p-adic L functions and trivial zeroes

Section 1. Notations

Fix an algebraic closure of ℚ, G _ℚ = Gal(/ℚ). In the following, M will designe Sym ²(h ₁(E)). The p-adic realization of M is V = M _p = Sym ²(V _p(E)) with V _p(E) = ℚ_p ⊗ lim←n E _pⁿ. It’s a p-adic representation of G _ℚ of dimension 3.

Let D _p(V) be the filtered ϕ-module associated to V by Fontaine’s theory. If D _dR(M) = Sym ²(H _dR ¹(E))[−2], there exists a natural isomorphism D _p(V) = ℚ_p ⊗ D _dR(M). We describe the action of ϕ and the filtration explicitly. Let (e ₀, e ₋₁, e ₋₂) be a basis such that ϕe ₋₁ = p ⁻¹ e ₋₁, ϕe ₀ = α⁻² e ₀, ϕe ₋₂ = β⁻² e ₋₂.

In the ordinary case, we can choose α to be in ℤ_p ^×; the filtration is given by where ω_e = e ₋₂ + e ₋₁ + e₀ for some λ ∈ ℚ_pthat we assume nonzero.

In the supersingular case (and a _p = 0, which is automatic if p > 3), V is a direct sum (as a G _ℚp-representation): V = W ₁ ⊕ W ₂ with and The filtration is given by with ω_e = e ₋₁ − e ₀ [for some suitable choice of (e ₀, e ₋₁, e ₋₂)].

In both cases, D _p(V)^ϕ=p−1 = ℚ_p e ₋₁. In supersingular case, take λ = −1/2.

Section 2. Greenberg Invariants

2.1. Ordinary Case.

On V, there exists a filtration of p-adic representations of G _ℚp = Gal(/ℚ_p): such that So there is a natural surjection Fil_p ¹ V → ℚ_p(1). We choose e ₋₁ such that the map sends e ₋₁ to 1.

It’s easy to see that H ¹(ℚ_p, Fil_p ¹ V) ≅ H ¹(ℚ_p, V)=H _g ¹(ℚ_p, V) (we use the notation H _f ¹, H _g ¹ of Bloch–Kato). Recall that there is an isomorphism The first one is just Kummer theory where (ℚ_p ^×)_p = lim←n ℚ_p ^×/ℚ_p ^×pn, the second one is given by q ↦ (log_p q, ord_p q) where log_p is the logarithm on ℚ_p ^× such that log_p p = 0. So there is a map Definition. If x ∈ H ¹ (ℚ_p, V), let it depends only on the line ℚ_p x.

Definition. If x ∈ H¹(ℚ, V) is a universal norm in the ℤ_p ^×-cyclotomic extension, define The universal norms are contained in H _f,{p} ¹(ℚ, V) [elements of H ¹(ℚ, V) which are unramified outside of p]. Thanks to Flach (7) and under technical conditions, (i) the universal norms are of dimension 1; (ii) H _f ¹(ℚ, V) = 0 and dim H _f,{p} ¹(ℚ, V) = 1. So in the above definition, ℓ_p(M) = ℓ(x) for any nonzero element x of H _f,{p} ¹(ℚ, V).

Section 3. p-adic L Functions

Let G _∞ = Gal(ℚ(μ_p∞)/ℚ) ≅ ℤ_p ^× and ℤ_p[[G _∞]] the continuous group algebra of G _∞. Define some algebras: Here ℋ(G _∞) is the algebra of elements in ℚ_p[[G _∞]] which are O(log^r) for a suitable r: it means that f ∈ ℋ(G _∞) can be written f = Σ_n a _n(γ − 1)ⁿ with sup_n>0 |a _n|/n ^r < ∞ (γ is a topological generator of the p-part of G _∞); 𝒦(G _∞) is the total fraction ring of ℋ(G _∞). If η is a continuous character from G _∞ with values ℚ̄_p ^×, we can evaluate η on any element of ℋ(G _∞).

