Kato's Euler System

Today I'll be talking about Kato's Euler System, which is a technical tool that was used to prove many deep results in the direction of the BSD conjecture. It was developed in Kato's 2004 Asterisque Paper. This paper is a 100+ page tour de force and would easily take several years to study fully. The purpose of this article is to give a high level overview of the main ideas of the paper.

A huge thank you to Fransesc Castella and David Loeffler for answering my questions about Kato's Euler Systems.

What is an Euler System?

Many problems in modern number theory revolve around the following theme: relate Galois cohomology classes to special values of LL-functions.

{Galois cohomology}{special values of L-functions}\left\{ \text{Galois cohomology} \right\} \leftrightarrow \{ \text{special values of $L$-functions} \}

For example, suppose you have an elliptic curve E/QE/\mathbb{Q}. On the left side of the diagram, you might want to study the Selmer group of EE over Q\mathbb{Q}. This is a group defined using Galois cohomology that essentially controls the arithmetic of EE. On the right side, you might want to study the Hasse-Weil LL-function L(E,s)L(E,s). Relating these two objects would give deep results in the direction of BSD.

An Euler System, simply put, is an object that links these two worlds. It is a technical tool that allows you to link Galois cohomology groups to special values of LL-functions.

{Galois cohomology}Euler Systems{special values of L-functions}\left\{ \text{Galois cohomology} \right\} \xleftrightarrow[\text{Euler Systems}] \, \{ \text{special values of $L$-functions} \}

Kato developed an Euler system and used it to prove the following amazing result (among many other results):

Theorem: (Kato) Let E/QE/\mathbb{Q} be an elliptic curve, and let L(E,s)L(E,s) be the Hasse-Weil LL-function of EE. If E(Q)E(\mathbb{Q}) is infinite, then L(E,1)=0L(E,1)=0.

In other words, "positive algebraic rank implies positive analytic rank." In the rest of the post, I will describe what Kato's Euler system is.

Kato's Euler System

Setup

For us, EE will be an elliptic curve over Q\mathbb{Q} and pp will be a prime of good reduction. We will write TpE=limnE[pn]T_pE = \varprojlim_n E[p^n] for the Tate module of EE. So TpET_pE is isomorphic to Zp×Zp\mathbb{Z}_p \times \Z_p as an abelian group and it has an action of Gal(Q/Q)\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}). Now define

VpE=TpEZpQp.V_pE = T_pE \otimes_{\mathbb{Z}_p} \mathbb{Q}_p.

Then VpEV_pE is a two-dimensional Qp\mathbb{Q}_p-vector space with an action of Gal(Q/Q)\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) inherited from the first factor TpET_pE of the tensor product. (We write VpEV_pE because the VV stands for Vector space.)

Kato's Euler System - the bottom layer

Kato considers the Galois cohomology group H1(Q,VpE)H^1(\mathbf{Q}, V_pE). With this group, Kato does two things. First, Kato defines a class zKatoH1(Q,VpE)z^{\text{Kato}} \in H^1(\mathbf{Q}, V_pE) called the Kato class.

Second, Kato defines a very mysterious map called the dual exponential map:

exp:H1(Qp,VpE)QpωE,\exp^*: H^1(\mathbf{Q}_p, V_pE) \to \mathbf{Q}_p \omega_E,

which is defined using pp-adic Hodge theory. We will not define explicitly and treat like a black box. Here ωE\omega_E is the invariant differential on EE.

Now let resp:H1(Q,VpE)H1(Qp,VpE)\text{res}_p: H^1(\mathbf{Q}, V_pE) \to H^1(\mathbf{Q}_p, V_pE) denote the restriction map. Kato showed that

exp(resp(zKato))=L{p}(E,1)ΩEωE.\exp^*(\text{res}_p(z^{\text{Kato}})) = \dfrac{L_{\{p\}}(E,1)}{\Omega_E}\omega_E.

The function L{p}(E,s)L_{\{p\}}(E,s) is the LL-function L(E,s)L(E,s) with the pp-th Euler Factor removed. And ΩE\Omega_E is the real Neron period of EE. This identity is called the explicit reciprocity law.

