Computations on the Tate-Shafarevich group using Iwasawa theory

cw

3rd December 2006

Abstract

A premature version of something like an article on shark.

1 Introduction

[?]

2 The Rank and the Tate-Shafarevich group

Let E be an elliptic curve defined over ℚ. Let

y^{2} + a_{1} x y + a_{3} y = x^{3} + a_{2} x^{2} + a_{4} x + a_{6}  (1)

be its global minimal Weierstrass equation. We will assume throughout the article that E does not admit complex multiplication. The theorem of Mordell-Weil asserts that the set of rational points E(ℚ) is an abelian group of finite rank r = rank(E(ℚ)). A way of giving a lower bound on the rank r consists of searching for linearly independent points of small height. Other methods include the method of Heegner points and its p-adic versions.

Our aim here is to give an upper bound on the rank r. Apart from the case of small analytic rank when the method of Heegner points can be applied, the only way of obtaining an upper bound so far is by doing a descent. The 2-descent implemented by J. Cremona [?] is particularly powerful. But it may fail to determine the rank completely due to the presence of 2-torsion elements in the so-called Tate-Shafarevich group.

The Tate-Shafarevich group, denoted by � (E/ℚ), is a torsion abelian group associated to E/ℚ. It is defined to be the kernel of the localisation map

0 → � (E/ℚ) → H^{1}(ℚ, E) → ∏_{υ} H^{1}(ℚ_{υ}, E)

where the product runs over all places υ in ℚ. The arithmetic importance of this group lies in its geometric interpretation. There is a bijection from � (E/ℚ) to the ℚ-isomorphism classes of principal homogeneous spaces C/ℚ of E which have points everywhere locally. In particular, C is a curve of genus 1 defined over ℚ whose Jacobian is isomorphic to E. Non-trivial elements in � (E/ℚ) correspond to curves C which defy the Hasse principle.

Conjecture 1. (Shafarevich and Tate) The Tate-Shafarevich group � (E/ℚ) is finite.

The finiteness of � (E/ℚ) is only known for curves of small analytic rank in which case the computation of Heegner points can give an explicit upper bound of its order. In general, the only known methods so far for computing an upper bound on the p-primary part � (E/ℚ)(p) of the Tate-Shafarevich group consists of doing a p-descent [?].

Our algorithm will give another method of computing an upper bound on the order of � (E/ℚ)(p), at least for most primes. For instance, we will have to exclude p = 2. The algorithm requires the previous determination of the full Mordell-Weil group E(ℚ).

These two invariants, the rank r and the Tate-Shafarevich group � (E/ℚ) are encoded in the Selmer group. By E(p), we denote the Gal(ℚ̄/ℚ)-module of all torsion points of E whose orders are powers of p. The Selmer group is defined to be the following kernel

0 → 𝒮_{p}(E/ℚ) → H^{1}(ℚ, E(p)) → ∏_{υ} H^{1}(ℚ_{υ}, E) .

It follows from the Kummer sequence that there is a short exact sequence

0 → E(ℚ) ⊗^{ℚ_{p}}/_{ℤ_{p}} → 𝒮_{p}(E/ℚ) → � (E/ℚ)(p) → 0 .

If the Tate-Shafarevich group is finite, then the corank of 𝒮_{p}(E/ℚ) is equal to the rank r of E(ℚ).

3 The Birch and Swinnerton-Dyer conjecture

A natural algorithm for computing an upper bound on the rank r and the size of the Tate-Shafarevich group would result from the resolution of the following conjecture. Let L(E, s) be the Hasse-Weil L-function associated to the ℚ-isogeny class of E. It is now known that this is a holomorphic function on the whole complex plane. Let ω_{E} be the invariant differential \frac{dx}{2y+a_{1}x+a_{3}} of the minimal Weierstrass equation (1) of E. We write Ω_{E} = ∫_{E(ℝ)} ω_{E} ∈ ℝ_{>0} for the Néron period of E.

Conjecture 2. (Birch and Swinnerton-Dyer)

• The order of vanishing of the Hasse-Weil function L(E, s) at s = 1 is equal to the rank r = rank(E(ℚ)).

• The leading term L^{∗}(E, 1) of the Taylor expansion of L(E, s) at s = 1 satisfies

\frac{L^{∗}(E, 1)}{Ω_{E}} = \frac{∏_{υ} c_{υ} · #� (E/ℚ)}{(#E(ℚ)_{tors})^{2}} · Reg(E/ℚ)  (2)

where the Tamagawa numbers are denoted by c_{υ} and Reg(E/ℚ) stands for the regulator associated to the canonical height of Néron and Tate.

The proof of this conjecture would only give a way of computing an upper bound on the rank since there is no known way of proving that higher order derivatives of L(E, s) at s = 1 vanish.

Note that the conjecture (2) is known to be invariant under isogenies defined over ℚ by a result of Cassels [?].

4 The p-adic L-function

In order to be able to formulate a p-adic analogue of the conjecture of Birch and Swinnerton-Dyer, one needs first a p-adic version of the analytic function L(E, s). Mazur and Swinnerton-Dyer [?] have found such a function. We refer to [?] for details on the construction and the historic references.

Let π: X_{0}(N) → E be the modular parametrisation of E and let c_{π} be the Manin constant, i.e. the integer satisfying c_{π} · π^{∗}ω_{E} = 2πif(τ)dτ with f being the newform associated to the isogeny class of E. It is conjectured that c_{π} = 1.

