Non-commutative p-adic L-functions for supersingular primes

# Non-commutative p-adic L-functions for supersingular primes

## Abstract.

Let be an elliptic curve with good supersingular reduction at with . We give a conjecture on the existence of analytic plus and minus -adic -functions of over the -cyclotomic extension of a finite Galois extension of where is unramified. Under some technical conditions, we adopt the method of Bouganis and Venjakob for -ordinary CM elliptic curves to construct such functions for a particular non-abelian extension.

###### Key words and phrases:
non-commutative Iwasawa theory; elliptic curves; supersingular primes; -adic -functions
###### 2000 Mathematics Subject Classification:
11R23 (primary), 11F67 (secondary)
The author is supported by a CRM-ISM fellowship.

## 1. Introduction

Let be an elliptic curve with good ordinary reduction at an odd prime . Coates et al. [CFK05, CH01] have recently developed the framework of the Iwasawa theory of over a -adic Lie extension of that contains the -cyclotomic extension . Let and . It is predicted that there exists a -adic -function , where is an appropriate localisation of an Iwasawa algebra of and interpolates the complex -values of in the following sense. For all Artin representations of , is expected to satisfy

 (1) LG,E(ρ)=ep(ρ)ufp(ρ)×Pp(ρ,u−1)Pp(ρ∨,w−1)×LR(E,ρ∨,1)Ωd+(ρ)+Ωd−(ρ)−.

Here, denotes the local epsilon factor at , and are respectively the -adic unit and non-unit roots of the quadratic , is the polynomial describing the Euler factor of at and is the set consisting of the prime and all primes at which has multiplicative reduction. We shall review some of the notation in §2. The main conjecture predicts that such a -adic -function, should it exist, generates the characteristic ideal of the dual Selmer group , which is conjectured to lie in the -category (see §2.4 below).

The existence of would generalise the work of Mazur and Swinnerton-Dyer [MSD74], where they constructed an element that interpolates the -values of twisted by finite characters of . More precisely, if is a character on with conductor , then

 LΓ,E(χ)=τ(χ)un×L(E,χ−1,1)Ω+,

where denotes the Gauss sum of .

When has good supersingular reduction, no such element exists. Instead, Amice and Vélu [AV75] constructed two admissible -adic -functions and , one for each of the two roots, and , to . Even though they have similar interpolating properties as their counterpart in the ordinary case, they do not lie in . Pollack [Pol03] resolved this by decomposing and into linear combinations of two elements when and it has been generalised to the case where by Sprung [Spr09]. We shall concentrate on the former case in this paper. Pollack’s -adic -functions exhibit the following interpolating properties. Let be a character on of conductor . If is even,

 (2) L+Γ,E(χ)=τ(χ)ω+(χ)×L(E,χ−1,1)Ω+,

whereas if is odd,

 (3) L−Γ,E(χ)=τ(χ)ω−(χ)×L(E,χ−1,1)Ω+.

Here, and are some non-zero factors that come from the plus and minus logarithms defined in [Pol03]. It is shown in op. cit. that are uniquely determined by (2) and (3) respectively. We shall review some of the details of Pollack’s work in §3.1.

The main goal of this paper is to formulate a conjecture on the existence of two elements with interpolating properties similar to (2) and (3) for , where is the -cyclotomic extension of a finite Galois extension of in which is unramified. Let be an irreducible Artin representation of with . We define in §3.2 the factors and , depending on the parity of . We then go on to predict that there exist that satisfy

 (4) L+G,E(ρ)=ep(ρ)ω+(ρ)×LR(E,ρ∨,1)Ωd+(ρ)+Ωd−(ρ)−

if is even, whereas for odd ,

 (5) L−G,E(ρ)=ep(ρ)ω−(ρ)×LR(E,ρ∨,1)Ωd+(ρ)+Ωd−(ρ)−.

Here is the set of primes as defined in the ordinary case above. Note that the Euler factors at are trivial for the representations which we consider, which explains why they are not present in our conjectural formulae. Roughly speaking, and are each interpolating the -values of twisted by “half” of all irreducible Artin representations of . Despite these seemingly weaker properties, we show that, as in the ordinary case, if exist, they are uniquely determined by (4) and (5) respectively as elements of modulo the kernel of a determinant map.