Kato's Euler System - the other layers

Kato realized that this class zKatoz^{\text{Kato}} is not alone, but is just the first class in an entire infinite collection of classes that interpolate LL-values in different ways.

For each prime power pmp^m, consider the cohomology group H1(Q(ζpm),VpE)H^1(\mathbf{Q}(\zeta_{p^m}), V_pE). Kato defined a class zpmKatoH1(Q(ζpm),VpE)z^{\text{Kato}}_{p^m} \in H^1(\mathbf{Q}(\zeta_{p^m}), V_pE). Kato also defined a dual exponential map using pp-adic Hodge theory:

exp:H1(Qp(ζpm),VpE)Qp(ζpm)ωE.\exp^*: H^1(\mathbf{Q}_p(\zeta_{p^m}), V_pE) \to \mathbf{Q}_p(\zeta_{p^m}) \, \omega_E.

The explicit reciprocity law here says: for any Dirichlet character χ:(Z/pm)×C×\chi: (\mathbf{Z}/p^m)^{\times} \to \mathbf{C}^{\times} of conductor pmp^m, we have

exp(σGal(Qp(ζpm)/Qp)resp(zpmKato)σ)=L{p}(E,χ,1)ΩE±ωE. \exp^* \left(\sum_{\sigma \in \text{Gal}(\mathbf{Q}_p(\zeta_{p^m})/\mathbf{Q}_p)} \text{res}_p(z^{\text{Kato}}_{p^m})^{\sigma} \right) = \dfrac{L_{\{p\}}(E,\chi, 1)}{\Omega_E^{\pm}}\omega_E.

Here resp:H1(Q(ζp),VpE)H1(Qp(ζp),VpE)\text{res}_p: H^1(\mathbf{Q}(\zeta_p), V_pE) \to H^1(\mathbf{Q}_p(\zeta_p), V_pE) denotes the restriction map. The periods ΩE+\Omega_E^{+} and ΩE\Omega_E^- are the real and imaginary periods of EE, respectively.

Kato's Euler System - the whole thing

The collection of classes {zpmKato}m1\{ z^{\text{Kato}}_{p^m} \}_{m \geq 1} called Kato's Euler System. The class zpmKatoz^{\text{Kato}}_{p^m} is called the pmp^m-th layer of Kato's Euler System.

Ok, this is a bit of a lie. Kato actually defined classes zmKatoz^{\text{Kato}}_m for every integer m1m \geq 1, not just prime powers pmp^m. But we only need the layers which are prime powers, so that's not a problem for us.

Construction of Kato's Euler System

In this section, I'll describe the construction of the bottom class of Kato's Euler system, the element zKato=zp0KatoH1(Q,VpE)z^{\text{Kato}} = z^{\text{Kato}}_{p^0} \in H^1(\mathbf{Q}, V_pE).

Modularity

Since every elliptic curve over Q\mathbf{Q} is modular, there exists a modular parametrization X0(N)EX_0(N) \to E, where NN is the conductor of EE. Compose this with the natural surjection X1(N)X0(N)X_1(N) \to X_0(N), to get a map

X1(N)EX_1(N) \to E.

This induces a map on the Jacobians: J1(N)EJ_1(N) \to E. This is turn induces a map on their Tate modules: Tp(J1(N))TpET_p(J_1(N)) \to T_pE . Tensoring with Qp\mathbb{Q}_p, we get a Galois equivariant:

Vp(J1(N))VpEV_p(J_1(N)) \to V_pE.

This induces a map on Galois cohomology, so we get a map which I'll call (a)(a):

H1(Q,Vp(J0(N)))(a)H1(Q,VpE)H^1(\mathbb{Q}, V_p(J_0(N))) \xrightarrow{(a)} \, H^1(\mathbb{Q}, V_pE).