Given a rational number r, we may regard the image π_{∗}({r}) in H_{1}(E, ℝ) of the path joining r to i ∞ in the upper half plane. Define

λ^{+}(r) = \frac{c_{π}}{2} · (∫_{π_{∗}(\{r\})} ω_{E} + ∫_{π_{∗}(\{−r\})} ω_{E}) = πi · (∫_{r}^{i∞} f(τ) dτ + ∫_{−r}^{i∞} f(τ) dτ)

There is a basis {γ_{+}, γ_{−}} of H_{1}(E, ℤ) such that ∫_{γ_{+}} ω_{E} is equal to Ω_{E} if E(ℝ) is connected and to \frac{1}{2} Ω_{E} otherwise. By a theorem of Manin [?], we know that λ^{+}(r) belongs to ℚ · Ω_{E}. We define the modular symbol [r]^{+} ∈ ℚ to be

[r]^{+} · Ω_{E} = λ^{+}(r)

for all r ∈ ℚ. In particular we have [0]^{+} = L(E, 1) · Ω−1_{E}.

Let p be a prime of semi-stable reduction. We write^{1} a_{p} for the trace of Frobenius. Suppose first that E has good reduction at p. Then N_{p} = p + 1 − a_{p} is the number of points on Ẽ(𝔽_{p}). Let X^{2} − a_{p} · X + p be the characteristic polynomial of Frobenius and let α ∈ ℚ̄_{p} be a root of this polynomial such that ord_{p}(α) < 1. There are two different possible choices if E has supersingular reduction and there is a single possibility for primes where E has good ordinary reduction. Now if E has multiplicative reduction at p, then a_{p} is 1 if it is split multiplicative and a_{p} is −1 if it is non-split multiplicative reduction. In either multiplicative case, we have to take α = a_{p}.

Define a measure on ℤ_{p}^{×} with values in ℚ(α) by

μ_{α}(a + p^{k}ℤ_{p}) = \frac{1}{α^{k}} · [ \frac{a}{p^{k}} ]^{+} − \frac{1}{α^{k+1}} · [ \frac{a}{p^{k−1}} ]^{+}

for any k ⩾ 1 and a ∈ ℤ_{p}^{×}. Given a continuous character χ on ℤ_{p}^{×} with values in the completion ℂ_{p} of the algebraic closure of ℚ_{p}, we may integrate χ against μ_{α}. Any invertible element x of ℤ_{p} can be written as ω(x) · ⟨x⟩ where ω(x) is a root of unity and ⟨x⟩ belongs to 1 + 2pℤ_{p}. We define the analytic p-adic L-function by

L_{α}(E, s) = ∫_{ℤ_{p}^{×}}⟨x⟩^{s−1} dμ_{α}(x) for all s ∈ ℤ_{p}.

where by ⟨x⟩^{s−1} we mean exp_{p}((s − 1) · log_{p}(x)). The function L_{α}(E, s) extends to a locally analytic function in s on the disc defined by |s − 1| < 1, see § 13 in [?].

Let _{∞}G be the Galois group of the cyclotomic extension obtained by adjoining all p-power roots of unities. By κ we denote the cyclotomic character _{∞}G → ℤ_{p}^{×}. Choosing a topological generator γ in Γ = _{∞}G^{4(p−1)} amounts to pick an element κ(γ) in 1+2pℤ_{p}^{×}. With this choice, we may convert the function L_{α}(E, s) into a p-adic power series in T = κ(γ)^{s−1} − 1. We write ℒ_{α}(E, T) for this series in ℚ_{p}(α)[[T]]. We have

ℒ_{α}(E, T) = ∫_{ℤ_{p}^{×}}(1 + T)^{\frac{log(x)}{log(κ(γ))}} dμ_{α}(x) .

As in [?], we define the polynomial

P_{n} = ∑_{a∈(ℤ/p^{k}ℤ)^{×}}[ \frac{a}{p^{k}} ]^{+} · (1 + T)^{\frac{log(a)}{log(κ(γ))}}

= ∑_{j=0}^{p^{k−1}−1} ∑_{b=1}^{p−1} [ \frac{ω(b) · κ(γ)^{j}}{p^{k}} ]^{+} · (1 + T)^{j}

where we changed the summation by putting a = ω(b) · κ(γ)^{j}. Then the approximation as a Riemann sum of the above integral for ℒ_{α}(E, T) can be written as

ℒ_{α}(E, T) = lim_{k→∞} ( \frac{1}{α^{k}} · P_{k} − \frac{1}{α^{k+1}} · P_{k−1}) .

4.1 The p-adic multiplier

For a prime of good reduction, we define the p-adic multiplier by

ϵ_{p} = (1 − \frac{1}{α} )^{2} .  (3)

But when the reduction is multiplicative, we put

ϵ_{p} = (1 − \frac{1}{α} ) = \begin{cases}0 & if p is split multiplicative and \\ 2 & if p is non-split. \end{cases}

1The context should make it clear if we speak about a_{p} or a_{2} and a_{3} as in (1).

4.2 Interpolation property

The p-adic L-function constructed above satisfies a desired interpolation property with respect to the complex L-function. For instance, we have that

ℒ_{α}(E, 0) = L_{α}(E, 1) = ∫_{ℤ_{p}^{×}} dμ_{α} = ϵ_{p} · \frac{L(E, 1)}{Ω_{E}} .

A similar formula holds when integrating non-trivial characters of ℤ_{p}^{×} against μ_{α}. If χ is the character on _{∞}G sending γ to a root of unity ζ of exact order p^{n}, then

ℒ_{α}(E, ζ) = \frac{1}{α^{n+1}} · \frac{p^{n+1}}{G(χ^{−1})} · \frac{L(E, χ^{−1}, 1)}{Ω_{E}} .

Here G(χ^{−1}) is the Gauss sum and L(E, χ^{−1}, 1) is the Hasse-Weil L-function twisted by χ^{−1}.

4.3 The good ordinary case

Suppose now that the reduction of the elliptic curve at the prime p is good and ordinary, i.e. that a_{p} is not divisible by p. As mentioned before, we have a unique choice of the root α of the characteristic polynomial satisfying ord_{p}(α) < 1. Here α is a unit in ℤ_{p}. We get therefore a unique p-adic L-function that we will denote simply by ℒ_{p}(E, T) = ℒ_{α}(E, T). It is proved in [?] that

Proposition 1. Let E be an elliptic curve with good and ordinary reduction at p > 2. Then the series ℒ_{p}(E, T) belongs to ℤ_{p}[[T]].