Conjecture (10): For any n ∈ ∧² D _p(V), there exists an element L _{p} ^p(n) ∈ ℋ(G _∞) such that for any nontrivial even character η of G_∞ of conductor p^a where (i) e is a basis of the ℚ-vector space det D_dR = ℚ(−3)_dR , and ω_ℚ is a basis of Fil⁰ D_dR; (ii) Ω _∞,ωℚ e = ω_ℚ ∧ n _B ⁺ ∈ ℂ ⊗ det D_dR with n_B ⁺ a basis of det M_B ⁺[for example of det Sym ²(H₁(E, ℤ))⁺]; (iii) Ω _p,ωℚ (n) = ω_ℚ ∧ n; and (iv) G(η) is a Gauss sum associated to η.

So η(L _{p} ^p(n))ω_ℚ ∧ n_B ⁺ = ½ G(η)² L _{p}(M, η, 0)ω_ℚ ∧ (pϕ)^−a(n). We may see L _{p} ^p(M) = L _{p} ^p as an element of Hom _ℚp(ℚ_p ⊗ ∧² D _dR(M), ℋ(G _∞)) and as a function of s ∈ ℤ_p: if χ is the cyclotomic character, with n ∈ ∧² D _p(V). For any f ∈ ℋ(G _∞), define ∂(f) = 〈 χ 〉^s (f))|_s=0.

Section 4. Logarithm

Let K _n = ℚ_p(μ_pⁿ⁺¹) and Z _∞ ¹(ℚ_p, T)= lim←n H ¹(K_n, T) with T = Sym ²(T _p(E)). It’s a ℤ_p[[G _∞]]-module of rank 3. Note π₀ the projection on H ¹(ℚ_p, T). One can construct a map (9) Recall only some properties of ℒ (the first one depends on a “reciprocity law” conjecture that seems to be proved now). If x ∈ Z _∞ ¹(ℚ_p, T) (11): If π₀(x) ∈ H _e ¹(ℚ_p, T),

Section 5. Special Systems and p-adic L Functions

There should exist a special element c _p ^spec ∈ ℚ_p ⊗ lim←n H ¹(ℚ(μ_pⁿ⁺¹), T) such that the p-adic L function should be defined by the formula for any n ∈ ∧² D _p(V).

Define c _p ^flach(p) = π₀ (c _p ^spec) ∈ H ¹(ℚ, V).

Conjecture: where π_[−1] is the projection on D _p(V)^ϕ=p−1 with respect to the other eigenspaces of ϕ.

In the ordinary case, it means that or

Section 6. Some Theorems

We assume the existence of c _p ^spec and the fact that the p-adic L function can be calculated by the formula L _{p} ^p(n)e = ℒ(c _p ^spec) ∧ n.

Theorem: The function L _{p} ^p is nonzero at the trivial character 1 if and only if c _p ^flach(p)∉ H_f ¹(ℚ, T) and one has In particular, by using Flach’s theorem (7), L _{p} ^p is nonzero if and only if c _p ^flach(p) is nonzero.

Assume c _p ^flach(p) ≠ 0. Let L _{p} ^p,sc = L _{p} ^p(e ₋₁ ∧ e ₋₂) ∈ ℋ (G _∞).

Theorem: The function L _{p} ^p,sc has a zero at 1 which is simple if and only if ℓ_p(M) ≠ 0 and one has Theorem: The following formulas are equivalent where Ω _p,ωℚ ^sc ∈ ℚ_p is defined by Ω _p,ωQ ^sc e = ω_ℚ ∧ e ₋₁ ∧ e ₋₂.

In the ordinary case, L _{p} ^p,sc should be the p-adic function already known, the last formula is then the formula conjectured by Greenberg.

Section 7. Even More Speculations

c _p ^flach(p) should come from a motivic element: so it would exist in any of the l-adic realizations of M; call it c _l ^flach(p) ∈ H ¹(ℚ, M _l), this element should again have good reduction outside of p. For l ≠ p, let D _l(M) = M _l ^I_p; there is a map and for l = p, We have A candidate of such an element has been constructed by Flach. On the other hand, there exists a natural ℚ-vector space 𝒟 such that It can be described in terms of the Néron–Severi group of the reduction E × E at p (8). We would like to compare λ_g ^l(c _l ^flach(p)) for different l and give a link with the p-adic L function (work in preparation). For l ≠ p, see calculations of Flach (7).