A formulation for p-adic versions of the Birch and Swinnerton-Dyer conjectures in the supersingular case

Given an elliptic curve E and a prime p of (good) supersingular reduction, we formulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions L^\sharp(E,T) and L^\flat(E,T). They are equivalent to the conjectures of Perrin-Riou and Bernardi. We also generalize work of Kurihara and Pollack to give a criterion for positive rank in terms of the value of the quotient between these functions, and derive a result towards a non-vanishing conjecture. We also generalize a conjecture of Kurihara and Pollack concerning the greatest common divisor of the two functions to the general supersingular case.


Introduction
Let E be an elliptic curve over the rational numbers Q. The Q-rational points form a finitely generated abelian group E(Q) = Z r ⊕ E(Q) tors , and the classical Birch and Swinnerton-Dyer conjectures predict that the order of vanishing r an C of the Hasse-Weil Lfunction L(E, s) at s = 1, an analytic quantity, should be equal to r, an algebraic quantity. The second part of their conjecture says that the leading Taylor coefficient of L(E, s) should encode E(Q) tors and the size of the Tate-Shafarevich group X(E/Q), among other algebraic quantities. More precisely, this conjecture says the following.
Denote by ω = ω E the Néron differential and by Ω E = E(R) ω ∈ R >0 the Néron period of E. We denote by L * (E) the leading coefficient of the Taylor expansion at s = 1.