Note that ord_{p}(ϵ_{p}) is equal to −2 ord_{p}(N_{p}) where N_{p} = p + 1 − a_{p} is the number of points in the reduction Ẽ(𝔽_{p}) at p.

4.4 Multiplicative case

We have to seperate the case of split from the case of non-split multiplicative reduction. In fact if the reduction is non-split, then the description of the good ordinary case applies just the same. But if the reduction is split multiplicative (the “exceptional case” in [?]), then the p-adic L-series must have a trivial zero, i.e. ℒ_{p}(E, 0) = 0 because ϵ_{p} = 0. By a result of Greenberg and Stevens [?] (see also [?] for a simple proof), we know that

\frac{d ℒ_{p}(E, T)}{d T} |_{T =0} = \frac{1}{log_{p} κ(γ)} · \frac{log_{p}(q_{E})}{ord_{p}(q_{E})} · \frac{L(E, 1)}{Ω_{E}}

where q_{E} denotes the Tate period of E over ℚ_{p}. This will replace the interpolation formula. Note that it is now known thanks to [?] that log_{p}(q_{E}) is non-zero. Hence we define the p-adic ℒ -invariant as

ℒ_{p} = \frac{log_{p}(q_{E})}{ord_{p}(q_{E})} ≠ 0 .  (4)

We refer to [?] for a detailed discussion of the different ℒ -invariants and their connections.

4.5 The supersingular case

In the supersingular case, that is when a_{p} ≡ 0 (mod p), we have two roots α and β both of valuation \frac{1}{2} . A careful analysis of the functions ℒ_{α} and ℒ_{β} can be found in [?]. The series ℒ_{α}(E, T) will not have integral coefficients in ℚ_{p}(α). Nevertheless one can still extract two integral series ℒ_{p}^{±}(E, T). We will not need this description.

There is a way of rewriting the p-adic L-series which relates more easily to the p-adic height defined in the next section. We follow Perrin-Riou’s description in [?].

As before ω_{E} denotes the chosen invariant differential on E. Let η_{E} = x · ω_{E}. The couple {ω_{E}, η_{E}} forms a basis of the Dieudonné module D_{p}(E) = ℚ_{p} ⊗H_{dR}^{1}(E/ℚ). This ℚ_{p}-vector space comes equipped with a (geometric) Frobenius φ acting on it linearly. Its characteristic polynomial is equal to X^{2} − p^{−1} a_{p} X + p^{−1}.

Write ℒ_{α}(E, T) as G(T) + α · H(T) with G(T) and H(T) in ℚ_{p}[[T]]. Then we define

ℒ_{p}(T) = G(T) · ω_{E} + a_{p} · H(T) · ω_{E} − p · H(T) · φ(ω_{E}) .

This is a formal power series with coefficients in D_{p}(E) ⊗ ℚ_{p}[[T]] which contains exactly the same information as ℒ_{α}(E, T). See [?] for a direct definition. The D_{p}-valued L-series satisfies again to certain interpolation properties,^{2} e.g.

(1 − φ)^{−2} ℒ_{p}(0) = \frac{L(E, 1)}{Ω_{E}} · ω_{E} ∈ D_{p}(E) .

4.6 Additive case

The case of additive reduction is much harder to treat. We have not tried to include the possibility of additive reduction in our algorithm. Note that there are two interesting paper of Delbourgo [?] and [?] on this subject. We will not refer to this case anymore throughout the paper.

5 The p-adic heights

The second term to be generalised in the Birch-Swinnerton-Dyer formula is the real valued regulator. It will be substituted by a p-adic regulator. We follow here the generalised version [?] and [?].

Let ν be an element of the Dieudonné module D_{p}(E). We will define a p-adic height function h_{ν}: E(ℚ) → ℚ_{p} which depends linearly on the vector ν. Hence it is sufficient to define it on the basis ω = ω_{E} and η = η_{E}.

If ν = ω, then we define

h_{ω}(P) = − log(P)^{2}

where log is the linear extension of the p-adic elliptic logarithm logÊ: Ê(pℤ_{p}) → pℤ_{p} defined on the formal group Ê.

For ν = η, we define first the p-adic sigma-function of Bernardi σ(z) as in [?]. Denote by t = − \frac{x}{y} the uniformiser at O_{E} and write z(t) = logÊ(t). Define the Weierstrass ℘- function as usual by

℘(t) = x(t) + \frac{a_{1}^{2} + 4 a_{2}}{12} ∈ ℚ((t))

Here a_{1} and a_{2} are the coefficients of the minimal Weierstrass equation (1) of E. The function ℘(t) is a solution to the usual differential equation. We define the sigmafunction of Bernardi to be a solution of the equation

−℘(t) = \frac{d}{ω_{E}} ( \frac{1}{σ} · \frac{dσ}{ω_{E}} )

such that σ(0) = 0 and σ(t(−P)) = −σ(t(P)). This provides us with a series

σ(t) = t + \frac{a_{1}}{2} t^{2} + \frac{a_{1}^{2} + a_{2}}{3} t^{3} + \frac{a_{1}^{3} + 2a_{1}a_{2} + 3a_{3}}{4} t^{4} + ⋯ ∈ ℚ((t)) .

As a function on the formal group Ê(pℤ_{p}) it converges for ord_{p}(t) > \frac{1}{p−1} .

Given a point P in E(ℚ) there exists a multiple m · P such that σ(t(P)) converges and such that m · P has good reduction at all primes. Denote by e(m · P) ∈ ℤ the square root of the denominator of the x-coordinate of m · P. Now define

h_{η}(P) = \frac{2}{m^{2}} · log_{p} ( \frac{σ(t(m · P))}{e(m · P)} )

2Perrin-Riou writes in [?] the multiplier as (1−φ)^{−1} ·(1−p^{−1}φ^{−1}) and she multiplies the right hand side with L(E/ℚ_{p}, 1)^{−1} = N_{p} · p^{−1}. It is easy to see that (1 − φ) · (1 − p^{−1}φ^{−1}) = 1 − φ − (φ − a_{p} · p^{−1}) + p^{−1} = N_{p} · p^{−1}.

