p-adic L functions and trivial zeroes

Abstract

The following is adapted from the notes for the lecture. It announces results and conjectures about values of the p-adic L function of the symmetric square of an elliptic curve.

First let us give some examples of trivial zeroes. Let K/ℚ be an imaginary quadratic field such that p splits in K, η the associated quadratic Dirichlet character; the Euler factor of L(η, s) at p is 1 − p^−s. Choose an ideal 𝒫 above p and a compatible embedding of an algebraic closure of ℚ in an algebraic closure _p of ℚ_p. There exists a Kubota–Leopoldt p-adic L function L_p(η, s) such that for n > 0 and even,

Theorem [Ferrero–Greenberg (1)].

with ℓ_p(η) = and q = π/π̄, 𝒫^h = (π).

Let E/ℚ be a modular elliptic curve with split multiplicative reduction at p. Mazur et al. (2) have constructed a p-adic L function L_p(E, s).

Theorem [Greenberg–Stevens (3)].

with ℓ_p(E) = and q_E the Tate parameter of E/ℚ_p.

It has been recently proved that ℓ_p(E) is nonzero: Barré-Sirieix et al. (12) proved that if E/ is a Tate curve at p, and if j_E is algebraic, then q_E is transcendental.

Finally, let E be a modular elliptic curve over ℚ and 1 − a_pp^s + p^1−2s the Euler factor at p of its L function. Let M = Sym²(h₁(E)) = Sym²(h¹(E)) (2). The Tate twist of M is M* (1) = sl(h₁(E)) = sl(h¹(E)). The Euler factor at p of M is

where α + β = a_p, αβ = p. The Euler factor at p of M*(1) is

When E has ordinary reduction, a p-adic L function has been constructed by interpolation of values of twists of L(M, s) at s = 0 (4). The complex L function L(M, s) is nonzero at s = 0 because 0 is inside the convergence domain of the Euler product.

Under a mild technical hypothesis, the following theorem has been proved:

Theorem [Greenberg–Tilouine (5)]. Assume E has multiplicative reduction at p. Then,

where Ω_∞ is some explicit complex period and ℓ_p(M) = ℓ_p(E).

So L_p(M, s) has a simple zero [recall ℓ_p(E) is nonzero].

In general, a trivial zero should appear when 1 or p⁻¹annihilates the p-Euler factor. It means that the p-adic L function should have a zero of multiplicity strictly bigger than the one of the complex L function.

The following work has been done by Greenberg (6) (in the ordinary situation). (i) He gives a definition of some ℓ_p(M) in a very general case. In particular, for M = Sym²(h₁(E)) with E having (good) ordinary reduction. (ii) He gives a conjecture for the behavior of the p-adic L function at the trivial zero (multiplicity order of the zero and behavior of the dominant coefficient of the expansion at this zero). (iii) He checks that one recovers theorems already proved.

In this talk, we look only at the case of the symmetric square of an elliptic curve with good reduction at p, we explain in this special case: (i) the construction of the Greenberg invariant in the ordinary case, (ii) a construction of a similar invariant in the supersingular case; (iii) the conjectural definition of the p-adic L function; (iv) a conjectural link between the p-adic L function and a conjectural special system, and (v) consequences on the p-adic L function and the trivial zero.

Section 1. Notations

Fix an algebraic closure of ℚ, G_ℚ = Gal(/ℚ). In the following, M will designe Sym²(h₁(E)). The p-adic realization of M is V = M_p = Sym²(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) = Sym²(H_dR¹(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 (e₀, e₋₁, e₋₂) be a basis such that ϕe₋₁ = p⁻¹e₋₁, ϕe₀ = α⁻²e₀, ϕe₋₂ = β⁻²e₋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 = W₁ ⊕ W₂ with

and

The filtration is given by

with ω_e = e₋₁ − e₀ [for some suitable choice of (e₀, e₋₁, e₋₂)].

In both cases, D_p(V)^ϕ=p−1 = ℚ_pe₋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_p¹V → ℚ_p(1). We choose e₋₁ such that the map

sends e₋₁ to 1.

It’s easy to see that H¹(ℚ_p, Fil_p¹V) ≅ H¹(ℚ_p, V)=H_g¹(ℚ_p, V) (we use the notation H_f¹, H_g¹ 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_pq, ord_pq) where log_p is the logarithm on ℚ_p^× such that log_pp = 0. So there is a map

Definition. If x ∈ H¹ (ℚ_p, V), let

it depends only on the line ℚ_px.

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 H¹(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_f¹(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).

2.2. Supersingular Case.

The canonical map

is an isomorphism. On the other hand, by Bloch–Kato, there is a natural map

Once having chosen log_p on ℚ_p^×(log_p p = 0), there is a canonical splitting of the inclusion

and so we obtain an extension of the Boch–Kato logarithm log_W1 to H_g¹(ℚ_p, W₁):

Definition.

If x = (x₁, x₂) ∈ H_g¹(ℚ_p, V) = H_g¹(ℚ_p, W₁) ⊕ H_f¹(ℚ_p, W₂), define ℓ(x) ∈ ℚ_p ∪ ∞ by

Definition. Define ℓ_p(M) = ℓ(x) with x a universal norm in H¹(ℚ, V) [again, we can just take a nonzero element in 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_na_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 ∈ ∧²D_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_∞,ωℚe = ω_ℚ ∧ n_B⁺ ∈ ℂ ⊗ det D_dR with n_B⁺ a basis of det M_B⁺[for example of det Sym²(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)²L_{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 ∈ ∧²D_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 H¹(K_n, T) with T = Sym²(T_p(E)). It’s a ℤ_p[[G_∞]]-module of rank 3. Note π₀ the projection on H¹(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_e¹(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 H¹(ℚ(μ_pⁿ⁺¹), T) such that the p-adic L function should be defined by the formula

for any n ∈ ∧²D_p(V).

Define c_p^flach(p) = π₀ (c_p^spec) ∈ H¹(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_f¹(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) ∈ H¹(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).