Conjecture 1.1 (BSD).
(1) We have r an C = r. Here, c v denotes the Tamagawa number for a place v, and the regulator Reg C (E/Q) is the discriminant of the Néron-Tate canonical height pairing on E(Q).
To construct a p-adic analogue, we identify algebraic numbers with p-adic numbers by fixing an embedding Q ֒→ C p . A p-adic analogue of this conjecture should look as follows. There should be a p-adic L-function L p (E, T ), whose order of vanishing r an at T = 0 should equal r, and whose leading Taylor coefficient should again encode algebraic quantities of E including E(Q) tors and #X(E/Q) as in the above formula, with the regulator Reg C replaced by a p-adic avatar. There are two types of such p-adic analogues.
The first type concerns the case when p is a prime of ordinary reduction (meaning that p is coprime to a p := p + 1 − #E(F p )). Here, Mazur, Tate, and Teitelbaum formulated a p-adic version of these conjectures. The p-adic L-function L p (E, T ) they employed gave rise to the expected version of p-adic BSD when p was of good ordinary reduction, but in the split multiplicative case r an corresponded to r + 1 in view of an extra zero.
The second type is the supersingular case (the case when p|a p ), which is more complex. There are two classical p-adic L-functions, denoted L α (E, T ) and L β (E, T ), constructed independently by Amice and Vélu, and Vishik. The subscripts α and β denote the roots of the Hecke polynomial Y 2 − a p Y + p. The formulation of a p-adic analogue of the Birch and Swinnerton-Dyer conjectures in terms of these L α (E, T ) and L β (E, T ) when p is of good reduction is due to Bernardi and Perrin-Riou. For questions of formulating Birch and Swinnerton-Dyer conjectures, this seems to suggest that the picture is complete except for some primes of bad reduction. The goal of this paper is to indicate that this is not the case.
We do this by formulating p-adic versions of BSD for supersingular primes p using a more natural pair of p-adic L-functions L ♯ (E, T ) and L ♭ (E, T ). This hints at a formulation of p-adic BSD in the ordinary case in terms of such a pair as well. These p-adic L-functions had been constructed by Pollack, Kobayashi, and the author in the supersingular case and are functions living in the power series ring Z p [[T ]], unlike L α (E, T ) and L β (E, T ) which have more complicated growth properties. Apart from being an ingredient for a natural formulation of the Iwasawa Main Conjecture, the appearance of the Iwasawa invariants of this pair of p-adic L-functions indicates that this pair is the natural choice: The Iwasawa invariants appear in analytic estimates for the sizes of the p-primary parts of the Tate-Shafarevich group along the cyclotomic Z p -extensions. The term "analytic estimates" refers to the corresponding special values of the Hasse-Weil L-functions twisted by characters of p-power conductor, which should encode these sizes. These analytic estimates can be found in [Po03] and [Sp15]. There are algebraic counterparts of L ♯ (E, T ) and L ♭ (E, T ), which are modified Selmer groups Sel ♯ (E) and Sel ♭ (E). Using their Iwasawa invariants one can estimate these sizes directly, see e.g. [Ko03] and [Sp13]. In addition to these estimates, the Iwasawa invariants appear in upper bounds for the rank of the elliptic curve in the cyclotomic Z p -extension of Q. For analytic estimates, see [Po03] and [Sp15], and for their algebraic counterparts, see [Ko03] and [Sp16].
Concretely, our conjecture says the following.
Conjecture 1.2 (Tandem p-adic BSD). Let E be an elliptic curve and p a prime of good supersingular reduction. Denote by − → L * p the first non-zero Taylor coefficient around T = 0 of the vector of p-adic L-functions (L ♯ (E, T ), L ♭ (E, T )).
(1) The minimum of the orders of vanishing of L ♯ (E, T ) and L ♭ (E, T ) at T = 0 is equal to r.
Our regulator Reg ♮ p (E/Q) is constructed explicitly from height functions that mirror the construction of L ♯ (E, T ) and L ♭ (E, T ). Theorem 1.3. This conjecture is equivalent to the conjectures of Bernardi and Perrin-Riou.
Since we have two functions at hand, we may consider their quotient, and give the following criterion for detecting non-zero rank: Theorem 1.4. Assume property ( * ) below holds and that X( This theorem is a generalization of [KP07, Corollary 0.5] who assumed p to be odd and a p = 0, which is automatically satisfied when p 5. We remark that the proof of Kurihara and Pollack almost immediately generalizes once the L ♯ (E, T ) and L ♭ (E, T ) have been defined in complete generality, as done in [Sp12]. There is only one proposition in their tools that assumes a p = 0, which is fixed in this paper by adhering to a generalization found in [Sp13].
As a corollary to this, we get: Corollary 1.5. Let rankE(Q) = 0. Then both L ♯ (E, T ) and L ♭ (E, T ) are non-zero functions, confirming [Sp12, Conjecture 6.15] in this case.
Another object to consider given two quantities is their greatest common divisor. Denote by d n the normalized jump in ranks 1 p n −p n−1 (rankE(Q n ) − rankE(Q n−1 )), where Q n is the nth layer in the cyclotomic Z p -extension numbered so that Q 0 = Q. We also let Φ p n be the p n th cyclotomic polynomial. Kurihara and Pollack have formulated the following conjecture in the case a p = 0 and p odd: Conjecture 1.6. Let E/Q be an elliptic curve, and p a good supersingular prime. Then Note that we excluded their assumptions. This is because we give some evidence towards their conjecture by proving the following proposition which works for general supersingular p. Denote by r an the order of vanishing of L(α, T ) (or L(β, T )) at T = 0: Proposition 1.7. Let E/Q be an elliptic curve and p a prime of good reduction. For some polynomial P X (E, T ) with P X (E, ζ p n − 1) = 0 for n 0, where ǫ an n = d an n or ǫ an n = d an n + 1. Finally, we ask a question on a possible generalization of the situation to the ordinary case. When p is ordinary, the functions L ♯ (E, T ) and L ♭ (E, T ) are not uniquely defined, but their values at T = 0 are. We find that when p is odd and a p = 2, L ♭ (E, 0) = 0.
Question 1. Where does this 'extra zero phenomenon' in the ordinary case come from?
In [LZ13], Loeffler and Zerbes find a pair of Iwasawa functions in the ordinary case using the theory of Wach modules. The above question suggests that an extra zero phenomenon should occur in terms of their pair as well.