It is proved in [?] that this function is quadratic and satisfies the parallelogram law. Finally, if ν = a ω + b η then put

h_{ν}(P) = a h_{ω}(P) + b h_{η}(P) .

This quadratic function induces a bilinear symmetric pairing ⟨·, ·⟩_{ν} with values in ℚ_{p}.

5.1 The good ordinary case

Since we have only a single p-adic L-function in the case that the reduction is good ordinary, we have also to pin down a canonical choice of a p-adic height function. This was first done by Schneider [?] and Perrin-Riou [?]. We refer to [?] and [?] for more details.

Let ν_{α} = a ω + b η be an eigenvector of φ on D_{p}(E) associated to the eigenvalue \frac{1}{α}. The value e_{2} = E_{2}(E, ω_{E}) = −12 · \frac{a}{b} is the value of the Katz p-adic Eisenstein series of weight 2 at (E, ω_{E}). Then, if P has good reduction at all primes and lies in the range of convergence of σ(t), we define the canonical p-adic height of P to be

ĥ_{p}(P) = \frac{1}{b} · h_{ν_{α}}(P)

= − \frac{a}{b} · z(P)^{2} + 2 log( \frac{σ(t(P))}{e(P)} )

= 2 log_{p}( \frac{exp( \frac{e_{2}}{24} log(P)^{2}) · σ(t(P))}{e(P)} ) = 2 log_{p}( \frac{σ_{p}(t(P))}{e(P)} )  (5)

The function σ_{p}(t), defined by the last line, is called the canonical sigma-function, see [?], it is known to lie in ℤ_{p}[[t]]. The p-adic height defined here is up to the factor 2 the same as in [?].^{3}

We write ⟨·, ·⟩_{p} for the canonical p-adic height pairing on E(ℚ) associated to ĥ_{p} and Reg_{p}(E/ℚ) for its determinant.

Conjecture 3. (Schneider [?]) The canonical p-adic height is non-degenerate on the free part of E(ℚ). In other words, the canonical p-adic regulator Reg_{p}(E/ℚ) is nonzero.

Apart from the special case treated in [?] of curves with complex multiplication of rank 1, there are hardly any results on this conjecture. See also [?].

5.2 The multiplicative case

In the case the case of multiplicative reduction, one may use Tate’s p-adic uniformisation (see [?]). We have an explicit description of the height pairing in [?]. If one wants to have the same closed formula in the p-adic version of the Birch and Swinnerton-Dyer conjecture for multiplicative primes as for other ordinary primes, the p-adic height has to be changed sligthly. We use here the description of the p-adic regulator given in section II.6 of [?]. Alas, their formula is not correct as explained by Werner in [?].

Let q_{E} be the Tate-parameter of the elliptic curve over ℚ_{p}, i.e. we have a homomorphism ψ: ℚ̄_{p}^{×} → E(ℚ̄_{p}) whose kernel is precisely qℤ_{E}. The image of ℤ_{p}^{×} under ψ is equal to the subgroup of points of E(ℚ_{p}) lying on the connected component of the Néron model of E. Now let C be the constant such that ψ^{∗}(ω_{E}) = C · \frac{du}{u} where u is a uniformiser of ℚ_{p}^{×} at 1. The value of the weight two p-adic Eisenstein series can then be computed as

e_{2} = E_{2}(E, ω_{E}) = C^{2} · (1 − 24 · ∑_{n⩾1} ∑_{d|n} d. · q^{n})

3This factor is needed if one does not want to modify the p-adic version of the Birch and Swinnerton-Dyer conjecture 5.

Then we use the formula of the good ordinary case to define the canonical sigma function σ_{p}(t(P)) = exp( \frac{e_{2}}{24} z(P)^{2})·σ(t(P)). If the reduction is non-split multiplicative, then we can simply use the formula (5) for the good ordinary case.

Suppose now that the reduction is split multiplicative. Let P be a point in E(ℚ) having good reduction at all finite places and with trivial reduction at p. Then

ĥ_{p}(P) = 2 log_{p} ( \frac{σ_{p}(t(P))}{e(P)} ) + \frac{log_{p}(u(P))^{2}}{log(q_{E})}

where u(P) is the unique element of ℤ_{p}^{×} mapping to P under the Tate parametrisation ψ. The p-adic regulator is formed as before but with this modified p-adic height ĥ_{p}.

5.3 The supersingular case

In the supersingular case, we can not find a canonical p-adic height with values in ℚ_{p}. Instead, the height will have values in the Dieudonné module D_{p}(E). The main references for this height is [?] and [?].

Suppose that ν = a ω + b η is any element of D_{p}(E) not lying in ℚ_{p} ω_{E}. It can be easily checked that the value of

H_{p}(P) = \frac{1}{b} · (h_{ν}(P) · ω − h_{ω}(P) · ν) ∈ D_{p}

is independent of the choice of ν. We will call this the D_{p}-valued height on E(ℚ).

On D_{p}(E) there is a alternating bilinear form [·, ·] characterised by the property that [ω_{E}, η_{E}] = 1. Write Reg_{ν} ∈ ℚ_{p} for the regulator of h_{ν} on a ℤ-basis of the free part of E(ℚ). Then

Reg_{p}(E/ℚ) = \frac{Reg_{ν} ·ν′ − Reg_{ν′} ·ν}{[ν′, ν]} ∈ D_{p}(E)

is independent of the choice of ν and ν′ in D_{p}(E), as long as they do not belong to ℚ_{p} ω_{E}. We call this the D_{p}-valued regulator of E/ℚ.