Etale Cohomology

There is an isomorphism of GQG_{\mathbf{Q}}-modules:

Het1(Y1(N),Qp(1))Vp(J1(N)))H^1_{\text{et}}(\overline{Y_1(N)}, \mathbb{Q}_p(1)) \xrightarrow{\sim} \, V_p(J_1(N))),

where Het1H^1_{\text{et}} denotes the etale cohomology group and Y1(N)\overline{Y_1(N)} is the base-change of Y1(N)Y_1(N) to Q\overline{\mathbb{Q}}. This induces a map on Galois cohomology which we denote (b)(b):

H1(Q,Het1(Y1(N),Qp(1)))(b)H1(Q,Vp(J1(N))).H^1(\mathbb{Q}, H^1_{\text{et}}(\overline{Y_1(N)}, \mathbb{Q}_p(1))) \xrightarrow{(b)} \, H^1(\mathbb{Q}, V_p(J_1(N))).

A Spectral Sequence

The next step is to use a spectral sequence to get a map from H2H1(Q,H1())H^2 \to H^1(\mathbf{Q}, H^1(\cdot\cdot\cdot)). Precisely, consider the spectral sequence

E2i,j=Hi(Q,Hetj(Y1(N),)    Hi+j(Y1(N),).E_2^{i,j}=H^i(\mathbb{Q}, H^j_{\text{et}}(\overline{Y_1(N)}, - ) \implies H^{i+j}(Y_1(N), -).

And use a fact from etale cohomology: Heti(C,)=0H^i_{\text{et}}(C, -) = 0 for i2i \geq 2 whenever CC is an affine curve over an algebraically closed field. Combining these two facts1, I am told that one obtains a map:

H2(Y1(N),Qp(1))(c)H1(Q,Het1(Y1(N),Qp(1)).H^2(Y_1(N), \mathbb{Q}_p(1)) \xrightarrow{(c)} H^1(\mathbb{Q}, H^1_{\text{et}}(\overline{Y_1(N)}, \mathbb{Q}_p(1) ).

Crucially, in the domain of the map (c)(c), we consider the curve Y1(N)Y_1(N) over Q\mathbb{Q}, whereas in the target, we consider Y1(N)Y_1(N) base-changed to the algebraic closure Q\overline{\mathbb{Q}}. So the group H2(Y1(N),Qp(1))H^2(Y_1(N), \mathbb{Q}_p(1)) does not have a Galois action. The benefit, therefore, of working over Q\mathbb{Q} is that we can use tools from geometry (namely, the Kummer map) to produce classes in H2(Y1(N),Qp(1))H^2(Y_1(N), \mathbb{Q}_p(1)).

Then we are going to apply a Tate twist to get a map (d)(d):

H2(Y1(N),Qp(2))(d)H2(Y1(N),Qp(1))H^2(Y_1(N), \mathbb{Q}_p(2)) \xrightarrow{(d)} H^2(Y_1(N), \mathbb{Q}_p(1))

Cup Product

There is a cup-product map in etale cohomology:

H1(Y1(N),Qp(1))×H1(Y1(N),Qp(1))(e)H2(Y1(N),Qp(2))H^1(Y_1(N), \mathbb{Q}_p(1)) \times H^1(Y_1(N), \mathbb{Q}_p(1)) \xrightarrow{(e)} H^2(Y_1(N), \mathbb{Q}_p(2)).

A useful mnemonic: when you apply the cup product, you add the twists (so Qp(1)\mathbb{Q}_p(1) and Qp(1)\mathbb{Q}_p(1) becomes Qp(2)\mathbb{Q}_p(2)) and add the degrees (so H1×H1H^1 \times H^1 becomes H2H^2).

The Kummer Map

The Kummer Map is a map κ:O(Y1(N))×H1(Y1(N),Qp(1))\kappa: \mathcal{O}(Y_1(N))^{\times} \to H^1(Y_1(N), \mathbb{Q}_p(1)). Applying the map κ×κ\kappa \times \kappa, we obtain a map (f)(f):

(O(Y1(N))×)2(f)H2(Y1(N),Qp(1))2\left( \mathcal{O}(Y_1(N))^{\times} \right)^{2} \xrightarrow{(f)} H^2(Y_1(N), \mathbb{Q}_p(1))^2.