Review of p-adic L-functions for elliptic curves
Let p be a prime of good reduction for our elliptic curve E. The work of Mazur, Swinnerton-Dyer, Amice and Vélu, and Višik gives constructions of p-adic L-functions that should encode the behavior of the Q n -rational points E(Q n ) of the elliptic curve E. Concretely, denote by α and β the roots of the Hecke polynomial Y 2 − a p Y + p ordered so that ord p (α) ord p (β) for any p-adic valuation ord p . We say that α resp. β is an allowable root if ord p (α) < ord p (p) resp. ord p (β) < ord p (p). Notice that α is always allowable by convention, while β is only when p is supersingular.
Notation 2.1. We denote by ζ p n a primitive p n th root of unity, and we let N = n + 1 when p is odd and N = n + 2 when p = 2. We also let χ u be a group morphism from 1 + 2pZ p into C × p sending a topological generator 1 + 2p to some u ∈ C p so that |u − 1| p < 1, where | | p is the normalized p-adic absolute value (i.e. | 1 p | p = 1).
Theorem 2.2 (Mazur and Swinnerton-Dyer, Amice and Vélu, Višik). [MTT86, Proposition in Section 14] Let E be an elliptic curve over Q and p a prime of good reduction. Regard χ ζ p n and χ ζ −1 p n as characters of Z p /p n Z p . There is a p-adic analytic function L α (E, T ) converging on the open unit disk with the following interpolation properties: When β is allowable (i.e. p is supersingular), there is a companion p-adic analytic function L β (E, T ) with the same interpolation properties with α replaced by β.
is a matrix of p-adic analytic functions converging on the open p-adic unit disk.
Theorem 2.4. [Sp15, Theorem 2.14] Let p be an ordinary good prime. Then the first column log α of Log α,β (1 + T ) converges and we have for two p-adic analytic functions converging on the closed p-adic unit disk L ♯ (E, T ) and L ♭ (E, T ).
We remark that these L ♯ (E, T ) and L ♭ (E, T ) are not uniquely defined in the ordinary case, but their values at T = 0 are unique: when p is even.
Proof. This follows from the table before Proposition 6.14. in [Sp12] in the supersingular case and the one before Conjecture 5.18 in [Sp15] for the general case. Q.E.D.
3. The behavior at T = 0: A criterion for non-zero rank The goal of this short section is to prove: Theorem 3.1. Assume property ( * ) below holds and that X( Let T p (E) be the Tate module for E.
Here is property ( * ): The composite of natural maps is not zero. Here, the limit is taken with respect to corestriction, and S is the product of the bad reduction primes and p.
We remark that property ( * ) should always be true.
Proof of Theorem. The arguments of Kurihara and Pollack of [KP07, Section 1.4] almost work with the appropriate modifications. In terms of notation, we write Col(z) = (Col ♯ (z), Col ♭ (z)) with the Coleman maps from [Sp12] instead of the functions h z (T ) and k z (T ). We note that their arguments work in spite of having to take into account the possibility of one of  Proof. We know that at least one of L ♯ (E, T ) or L ♭ (E, T ) is non-zero by [Sp12, Proposition 6.14], so the theorem tells us they both have to be. Q.E.D.

The behavior at T = 0: p-adic versions of the BSD conjectures
The goal of this section is to formulate p-adic versions of BSD in terms of the vector (L ♯ (E, T ), L ♭ (E, T )) in the supersingular case. We thus assume that p is supersingular for this section.
4.1. Dieudonné modules and p-adic heights. The Dieudonné module is the following two-dimensional Q p -vector space: There is a Frobenius endomorphism ϕ which acts on D p (E) linearly. We refer the reader to [BPR93, Paragraph 2] for a concrete definition, but let us record that its characteristic polynomial is Y 2 − ap p Y + 1 p , as opposed to the definition of [MST06] or [Ke01] (where it is Y 2 −a p Y +p). This vector space admits a basis ω and ϕ(ω), where ω is the invariant/Néron differential of E. We want to define eigenvectors ν A := ν 1 α and ν B := ν 1 β of ϕ with eigenvalues 1 α and 1 β which live in the Q p (α)-vector space D p (E)(α) := Q p (α) ⊗ H 1 dR (E/Q). Definition 4.1. We define (scale) both eigenvectors as follows:  Given a globally minimal Weierstrass equation over Z y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 for E, recall that the associated Néron/invariant differential is ω = dx 2y+a 1 x+a 3 (see e.g. [Sil, Chapter III.1]). The Q-vector space H 1 dR (E/Q) admits a basis {ω, xω} and is equipped with a canonical alternating bilinear form [·, ·] so that [ω, xω] = 1. We extend it linearly to the Dieudonné modules above and denote these extensions by [·, ·] as well.
The bilinear form associated to this height function has values in Q p (resp.Q p (α)): Definition 4.5. Let Reg ν be the discriminant of this height pairing on E(Q)/E(Q) tors .
Definition 4.6. Let ν A and ν B be as above. We define normalized height functionŝ Denote by r(E) the rank of E(Q). In the supersingular case, Perrin-Riou defines the regulator Reg BP R p (E/Q) as the unique element in D p (E) so that for any ν ∈ D p (E) with ν ∈ Q p ω, we have 1  Proof. When r(E) = 0, this follows from the fact that Reg 1 α = 1 and Reg 1 β = 1.
When r(E) > 0, note that and similarly, [ω,ν B ] r−1 . Since ν A and ν B form a basis for D p (E)(α), linearity tells us that the property described in equation (1)  (1) We have r an = r(E).
Remark 4.10. Note that while the objects in the second part of this conjecture depend on the choice of Weierstraß equation, their coordinates with respect to the basis {ν α , ν β } don't. In fact, one can formulate Bernardi's and Perrin-Riou's conjecture in a form that resembles more closely that of the one given by Mazur, Tate, and Teitelbaum: Conjecture 4.11 (Equivalent formulation of above). Let r an be as in Lemma 4.12 below, and L * α and L * β be the leading coefficients in the Taylor expansion. Then (1) r an = r(E).
This version can be found in [Co04, Conjecture 0.12], where it is attributed to Mazur, Tate and Teitelbaum.
Proof. The same proof as in [Po03, Lemma 6.6] works.