It is not difficult to see that Reg_{p}(E/ℚ) = H_{p}(P) if the curve is of rank 1 with generator P. If E(ℚ) is finite, then Reg_{p}(E/ℚ) is simply ω_{E}. In both these cases the D_{p}-valued regulator can not vanish.

If one restricts any p-adic height h_{ν} to the fine Mordell-Weil group defined in [?] to be the kernel

𝔐(E/ℚ) = ker (E(ℚ) ⊗ ℤ_{p} → E(ℚ_{p})^{p-adic completed}),

then the restricted height is independent of the chosen element ν in D_{p}(E). We call its regulator the fine regulator which is an element of ℚ_{p} defined up to multiplication by a unit in ℤ_{p}.

In general, the D_{p}-valued regulator is zero if and only if the fine regulator vanishes.

Conjecture 4. (Perrin-Riou [?, Conjecture 3.3.7.i]) The fine regulator of E/ℚ is non zero for all primes p. In particular, Reg_{p}(E/ℚ) ≠ 0 for all primes where E has supersingular reduction.

5.4 Normalisation

In view of Iwasawa theory, it is actually natural to normalise the heights and the regulators depending on the choice of the generator γ. In this way the heights depend on the choice of an isomorphism Γ → ℤ_{p} rather than on the ℤ_{p}-extension only.

This can be achieved by simply dividing ĥ_{p}(P) and h_{ν}(P) by κ(γ). The regulators will be divided by κ(γ)^{r} where r is the rank of E(ℚ). Hence we will write

Reg_{γ}(E/ℚ) = \frac{Reg_{p}(E/ℚ)}{κ(γ)^{r}}

6 The p-adic Birch and Swinnerton-Dyer conjecture

6.1 The ordinary case

The following conjecture is due to Mazur, Tate and Teitelbaum [?]. Rather than formulating it for the function L_{α}(E, s), we state it directly for the series ℒ_{p}(E, T). It is then a statement about the development of this function at T = 0 rather than at s = 1.

Conjecture 5. (Mazur, Tate and Teitelbaum [?]) Let E be an elliptic curve with good ordinary reduction or with multiplicative reduction at a prime p.

• The order of vanishing of the p-adic L-function ℒ_{p}(E, T) at T = 0 is equal to the rank r, unless E has split multiplicative reduction at p in which case the order of vanishing is equal to r + 1.

• The leading term ℒ_{p}^{∗}(E, 0) satisfies

ℒ_{p}^{∗}(E, 0) = ϵ_{p} · \frac{∏_{υ} c_{υ} · #� (E/ℚ)}{(#E(ℚ)_{tors})^{2}} · Reg_{γ}(E/ℚ)  (6)

unless the reduciton is split multiplicative in which case the leading term is

ℒ_{p}^{∗}(E, 0) = ℒ_{p} · \frac{∏_{υ} c_{υ} · #� (E/ℚ)}{(#E(ℚ)_{tors})^{2}} · Reg_{γ}(E/ℚ)  (7)

6.2 The supersingular case

The conjecture in the case of supersingular reduction is due to Bernardi and Perrin-Riou in [?] and [?]. The conjecture relates here an algebraic and an analytic values in the ℚ_{p}-vector space D_{p}(E) of dimension two. The fact of having two coordinates was used cleverly by Kurihara and Pollack in [?] to construct global points via a p-adic analytic computation.

For an element a(T) · ω_{E} + b(T) · η_{E} in D_{p}(E) ⊗ ℚ_{p}[[T]], we say that it has order d at T = 0 if d is equal to the minimum of the orders of a(T) and b(T).

Conjecture 6. (Bernardi and Perrin-Riou [?]) Let E be an elliptic curve with good supersingular reduction at a prime p.

• The order of vanishing of the D_{p}-valued L-function ℒ_{p}(E, T) at T = 0 is equal to the rank r.

• The leading term ℒ_{p}^{∗}(E, 0) satisfies

(1 − φ)^{−2} · ℒ_{p}^{∗}(E, 0) = \frac{∏_{υ} c_{υ} · #� (E/ℚ)}{(#E(ℚ)_{tors})^{2}} · Reg_{γ}(E/ℚ) ∈ D_{p}(E) (8)

7 Iwasawa Theory of Elliptic Curves

We suppose from now on that p > 2. Let _{∞}ℚ be the Galois extension of ℚ whose Galois group is Γ. It is the unique ℤ_{p}-extension of ℚ_{p}. Let Λ be the completed group algebra ℤ_{p}[[Γ]]. We use the fixed topological generator γ of Γ to identify Λ with ℤ_{p}[[T]] by sending γ to 1 + T. It is well-known that Λ-modules of finite generation admit a decomposition in elementary Λ-modules. Denote by _{n}ℚ the n^{th} layer of the ℤ_{p}-extension. As before, we may define the p-Selmer group over _{n}ℚ by the exact sequence

0 → 𝒮_{p}(E/_{n}ℚ) → H^{1}(_{n}ℚ, E(p)) → ∏_{υ} H^{1}(_{n}ℚ_{υ}, E)

with the product running over all places υ of _{n}ℚ. Moreover, we define 𝒮_{p}(E/_{∞}ℚ) to be the limit \underrightarrow{lim} 𝒮_{p}(E/_{n}ℚ) following the maps induced by the restriction maps H^{1}(_{n}ℚ, E(p)) → H^{1}(_{n+1}ℚ, E(p)). The group 𝒮_{p}(E/_{∞}ℚ) contains essentially the information about the growth of the rank of E(_{n}ℚ) and of the size of � (E/_{n}ℚ)(p) as n tends to infinity. We will consider the Pontryagin dual

X(E/_{∞}ℚ) = Hom (𝒮_{p}(E/_{∞}ℚ),^{ℚ_{p}}/_{ℤ_{p}})

which is a finitely generated Λ-module, see [?].