Kobayashi [Kob03] defined the plus and minus Selmer groups for over and formulate a main conjecture predicting that the characteristic ideals of their Pontryagin duals should be generated by (this has been proved by Pollack and Rubin [PR04] when has complex multiplication). In [LZ11], we have generalised Kobayashi’s construction to arbitrary -adic Lie extensions. Therefore, analogous to the ordinary case, the existence of would allow us to formulate a main conjecture for these plus and minus Selmer groups, which relates the characteristic ideals of to . See §3.3 for details.

In [BV10], Bouganis and Venjakob proved the existence of the -adic -function that satisfies (1), where is an elliptic curve with complex multiplication that has good ordinary reduction at and assuming that the Selmer group satisfies the -conjecture. Their construction makes use of the -variable -adic -function of Yager [Yag82], which does not exists in the supersingular case. We have nonetheless managed to adopt their method, together with some of the ideas from [KPZ10], to construct two elements that satisfy (4) and (5) respectively under some technical assumptions when is an elliptic curve with complex multiplication by which has good supersingular reduction at and is the Galois group of the extension of by , where is an abelian extension of in which is unramified. Note that could be abelian, but there do exist examples for which this is not the case, e.g. and . The details of our construction are given in §4.

## Acknowledgement

The idea of this paper was originally developed when the author was a member of Monash University. He is extremely grateful for many helpful discussions with Daniel Delbourgo and Lloyd Peters on the subject while he was there. He is also indebted to David Loeffler for pointing out a few mistakes in an earlier version of this paper and to Henri Darmon and Alex Bartel for answering many of his questions. Finally, the author would like to thank the anonymous referee for suggestions that help improving the paper.

## 2. Notation and setup

Throughout this paper, is an odd prime. If is a field of characteristic , either local or global, denotes its absolute Galois group, the -cyclotomic character on and the ring of integers of . We write for the complex conjugation in .

We let denote the -cyclotomic extension of . The Galois group is written as . When , we simply write for . We fix a topological generator of .

From now on, we fix a finite Galois extension of with Galois group . We assume that is unramified in . Let . Our assumption on implies that . Hence, is Galois. We write for its Galois group. Then

 (6) G≅H×Γ,

which allows us to view both and as subgroups of (e.g. could be the Galois group studied in [Har10], where is the group of upper-triangular unipotent matrices over ). We fix a family of subgroups of , written as , such that the following is true.

###### Condition 2.1.

For all irreducible representations of , the trace of is equal to a -linear combinaton of representations of the form where and is an one-dimensional character on .

###### Remark 2.2.

Note that such a family exists by Brauer’s theorem on induced characters.

By (6), for all irreducible representations of , there exists an one-dimensional character of such that the trace of is equal to a -linear combinaton of representations of the form

 IndGUi×Γ(χρ,i⊗ρ0)≅(IndGUiχρ,i)⊗ρ0

where and is an one-dimensional character on .

We write for all .

### 2.2. Iwasawa algebras and power series

Given a finite extension of and a -adic Lie group , denotes the Iwasawa algebra of over , i.e.

 lim←−NOK[G/N],

where the inverse limit runs over the open normal subgroups of . When (so ), we suppress from the notation and write for . We denote by .

Let . We define

 Hr={∑n≥0cnXn∈Cp[[X]]:supn|cn|pnr<∞ },

where is the -adic norm on such that . We write . In other words, the elements of (respectively ) are the power series in (respectively ) over with growth rate .

Given a subfield of , we write and similarly for . In particular, .

If and , we write

 h(λ)=∑n≥0cn(λ(γ)−1)n∈Cp.

More generally, if is an Artin representation on , then may be decomposed into a finite sum of characters on , say

 ρ≅r⨁i=1χi.

We then write

 h(ρ)=r∏i=1h(χi).

### 2.3. Artin representations

Let be a finite dimensional vector space over . If is an Artin representation, we write (respectively ) for the dimension of (respectively the subspace of on which the complex conjugation acts as ). The contragredient representation of is denoted by .

We now review the definition of the local epsilon factor at (see [Tat79, §3] and [DD07, §6.10] for details). It depends on a choice of Haar measure and an additive character on . In this article, we take the canonical measure on with and the additive character that sends to for . It is multiplicative in the sense that , so it is enough to define for irreducible .