4.3.
A version of the conjectures via L ♯ and L ♭ in the supersingular case. We now reformulate Conjecture 4.9 using L ♯ and L ♭ .
Definition 4.13. We call − → L p (E, T ) := − → L p := (L ♯ , L ♭ ) the p-adic L-vector of E, and denote by r ♮ p the minimum of the orders of vanishing of L ♯ and L ♭ . We would now like to find a pair of elements ν ♯ and ν ♭ in D p (E) that give rise to regulators corresponding to our p-adic L-functions. Recall that Definition 4.14. Let Z := Log α,β (1 + T )| T =0 = Log α,β (1). We define be the regulators for the normalized heights associated to N ♯ and N ♭ . Also, we let We are now ready to give our p-adic version of BSD: Conjecture 4.17 (Tandem p-adic BSD). Let E be an elliptic curve and p a prime of good supersingular reduction. Denote by − → L * p the first non-zero leading Taylor coefficient around (1) We have r ♮ p = r(E).
Remark 4.18. The term Reg ♮ p (E/Q) is independent from the choice of Weierstraß equation. This follows from the proof of part 2 of Theorem 4.19, which only compares coordinates with respect to the basis ν α , ν β .
Theorem 4.19. This conjecture is equivalent to that of Bernardi and Perrin-Riou (i.e. Conjecture 4.9).
Definition 4.20. Let r = r(E) > 0. Given a vector ν ∈ D p (E) (or in D p (E)(α)) that is not a linear multiple of ω, we put Proof of equivalence for part 1. This follows from Lemma 4.12 and the product rule. Q.E.D.
Proof for part 2. From the equivalence of part 1 and the product rule, we have . But we also have, for r > 0, and Reg ν is linear in ν, and by Lemma 4.15, this is equal to The rest follows from explicit calculation of the factors preceding the regulators. Q.E.D.
What is know so far is the following theorem of Kato:  (1) We have r an = r(E).
Remark 4.25. We encounter the term Reg 1 β (rather than Reg 1 α ) because of our choice of Frobenius ϕ = F p , where F is the Frobenius as chosen in [MST06] or [Ke01]. The regulator comes from the normalized height associated to the unit-eigenvector α of F on D p (E), so that the eigenvalue for ϕ becomes α p = 1 β . 2 In the ordinary case, recall that L ♯ and L ♭ are not well-defined, but their values at T = 0 are. In particular this means that when a p = 2 and p is odd, we may have L ♭ (E, 0) = 0 while L(E, 1) = 0, reminiscent of an extra zero phenomenon. This leads us to ask: Where does this extra-zero phenomenon come from?

The greatest common divisor
We now generalize and give some evidence for the following conjecture found in [KP07, Problem 3.2].
Recall that d n denoted the normalized jump in the ranks at the nth level of the cyclotomic tower: Conjecture 5.1 (The problem of Kurihara and Pollack). Let E/Q be an elliptic curve so p is an odd prime of good supersingular reduction and a p = 0. Then Note that this is an equality of ideals, since the greatest common divisor of two functions of Q ⊗ Z p [[T ]] is only well-defined as a Z p [[T ]]-ideal. We can give the following proposition: Proposition 5.2. Let E/Q be an elliptic curve and p a prime of good supersingular reduction. For some polynomial P X (E, T ) with P X (E, ζ p n − 1) = 0 for n 0, where ǫ an n − 1 = d an n − 1 or ǫ an n − 1 = d an n . Convention 5.3. Given a vector (f (T ), g(T )) of p-adic analytic functions, we define its order of vanishing at s by ord T =s (f (T ), g(T )) := min(ord T =s f (T ), ord T =s g(T )).
Corollary 5.5. The exact power of T dividing gcd (L ♯ (E, T ), L ♭ (E, T )) is T r an .
In view of the problem of Kurihara and Pollack (Conjecture 5.1), we make the following conjecture: Conjecture 5.7. Let E/Q be an elliptic curve, and p a good supersingular prime. Then