7.1 The ordinary case

Assume now that the reduction at p is good and ordinary or of multiplicative type. It was shown by Kato in [?] that X(E/_{∞}ℚ) is a torsion Λ-module. Hence by the decomposition theorem, we may associated to it a characteristic series f_{E}(T) in Λ. The series f_{E}(T) ∈ ℤ_{p}[[T]] is well-defined up to multiplication by a unit in Λ^{×}.

Analogue to the zeta-function of a variety over a finite field, one should think of f_{E}(T) as a generating function encoding the growth of the rank and the Tate-Shafarevich group. For instance, the zeros of f_{E}(T) at roots of unity whose orders are powers of p are describing the growth of the rank. Since power series with coefficients in ℤ_{p} may only have finitely many zeros, one can show that the rank of E(_{n}ℚ) has to stabilize in the tower ℚ_{n}. In other words, the Mordell-Weil group E(_{∞}ℚ) is still of finite rank.

A relatively old result is due to Schneider [?] and Perrin-Riou [?]. The multiplicative case is due to Jones [?].

Theorem 2. The order of vanishing of f_{E}(T) at T = 0 is at least equal to the rank r. It is equal to r if and only if the p-adic height pairing is non-degenerate (conjecture 3) and the p-primary part of the Tate-Shafarevich group � (E/ℚ)(p) is finite (conjecture 1). In this case the leading term of the series f_{E}(T) has the same valuation as

ϵ_{p} · \frac{∏_{υ} c_{υ} · #� (E/ℚ)(p)}{(#E(ℚ)(p))^{2}} · Reg_{γ}(E/ℚ)

unless the reduction is split multiplicative in which case the same formula holds with ϵ_{p} replaced by ℒ_{p}.

7.2 The supersingular case

The supersingular case is much more complicated, since the Λ-module X(E/_{∞}ℚ) is not torsion. A very beautiful approach to the supersingular case has been found by Pollack [?] and Kobayashi [?]. As mentioned above there exists two p-adic series ℒ_{p}^{±}(E, T) to which will correspond to two new Selmer groups X^{±}(E/_{∞}ℚ) which now are Λtorsion. Despite the advantages of this ±-theory, we are using the approach of Perrin-Riou here. See section 3 in [?].

Let T_{p}E be the Tate module and define _{∞}H_{loc}^{1} to be the projective limit of the cohomology groups H^{1}(_{n}ℚ_{𝔭}, T_{p}E) following the corestriction maps. Here _{n}ℚ_{𝔭} is the localisation of _{n}ℚ at the unique prime 𝔭 above p. Perrin-Riou [?] has constructed a Λ-linear Colman map Col from _{∞}H_{loc}^{1} to a sub-module of ℚ_{p}[[T]] ⊗ D_{p}(E).

Define the fine Selmer group to be the kernel

ℛ(E/_{n}ℚ) = ker (𝒮(E/_{n}ℚ) → E(_{n}ℚ_{𝔭}) ⊗^{ℚ_{p}}/_{ℤ_{p}}) .

It is again a consequence of the work of Kato that the Pontryagin dual Y (E/_{∞}ℚ) of ℛ(E/_{∞}ℚ) is a Λ-torsion module. Denote by g_{E}(T) its characteristic series.

Let Σ be any finite set of places in ℚ containing the places of bad reduction for E and the places ∞ and p. By G_{Σ}(_{n}ℚ), we denote the Galois group of the maximal extension of _{n}ℚ unramified at all places which do not lie above Σ. Next we define ∞H_{glob}^{1} as the projective limit of H^{1}(G_{Σ}(_{n}ℚ), T_{p}E). It is a Λ-module of rank 1 and it is actually independent of the choice of Σ.

Choose now any element _{∞}c in _{∞}H_{glob}^{1} such that Z_{c} = _{∞}H_{glob}^{1} /(Λ·_{∞}c) is Λ-torsion. Typically the “zeta element” of Kato could be such a choice. Write h_{c}(T) for the characteristic series of Z_{c}. Then we define an algebraic equivalent of the D_{p}(E)-valued L-series by

f_{E}(T) = g_{E}(T) · Col(_{∞}c) · h_{c}(T)^{−1} ∈ ℚ_{p}[[T]] ⊗ D_{p}(E)

where by Col(_{∞}c) we mean the image of the localisation of _{∞}c to _{∞}H_{loc}^{1} under the Colman map Col. The resulting series f_{E}(T) is independent of the choice of _{∞}c. Of course, f_{E}(T) is again only defined up to multiplication by a unit in Λ^{×}.

Again we have a Euler-characteristic result due to Perrin-Riou [?] :

Theorem 3. The order of vanishing of f_{E}(T) at T = 0 is at least equal to the rank r. It is equal to r if and only if the D_{p}(E)-valued regulator Reg_{p}(E/ℚ) is non-zero (conjecture 4) and the p-primary part of the Tate-Shafarevich group � (E/ℚ)(p) is finite (conjecture 1). In this case the leading term of the series (1 − φ)^{−2} f_{E}(T) has the same valuation as

∏_{υ} c_{υ} · #� (E/ℚ)(p) · Reg_{γ}(E/ℚ)

Note that we simplified the right hand term in comparison to (8), because N_{p} ≡ 1 (mod p) and hence #E(ℚ)_{tors} must be p-adic units if the reduction at p is supersingular.

8 The Main Conjecture

The main conjecture links the two p-adic power series presented in the previous sections. We formulate everything now simultaneously for the ordinary and the supersingular case, even if they are of quite different nature. We still assume that p ≠ 2.

Conjecture 7. (Main conjecture of Iwasawa theory for elliptic curves) If E has good or non-split multiplicative reduction at p then there exists an element u(T) in Λ^{×} such that ℒ_{p}(E, T) = f_{E}(T) · u(T). If the reduction of E at p is split multiplicative, then there exists such a u(T) in T · Λ^{×}.