If is one-dimensional, we may factor into , where is an unramified character and is a Dirichlet character of conductor . In this case, the epsilon factor is defined to be

 ep(ρ)=ρ0(p)n×τ(ρ′),

where denotes the Gauss sum of . If for some character on , where is a number field, we may define for each place of above in a similar way. The epsilon factor of at is then defined to be

 ep(ρ)=ep(IndQF(1))∏p|pep(χ)ep(1).

### 2.4. A canonical Ore set

We recall the definition of an Ore set from [CFK05]. Let be the ring of integer of a finite extension of . If and are as in §2.1, we define

 S(O)={f∈ΛO(G):ΛO(G)/ΛO(G)f is a finitely % generated ΛO(H)-module}

and . We write for the localisation of at .

Let be the subgroups as fixed in Section 2.1. For each , we have a natural map

 θi:K1(ΛO(G)S∗)→K1(ΛO(Ui×Γ)S∗)→K1(ΛO(Ui/Vi×Γ)S∗)=ΛO(Ui/Vi×Γ)×S∗.

Let be the additive group generated by isomorphic classes of -adic representations of . There is a determinant map

 Det:K1(ΛO(G)S∗) → Maps(Irrp(G),¯¯¯¯¯¯Qp∪{∞}) x ↦ (ρ↦x(ρ)).

We write for the category of all finitely generated -modules which are -torsion. There is a connecting map

 ∂G:K1(Λ(G)S∗)→K0(MH(G)).

It is surjective when has no -torsion. Given an element in , a characteristic element for is any such that .

## 3. Conjectures

### 3.1. Pollack’s plus and minus p-adic L-functions

We first recall Pollack’s construction of plus and minus -adic -functions for the -cyclotomic extension of in [Pol03]. The plus and minus logarithms are defined as follows.

###### Definition 3.1.

Let be an integer, define

 log+r = r∏s=01p∞∏n=1Φ2n(κ(γ)−sγ)p, log−r = r∏s=01p∞∏n=1Φ2n−1(κ(γ)−sγ)p,

where denotes the -cyclotomic polynomial.

###### Lemma 3.2.

Let be a character on . Then if and only if , where and is a Dirichlet character whose conductor is an odd power of . Similarly, if and only if , where and is a Dirichlet character whose conductor is an even power of .

###### Proof.

[Pol03, Lemma 4.1]. ∎

###### Remark 3.3.

Both and are elements of as given by [Pol03, Lemma 4.5].

Let be a normalised eigen-newform of weight , level and Nebentypus character . Throughout, we assume . We write for the completion of the coefficient field at a prime above . Let be a root of with . Amice and Vélu [AV75] constructed (written as in the introduction) with the following interpolating properties. For all integers and Dirichlet characters of conductor ,

 (7) Lp,α(θκj)=eα(θ,j)×pn(j+1)j!τ(θ−1)(−2πi)j×L(f,θ−1,j+1)Ωδf,

where , denotes the Gauss sum of ,

 eα(θ,j)=1αn(1−θ−1(p)ϵ(p)pk−2−jα)(1−θ(p)pjα)

and are some choice of periods. Moreover, is uniquely determined by its values at as given by (7).

###### Remark 3.4.

We are only considering the -cyclotomic extension here, rather than the whole extension by -power roots of unity as studied in [AV75]. In particular, is always .

###### Theorem 3.5.

Let and be as above with . There exist (written as in the introduction) such that

 Lf,α=log+k−1L+f+αlog−k−1L−f.
###### Proof.

This is the main result of [Pol03]. ∎

In particular, if and are the two roots to , we have and

 (8) L+f = Lp,α1+Lp,α22log+k−1, (9) L−f = Lp,α2−Lp,α1(α2−α1)log−k−1.

Therefore, we can readily combine (7) with (8) and (9) to obtain the interpolating formulae of and respectively as given below.

###### Lemma 3.6.

Let be an integer and a Dirichlet conductor . Write as in (7). If is even, then

 L+f(θκj)=1(−ϵ(p)pk−1)n/2log+k−1(θχj)×pn(j+1)j!τ(θ−1)(−2πi)j×L(f,θ−1,j+1)Ωδf,

whereas if . For all odd , we have

 L−f(θκj)=1(−ϵ(p)pk−1)(n+1)/2log−k−1(θχj)×pn(j+1)j!τ(θ−1)(−2πi)j×L(f,θ−1,j+1)Ωδf.