Putting it all together

Composing the maps (a)(a) through (f)(f), we have:

(O(Y1(N))×)2(f)H2(Y1(N),Qp(1))2(e)H2(Y1(N),Qp(2))(d)H2(Y1(N),Qp(1))(c)H1(Q,Het1(Y1(N),Qp(1))(b)H1(Q,Vp(J1(N)))(a)H1(Q,VpE).\begin{align*} \left( \mathcal{O}(Y_1(N))^{\times} \right)^{2} &\xrightarrow{(f)} H^2(Y_1(N), \mathbb{Q}_p(1))^2 \\ &\xrightarrow{(e)} H^2(Y_1(N), \mathbb{Q}_p(2)) \\ &\xrightarrow{(d)} H^2(Y_1(N), \mathbb{Q}_p(1))\\ &\xrightarrow{(c)} H^1(\mathbb{Q}, H^1_{\text{et}} (\overline{Y_1(N)}, \mathbb{Q}_p(1) ) \\ &\xrightarrow{(b)} \, H^1(\mathbb{Q}, V_p(J_1(N))) \\ &\xrightarrow{(a)} \, H^1(\mathbb{Q}, V_pE). \end{align*}

This is confusing! To recap, the maps are given as follows:

  • (a)(a) is induced by the map Vp(J1(N))Vp(E)V_p(J_1(N)) \to V_p(E),
  • (b)(b) is induced by the isomorphism Het1(Y1(N),Qp(1)Vp(J1(N))H^1_{\text{et}} (\overline{Y_1(N)}, \mathbb{Q}_p(1) \simeq V_p(J_1(N)),
  • (c)(c) comes from a certain spectral sequence,
  • (d)(d) is a twist,
  • (e)(e) is the cup-product map in etale cohomology, and
  • (f)(f) is the (square of the) Kummer map.

To produce an element of H1(Q,VpE)H^1(\mathbf{Q}, V_pE), Kato defines an explicit element of (O(Y1(N))×)2\left( \mathcal{O}(Y_1(N))^{\times} \right)^{2} and lets zKatoz^{\text{Kato}} be its image under the maps (a)(a) through (f)(f).

Kato-Seigel Units

Elements of O(Y1(N))×\mathcal{O}(Y_1(N))^{\times} are called modular units, and we can write some of them down using nothing but classical 19-th century elliptic function theory.

Definition. Let N1N \geq 1 and 0a,b<N0 \geq a,b < N be integers. For each pair (aM,bM)Q2(0,0)\left( \dfrac{a}{M}, \dfrac{b}{M} \right) \in \mathbb{Q}^2 \setminus (0,0), define the function gaM,bM:HCg_{\frac{a}{M}, \frac{b}{M}}: \mathbb{H} \to \mathbb{C} as follows:

gaN,bN(z)=qwn=0(1qn+a/NζNb)n=1(1qna/NζNb),\begin{equation*} g_{\frac{a}{N}, \frac{b}{N}}(z) = q^w \prod_{n=0}^{\infty} \left(1-q^{n + a/N} \zeta_N^b \right) \prod_{n=1}^{\infty} \left(1-q^{n - a/N} \zeta_N^{-b} \right), \end{equation*}

where q=e2πizq = e^{2\pi i z} and w=112aN+a22N2w = \frac{1}{12} - \frac{a}{N} + \frac{a^2}{2N^2}.

This is well-defined (independent of the choice of common denominator NN). We would like to say that it is modular of level NN, but this is not quite true. Acting on it by an element of Γ(N)\Gamma(N) multiplies it by a root of unity so it defines an element of O(Y(N))ZQ\mathcal{O}(Y(N)) \otimes_{\Z} \mathbf{Q}. We can kill the denominator by making a very simple modification:

Definition. Let c>1c>1 be an integer coprime to 6N6N. Put

cgaN,bN=(gaN,bN)c2gcaN,cbN\begin{equation*} {}_cg_{\frac{a}{N}, \frac{b}{N}} = \dfrac{(g_{\frac{a}{N}, \frac{b}{N}})^{c^2}}{g_{\frac{ca}{N}, \frac{cb}{N}} } \end{equation*}

Then cgaN,bN(z){}_cg_{\frac{a}{N}, \frac{b}{N}}(z) is Γ(N)\Gamma(N)-invariant so it belongs to O(Y(N)C)×\mathcal{O}(Y(N)_{\mathbf{C}})^{\times}. However, we can do better: it actually descends to the number field Q(ζN)\mathbf{Q}(\zeta_N). Since the affine curve Y(N)Y(N) is defined over Q(ζN)\mathbf{Q}(\zeta_N), we have:

Proposition. The units cgaN,bN(z){}_cg_{\frac{a}{N}, \frac{b}{N}}(z) belong to O(Y(N))×\mathcal{O}(Y(N))^{\times}.