A good deal of things is now known about this conjecture. Denote by ρ̄_{p} the mod-p representation

ρ̄_{p}: Gal(ℚ̄/ℚ) → Aut(E[p]) ≃ GL_{2}(𝔽_{p})

of the absolute Galois group attached to the elliptic curve. Serre has shown that ρ̄_{p} is almost always surjective and that for semi-stable curve it can only fail when there is an isogeny of degree p defined over ℚ. See [?] and [?].

Kato’s Theorem 4. Suppose that E has semi-stable reduction at p and that ρ̄_{p} is either surjective or that its image is contained in a Borel subgroup. Then there exists a series d(T) in Λ such that ℒ_{p}(E, T) = f_{E}(T) · d(T). If the reduction is split multiplicative then T divides d(T).

The main ingredient for this theorem is in theorem 17.4 in [?] for the good ordinary case when ρ̄_{p} is surjective, or in [?] when there is a p-isogeny. The exceptional case refers to [?] and [?]. The statement of the main conjecture for supersingular primes is known to be equivalent to Kato’s formulation in Conjecture 12.10 in [?] and to Kobayashi’s version in [?].

In particular the theorem applies to all odd primes p if E is a semi-stable curves. For the remaining cases, e.g. if the image of ρ̄_{p} is contained in the normalizer of a Cartan subgroup, one obtains only a weaker statement :

Kato’s Theorem 5. Suppose the image of ρ̄_{p} is not contained in a Borel subgroup of GL_{2}(𝔽_{p}) and that ρ̄_{p} is not surjective, then there is an integer m ⩾ 0 such that f_{E}(T) divides p^{m} · ℒ_{p}(E, T).

Greenberg and Vatsal [?] have shown that in certain cases the main conjecture holds. There is hope that the main conjecture will be proved soon for primes p subject to certain conditions. We are awaiting the forthcoming paper of Skinner and Urban.

9 If the L-series does not vanish

Suppose the Hasse-Weil L-function L(E, s) does not vanish at s = 1. In this case it is known by the result of Kolyvagin that E(ℚ) and � (E/ℚ) are finite. In particular conjecture 1 is valid; the conjectures 3 and 4 are trivially true in this case.

Let p > 2 be a prime of semi-stable reduction such that the representations ρ̄_{p} is either surjective or has its image contained in a Borel subgroup of GL_{2}(𝔽_{p}). By the interpolation property, we know that ℒ_{p}(E, 0) is non zero, unless the reduction is split multiplicative.

9.1 The good ordinary case

In the ordinary case it simply equal to

ϵ_{p}^{−1} · ℒ_{p}(E, 0) = \frac{L(E, 1)}{Ω_{E}} = [0]^{+}

which is a non-zero rational number by [?]. Using the theorem^{4} of Perrin-Riou and Schneider 2 in the first line and Kato’s theorem 4 on the main conjecture in the second line, we find that

ord_{p} (ϵ_{p} · \frac{∏_{υ} c_{υ} · #� (E/ℚ)(p)}{(#E(ℚ)(p))^{2}}) = ord_{p}(f_{E}(0))
⩽ord_{p}(ℒ_{p}(E, 0))

= ord_{p}( \frac{L(E, 1)}{Ω_{E}} ) + ord_{p}(ϵ_{p}) .

Hence, we have an upper bound on the p-primary part of the Tate-Shafarevich group which is sharp under the assumption of the main conjecture.

ord_{p}(� (E/ℚ)(p)) ⩽ ord_{p}( \frac{L(E, 1)}{Ω_{E}} ) − ord_{p}( \frac{∏ c_{υ}}{(#E(ℚ)_{tors})^{2}}) .  (9)

This bound agrees with the Birch and Swinnerton-Dyer conjecture.

9.2 The multiplicative case

If the reduction is not split, then the above holds just the same. If instead the reduction is split multiplicative, we have that ℒ_{p}(E, 0) = 0 and

ℒ′_{p}(E, 0) = ℒ_{p} · \frac{L(E, 1)}{Ω_{E}} = ℒ_{p} · [0]^{+} ≠ 0 .

Since the p-adic multiplier is the same on the algebraic as on the analytic side, we can once again compute it as above to obtain the same bound (9) again.

9.3 The supersingular case

For the supersingular D_{p}(E)-valued series, we have

(1 − φ)^{−2} · ℒ_{p}(E, 0) = \frac{L(E, 1)}{Ω_{E}} · ω_{E} = [0]^{+} · ω_{E}

which is a non-zero element of D_{p}(E). The D_{p}(E)-valued regulator Reg_{p}(E/ℚ) is equal to ω_{E}. We may therefore concentrate solely on the coordinate in ω_{E}. Write ord_{p}(f_{E}(0)) for the p-adic valuation of the leading coefficient of the ω_{E}-coordinate of f_{E}(T). Again we obtain an inequality by using theorem 3

4In the case of rank 0, the theorem is actually relatively easy and well explained in [?].

ord_{p}(∏_{υ} c_{υ} · #� (E/ℚ)(p)) =ord_{p}((1 − φ)^{−2} f_{E}(0))
⩽ord_{p}((1 − φ)^{−2} ℒ_{p}(E, 0))

= ord_{p}( \frac{L(E, 1)}{Ω_{E}} ) .

9.4 Conclusion

Summarising the above computations, we have

Theorem 6. Let E be an elliptic curve such that L(E, 1) ≠ 0. Then � (E/ℚ) is finite and

#� (E/ℚ) ⩽ C · \frac{L(E, 1)}{Ω_{E}} · \frac{(#E(ℚ)_{tors})^{2}}{∏ c_{υ}}

where C is a product of a power of 2 and of primes of additive reduction and of primes for which the representation ρ̄_{p} is not surjective and there is no isogeny of degree p on E defined over ℚ.

In particular if E is semi-stable, then C is a power of 2.

This improves Corollary 3.5.19 in [?].