Moreover,

 L+f(κj) = 1−p−1log+k−1(χj)×j!(−2πi)j×L(f,j+1)Ωδf, L−f(κj) = p−j−1+ϵ(p)−1pj−k+1log−k−1(χj)×j!(−2πi)j×L(f,j+1)Ωδf.
###### Definition 3.7.

Let be a Dirichlet character of conductor . For even , define

 ω+(θ)=(−p)n/2log+1(θ).

For odd , define

 ω−(θ)=(−p)(n+1)/2log−1(θ).

For a weight modular form, we have the following simplified version of Lemma 3.6.

###### Lemma 3.8.

Assume that is of weight . Let be a Dirichlet conductor . If is even, then

 L+f(θ)=τ(θ)ϵ(p)n/2ω+(θ)×L(f,θ−1,1)Ω+f,

whereas if . For all odd , we have

 L−f(θ)=τ(θ)ϵ(p)(n+1)/2ω−(θ)×L(f,θ−1,1)Ω+f.

Moreover,

 L+f(1) = (p−1)×L(f,1)Ω+f, L−f(1) = (1+ϵ(p)−1)×L(f,1)Ω+f.
###### Proof.

Recall from Remark 3.4 that , so we have . The result is then immediate from Lemma 3.6. See also [Kob03, (3.4)-(3.7)]. ∎

### 3.2. Conjectural analytic p-adic L-functions

In this section, we fix an elliptic curve defined over . We assume that has good supersingular reduction at with . As in the introduction, we write for the set consisting of the prime and the primes where has multiplicative reduction. Let and denote the real and complex periods of respectively. Let and be as defined in Section 2.1.

###### Definition 3.9.

Let be an Artin representation on . We say that is of even (respectively odd) -conductor if, via (6),

 ρ≅ρ0⊗ρ′

for some representation of and some one-dimensional character of whose conductor is an even (respectively odd) power of .

###### Remark 3.10.

By (6), an irreducible representation of is of the form where is an irreducible representation of and is an one-dimensional character of . In particular, if , it has either even or odd -conductor. Moreover, all Artin representations of that have even (respectively odd) -conductors are direct sums of such irreducible representations.

###### Lemma 3.11.

Let be an Artin representation on . If is of even -conductor, then

 ω+(ρ|Γ)=ω+(ρ′)d(V)≠0.

Similarly, if is of odd -conductor,

 ω−(ρ|Γ)=ω−(ρ′)d(V)≠0.
###### Proof.

If is of even or odd -conductor, we have

 ρ|Γ≅(ρ′)⊕d(V).

By definition, and differ by a power of . Hence the result by Lemma 3.2. ∎

As in Definition 3.7, we make the following definition to simplify our notation.

###### Definition 3.12.

Let be an Artin representation on that is of even -conductor, we write

 ω+(ρ)=ω+(ρ|Γ),

which is non-zero by Lemma 3.11. If on the other hand is of odd -conductor, we write

 ω−(ρ)=ω−(ρ|Γ),

which again is non-zero by Lemma 3.11.

We now formulate our conjecture on the existence of plus and minus -adic -functions of over .

###### Conjecture 3.13.

There exist two elements for the ring of integer of some finite unramified extension of such that and

 (10) LνG,E(ρ)=ep(ρ)ων(ρ)×LR(E,ρ∨,1)Ωd+(ρ)+Ωd−(ρ)−

for all Artin representations on that has even (respectively odd) -conductor with (respectively ).

This is analogous to [CFK05, Conjecture 5.7].

###### Remark 3.14.

Remark 3.10 implies that we might assume that the representations considered in the statement of Conjecture 3.13 are irreducible.

We now show that despite having much weaker interpolating properties than their counterpart in the good ordinary case, (10) uniquely determines modulo .

###### Theorem 3.15.

If Conjecture 3.13 holds, are uniquely determined by (10) modulo the kernel of .

###### Proof.

For all and monomial representations on , with , we have

 Det(ξ)(ρ)=ξ(ρ)=θi(ξ)(χρ).

Hence, by Condition 2.1, is uniquely determined by its image under the map . In other words,

 ker(Det)=ker(∏i∈Iθi).

Therefore, it suffices to show that (10) uniquely determines an element in

 K1(ΛO(G)S∗)/ker(∏i∈Iθi).