Definition of zKatoz^{\text{Kato}} (wrong version)

I say "wrong version" because this is morally correct and conveys the overall idea, but is technically false. The basic idea is that Kato picks parameters M,NM,N appropriately and gets a pair of modular units cg1M,0,dg0,1N{}_cg_{\frac{1}{M}, 0} , {}_dg_{0, \frac{1}{N}} in (O(Y1(N))×)2\left( \mathcal{O}(Y_1(N))^{\times} \right)^{2}. Then Kato lets zKatoz^{\text{Kato}} be its image under the maps (a)(a) through (f)(f).

Definition of zKatoz^{\text{Kato}} (correct version)

The full details here are very overwhelming (at least I found them so). From this point onwards, I'm skipping over a bunch of technical details to get the main point across. If you'd like technical details, then I'd suggest looking Sections 2 and 5 of Kato's paper as a reference.

Kato fixes two integers c,d>1c,d > 1 such that (cd,6N)=1(cd, 6N)=1. He considers a pair of Seigel units (cg1M,0,dg0,1N)\left({}_cg_{\frac{1}{M}, 0} , {}_dg_{0, \frac{1}{N}} \right) where M,NM,N are chosen appropriately. After doing this, Kato obtains an element which he denotes c,dzM,N=(cg1/M,0,dg0,1/N){}_{c,d}z_{M,N} = \left({}_c g_{1/M, 0}, {}_d g_{0, 1/N} \right). This element c,dzM,N{}_{c,d}z_{M,N} belongs to (O(Y(M,N))×)2\left(\mathcal{O}(Y(M,N))^{\times} \right)^2, where Y(M,N)Y(M,N) is a certain "two-level" modular curve and M,NM,N are parameters to be chosen later.

Then for any matrix ξSL2(Z)\xi \in \textrm{SL}_2(\mathbf{Z}), Kato "twists" this pair by ξ\xi and applies a norm map down to Y1(N)Y_1(N) to get an element which Kato denotes by

c,dz1,N,m(k=2,r=1,r=1,ξ,S=prime(Np))(O(Y1(N))×)2\begin{equation*} {}_{c,d}z_{1,N,m}(k=2,r=1,r'=1,\xi,S=\text{prime}(Np)) \in \left(\mathcal{O}(Y_1(N))^{\times} \right)^2 \end{equation*}

(See page 153 of Kato's paper for the precise definition).

The image of the above element c,dz1,N,1(2,1,1,ξ,S){}_{c,d}z_{1,N,1}(2,1,1,\xi,S) under the maps (a)(a) through (f)(f) is denoted (see Sections 8.9, 8.11 of Kato's paper)

c,dz(f,ξ)=c,dzm=1(p)(f,r=1,r=1,ξ,S=prime(pN)).\begin{equation*} {}_{c,d} z(f, \xi) = {}_{c,d} z^{(p)}_{m=1}(f, r=1,r'=1,\xi,S=\text{prime}(pN)). \end{equation*}

Then in 13.9 of Kato's paper, the zeta element zKatoz^{\text{Kato}} (denoted zγ(p)z_{\gamma}^{(p)} in Kato's paper for some auxiliary parameter γ\gamma) is defined as quotients of elements of the form c,dz(f,ξ){}_{c,d} z(f,\xi) by certain elements μ(c,d)\mu(c,d). It is this element zγ(p)z_{\gamma}^{(p)} which is related to LL-values.

This is clearly very technical. And to be fully honest, I don't intuitively understand a lot of the motivation behind Kato's constructions. But hopefully this gives at least a vague idea about how Kato's Euler System is constructed.

Footnotes

  1. I do not know spectral sequences, so have not (and don't know how to) verify this fact. Kato says that you get this map, so I'm taking it for granted.