof: long exact sequence of cohomology for X (Γ), Y (Γ); observe that X (Γ) Y (Γ) = P 1 (Q)/Γ

21 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.

22 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.

23 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)

24 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) = S k+2 (Γ) S k+2 (Γ) E k+2 (Γ)

25 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) = 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).

26 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).

27 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.)

28 Stevens Overconvergent Modular Symbols

29 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).

30 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

31 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 A k = A with this action. 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

32 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 A k = A with this action. D k = dual of A k. } M 2 (Z p ) GL 2 (Q), p a, p c, ( ) b + dx. a + cx

33 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

34 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: 0 P k A k d k+1 dx k+1 A 2 k 0.

35 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.)

36 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.) Dual exact sequence: 0 D 2 k D k V k 0.

37 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.

38 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.

39 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. Corollary (Stevens control theorem (90 s)) The induced map is an isomorphism. Symb Γ (D k ) slope<k+1 Symb Γ (V k ) slope<k+1

40 End of Stevens Construction:

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

42 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.

43 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).

44 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.

45 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.

46 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.

47 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).

48 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). Remark: One can do this with f Eisenstein as well (still with v p (α) < k + 1, so v p (α) = 0), at least for one choice of ±. Get a non-zero µ fα, but the p-adic L-function is 0.

49 B. The missing p-adic L-functions

50 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.

51 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.

52 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?

53 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.

54 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.

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

56 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

57 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.

58 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.

59 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.

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

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

62 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

63 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.

64 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.

65 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 ).

66 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 ). Since this algebra is an algebra over W, and is torsion-free, C h Symb,W has a flat map toward W. To construct the full eigencurve C Symb, glue those open parts according to the standard procedure, axiomatized by Buzzard.

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

68 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.

69 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).

70 Geometry of the eigencurve:

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

72 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.

73 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. Remark: this is false if k = 1, i.e. f of weight 1. (work in progress with Mladen Dimitrov).

74 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. Remark: this is false if k = 1, i.e. f of weight 1. (work in progress with Mladen Dimitrov).

75 End of the proof

76 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))

77 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.

78 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.

79 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 (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

80 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

81 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 Theorem H f,α is isomorphic to Q p [T ]/T e and Symb ± Γ (D k) (fα) is free of rank 1 on it.

82 C. Study of critical p-adic L-functions

83 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).

84 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:

85 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).

86 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.

87 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. Case 3: f cuspidal non CM, p-ordinary. Then f α is of critical slope, and is conjectured to be non-critical.

88 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.

89 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).

90 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

91 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 It is easy to see that log p [k] is independent of the choice of γ. This function vanishes at order 1 on every character of the form σ(t) = ψ(t)t i, ψ of finite image, 0 i k.

92 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.

93 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.

94 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.

95 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.

96 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.

97 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. Recently a student of Stevens has made an important progress toward this conjecture. It is possible that using this progress, Stevens and Pollack will be able to prove the conjecture soon.

98 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:

99 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.

100 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.

101 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.

102 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 ).

103 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.

104 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)

105 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.

106 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.

107 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.