A character on decomposes into , where and . Therefore, by Weierstrass’ preparation theorem, an element in is uniquely determined by its values at characters of the form for all and an infinite number of . Artin representations induced from characters on that have even (or odd) -conductor provide such a set of characters because Definition 3.9 does not impose any restrictions on and can send to an infinite number of primitive roots of unity. The values predicted by (10) therefore uniquely determine for all , hence the result.

### 3.3. Main conjecture

In [LZ11], we have defined the signed Selmer groups , which are subgroups of the usual Selmer group . Let be their respective Pontryagin duals. We conjecture that the following holds.

###### Conjecture 3.16.

The dual Selmer groups belong to the category .

This corresponds to [CFK05, Conjecture 5.1]. If Conjecture 3.16 holds and contains no -torsion, then there exist characteristic elements as discussed in § 2.4. We may then formulate a main conjecture that relates these characteristic elements to the conjectural analytic -adic -functions predicted by Conjecture 3.13.

###### Conjecture 3.17 (Main conjecture).

Let be the natural homomorphism. Assume that does not contain an element of order and both Conjectures 3.13 and 3.16 hold. Then

 L±G,E≡i(ξX±(E/F))modi(K1(ΛO(G))).

This is analogous to [CFK05, Conjecture 5.8].

## 4. A special case

In this section, we assume that our elliptic curve has complex multiplication by , where is an imaginary quadratic extension of . If is a character on , we write for the character that sends to .

We recall that a Grossencharacter of is simply a continuous homomorphism , where is the idele class group of . It has complex -function

 L(ϕ,s)=∏v(1−ϕ(v)N(v)−s)−1,

where the product runs through the finite places of at which is unramified, is the image of the uniformiser of under and is the norm of . We say that is of type for some if the restriction of to the archimedean part of is of the form . By Class Field Theory, we may equally view as a character on the Galois group .

Since we assume that has complex multiplication, we have . Moreover, there exist a Grossencharacter be of type over and a weight modular form such that

 L(E,s)=L(ϕ,s)=L(fϕ,s).

Then . We continue to assume that has good supersingular reduction at . This implies that is inert in and is automatically .

Let be a finite abelian extension of in which is unramified. Then, . Moreover, is abelian and

 Gal(F∞/K)≅Gal(F/K)×Gal(K∞/K).

Let , and . We further assume that is Galois and write . Then

 (11) G≅Δ⋉G.

As remarked in the introduction, can be either abelian or non-abelian, depending on whether acts on trivially.

For the rest of this section, we shall show under some technical conditions that there exist two elements satisfying the interpolating properties predicted by Conjecture 3.13 for some finite extension of . Let us briefly outline our strategy here. In [BV10], Bouganis and Venjakob constructed a non-commutative -adic -function over the extension for a -ordinary elliptic curve with complex multiplication by . They made use of the -variable -adic -function of Yager [Yag82] for the extension and consider its image in under a natural map, which turns out to satisfy (1) up to some correction factor . In the supersingular case, a corresponding -variable -adic -function has not yet been constructed. We instead construct plus and minus -adic -functions for the extension using ideas from [KPZ10]. Once this is done, we can apply the machineries developed by Bouganis and Venjakob to construct our desired elements for the extension .

Let be an one-dimensional character on . We write for the field . Note that is once again a Grossencharacter of type over . Hence, by [Rib77, Theorem 3.4], there exists a CM modular form such that . Its conductor is coprime to by our assumption on , so the plus and minus -adic -functions exist, as given by Theorem 3.5. Note that the periods are not unique, but we may choose to be for all by the following lemma.

###### Lemma 4.1.

For any choice of , there exists a constant such that .

###### Proof.

Given a modular form , let be the -vector space generated by the -values where runs through the set of Dirichlet characters.

By the algebraicity property of the periods, the lemma would follow from the inclusion

 Lfϕη⊂Fη⋅Lfϕ,

which is a consequence of the Fourier inversion formula for and Birch’s lemma (c.f. [Har87, §4]). ∎

Under our choice of periods, we define the following.

###### Definition 4.2.

Let and if , whereas if . Define

 LνG,ϕ=∑η∈^Aeη⋅¯¯¯η(p)δLνfϕ¯η∈ΛFA(Γ)[A],

where and is the idempotent