10 If the L-series vanishes to the first order

We suppose for this section that E has good and ordinary reduction at p and that the complex L-series L(E, s) has a zero of order 1 at s = 1. The method of Heeger points and the theorem of Kolyvagin show again that � (E/ℚ) is finite and that the rank is equal to 1. Let P be a generator of the free part of the Mordell-Weil group. Suppose that the p-adic height ĥ_{p}(P) is non-zero. Thanks to a theorem of Perrin-Riou in [?], we must have the following equality of rational numbers

\frac{1}{Reg(E/ℚ)} · \frac{L′(E, 1)}{Ω_{E}} = \frac{1}{Reg_{p}(E/ℚ)} · \frac{ℒ′_{p}(E, 0)}{(1 − \frac{1}{α})^{2} · log(κ(γ))}

where, on the left hand side, we have the canonical real-valued regulator Reg(E/ℚ) = ĥ(P) and the leading coefficient of L(E, s), while, on the right hand side, we have the p-adic regulator Reg_{p}(E/ℚ) = ĥ_{p}(P) and the leading term of the p-adic L-series. By the conjecture of Birch and Swinnerton-Dyer (or its p-adic analogue), this rational number should be equal to ∏ c_{υ} · #� (E/ℚ) · (#E(ℚ)_{tors})^{−2}. By Kato’s theroem, one knows that the characteristic series f_{E}(T) of the Selmer group divides ℒ_{p}(E, T); at least up to a power of p. Hence the series f_{E}(T) has a zero of order 1 at T = 0 and its leading term divides the above rational number in ℚ_{p}. Hence we have

Theorem 7. Let E/ℚ be an elliptic curve with good ordinary reduction at the odd prime p. Suppose that the representation ρ̄_{p} is surjectif onto GL_{2}(𝔽_{p}) or that the curve admits an isogeny of degree p defined over ℚ. If L(E, s) has a simple zero at s = 1, then the p-primary part of � (E/ℚ) is finite and its valuation is bounded by

ord_{p}(#� (E/ℚ)(p)) ⩽ ord_{p} ( \frac{(#E(ℚ)_{tors})^{2}}{∏ c_{υ}} · \frac{1}{Reg(E/ℚ)} · \frac{L′(E, 1)}{Ω_{E}} )

In other words the Birch and Swinnerton-Dyer conjecture if true up to a factor involving only bad and supersingular primes, and primes for which the representation is neither surjective nor has its image contained in a Borel subgroup.

11 The algorithm

11.1 The rank

Let E/ℚ be an elliptic curve. Suppose we are in the situation that we have found n linearly independent points. We wish to prove that n is equal to the rank r = rank(E(ℚ)).

For this purpose, we choose a prime p satisfying the following conditions

• p > 2,

• E has good reduction at p.

By computing the analytic p-adic L-function ℒ_{p}(E, T) to a certain precision, we find an upper bound, say b, on the order of vanishing of ℒ_{p}(E, T) at T = 0. Then

b ⩾ ord_{T =0} ℒ_{p}(E, 1) ⩾ ord_{T =0} f_{E}(T) ⩾ r

by Kato’s theorem and by the theorems 2 and 3. Hence we have an upper bound on the rank r. In case b is different from n, we can either increase the precision or we can change the prime p. Note that the p-adic Birch and Swinnerton-Dyer conjecture tells us exactly what the needed precision should be for being able to distinguish the leading coefficient from zero.

11.2 The Tate-Shafarevich group

Suppose now that E is an elliptic curve and p is a prime satisfying the following conditions

• p > 2,

• E has good reduction at p.

• The image of ρ̄_{p} is either the full group GL_{2}(𝔽_{p}) or it is contained in a Borel subgroup.

Note that these conditions apply to all but finitely many primes p.

Suppose further that the rank computation presented in the previous part of the algorithm was successful (for any prime not necessarily p). We may assume that we are able to compute a basis of the full Mordell-Weil group E(ℚ) modulo torsion.

Using the explicit basis of E(ℚ) we can compute the p-adic regulator of E over ℚ using the efficient algorithm in [?].

We compute the leading coefficient ℒ_{p}^{∗}(E, 0) of the analytic p-adic L-function. If the order of vanishing of ℒ_{p}(E, T) at T = 0 is equal to r then we know already that the p-primary part of the Tate-Shafarevich group is finite. Moreover, we get an upper bound.

11.3 The ordinary case

If E has ordinary reduction at p, good or multiplicative, then

ord_{p}(#� (E/ℚ)(p)) =ord_{p}(f ∗_{E}(0)) + ord_{p} ( \frac{(#E(ℚ)(p))^{2}}{ϵ_{p} · ∏_{υ} c_{υ}} · \frac{1}{Reg_{γ}(E/ℚ)})
⩽ord_{p}(ℒ_{p}^{∗}(E, 0)) + 2 · ord_{p}(#(E(ℚ)(p)) − ord_{p}(ϵ_{p})
− ∑_{υ} ord_{p}(c_{υ}) − ord_{p}(Reg_{γ}(E/ℚ))

The inequality uses Kato’s theorem 4.

Note that if the main conjecture holds this inequality will be an equality. It should also be mentioned that Grigorov [?] has found lower bounds on the order of the Tate-Shafarevich group in certain cases.

11.4 The supersingular case

Suppose now that E has supersingular reduction at p. Then we may use theorem 3 and theorem 4 to obtain

ord_{p}(#� (E/ℚ)(p)) =ord_{p}((1 − φ)^{−2} f ∗_{E}(0)) − ord_{p}(Reg_{p}(E/ℚ)) − ord_{p}(∏_{υ} c_{υ})
⩽ord_{p}((1 − φ)^{−2} ℒ_{p}^{∗}(E, 0)) − ord_{p}(Reg_{p}(E/ℚ)) − ∑_{υ} ord_{p}(c_{υ})

where the convention on ord_{p}(d(T)) for an element d(T) ∈ ℚ_{p}[[T]] ⊗ D_{p}(E) is as before. Again the inequality can be replaced by an equality if the main conjecture holds for E at p.

12 Technical details

13 Numerical results