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.
Many problems in modern number theory revolve around the following theme: relate Galois cohomology classes to 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 $E/\mathbb{Q}$. On the left side of the diagram, you might want to study the Selmer group of $E$ over $\mathbb{Q}$. This is a group defined using Galois cohomology that essentially controls the arithmetic of $E$. On the right side, you might want to study the Hasse-Weil $L$-function $L(E,s)$. 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 $L$-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 $E/\mathbb{Q}$ be an elliptic curve, and let $L(E,s)$ be the Hasse-Weil $L$-function of $E$. If $E(\mathbb{Q})$ is infinite, then $L(E,1)=0$.
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.
For us, $E$ will be an elliptic curve over $\mathbb{Q}$ and $p$ will be a prime of good reduction. We will write $T_pE = \varprojlim_n E[p^n]$ for the Tate module of $E$. So $T_pE$ is isomorphic to $\mathbb{Z}_p \times \Z_p$ as an abelian group and it has an action of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Now define
$V_pE = T_pE \otimes_{\mathbb{Z}_p} \mathbb{Q}_p.$
Then $V_pE$ is a two-dimensional $\mathbb{Q}_p$-vector space with an action of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ inherited from the first factor $T_pE$ of the tensor product. (We write $V_pE$ because the $V$ stands for Vector space.)
Kato considers the Galois cohomology group $H^1(\mathbf{Q}, V_pE)$. With this group, Kato does two things. First, Kato defines a class $z^{\text{Kato}} \in H^1(\mathbf{Q}, V_pE)$ called the Kato class.
Second, Kato defines a very mysterious map called the dual exponential map:
$\exp^*: H^1(\mathbf{Q}_p, V_pE) \to \mathbf{Q}_p \omega_E,$
which is defined using $p$-adic Hodge theory. We will not define explicitly and treat like a black box. Here $\omega_E$ is the invariant differential on $E$.
Now let $\text{res}_p: H^1(\mathbf{Q}, V_pE) \to H^1(\mathbf{Q}_p, V_pE)$ denote the restriction map. Kato showed that
$\exp^*(\text{res}_p(z^{\text{Kato}})) = \dfrac{L_{\{p\}}(E,1)}{\Omega_E}\omega_E.$
The function $L_{\{p\}}(E,s)$ is the $L$-function $L(E,s)$ with the $p$-th Euler Factor removed. And $\Omega_E$ is the real Neron period of $E$. This identity is called the explicit reciprocity law.
Kato realized that this class $z^{\text{Kato}}$ is not alone, but is just the first class in an entire infinite collection of classes that interpolate $L$-values in different ways.
For each prime power $p^m$, consider the cohomology group $H^1(\mathbf{Q}(\zeta_{p^m}), V_pE)$. Kato defined a class $z^{\text{Kato}}_{p^m} \in H^1(\mathbf{Q}(\zeta_{p^m}), V_pE)$. Kato also defined a dual exponential map using $p$-adic Hodge theory:
$\exp^*: H^1(\mathbf{Q}_p(\zeta_{p^m}), V_pE) \to \mathbf{Q}_p(\zeta_{p^m}) \, \omega_E.$
The explicit reciprocity law here says: for any Dirichlet character $\chi: (\mathbf{Z}/p^m)^{\times} \to \mathbf{C}^{\times}$ of conductor $p^m$, we have
$\exp^* \left(\sum_{\sigma \in \text{Gal}(\mathbf{Q}_p(\zeta_{p^m})/\mathbf{Q}_p)} \text{res}_p(z^{\text{Kato}}_{p^m})^{\sigma} \right) = \dfrac{L_{\{p\}}(E,\chi, 1)}{\Omega_E^{\pm}}\omega_E.$
Here $\text{res}_p: H^1(\mathbf{Q}(\zeta_p), V_pE) \to H^1(\mathbf{Q}_p(\zeta_p), V_pE)$ denotes the restriction map. The periods $\Omega_E^{+}$ and $\Omega_E^-$ are the real and imaginary periods of $E$, respectively.
The collection of classes $\{ z^{\text{Kato}}_{p^m} \}_{m \geq 1}$ called Kato's Euler System. The class $z^{\text{Kato}}_{p^m}$ is called the $p^m$-th layer of Kato's Euler System.
Ok, this is a bit of a lie. Kato actually defined classes $z^{\text{Kato}}_m$ for every integer $m \geq 1$, not just prime powers $p^m$. But we only need the layers which are prime powers, so that's not a problem for us.
In this section, I'll describe the construction of the bottom class of Kato's Euler system, the element $z^{\text{Kato}} = z^{\text{Kato}}_{p^0} \in H^1(\mathbf{Q}, V_pE)$.
Since every elliptic curve over $\mathbf{Q}$ is modular, there exists a modular parametrization $X_0(N) \to E$, where $N$ is the conductor of $E$. Compose this with the natural surjection $X_1(N) \to X_0(N)$, to get a map
$X_1(N) \to E$.
This induces a map on the Jacobians: $J_1(N) \to E$. This is turn induces a map on their Tate modules: $T_p(J_1(N)) \to T_pE$. Tensoring with $\mathbb{Q}_p$, we get a Galois equivariant:
$V_p(J_1(N)) \to V_pE$.
This induces a map on Galois cohomology, so we get a map which I'll call $(a)$:
$H^1(\mathbb{Q}, V_p(J_0(N))) \xrightarrow{(a)} \, H^1(\mathbb{Q}, V_pE)$.
There is an isomorphism of $G_{\mathbf{Q}}$-modules:
$H^1_{\text{et}}(\overline{Y_1(N)}, \mathbb{Q}_p(1)) \xrightarrow{\sim} \, V_p(J_1(N)))$,
where $H^1_{\text{et}}$ denotes the etale cohomology group and $\overline{Y_1(N)}$ is the base-change of $Y_1(N)$ to $\overline{\mathbb{Q}}$. This induces a map on Galois cohomology which we denote $(b)$:
$H^1(\mathbb{Q}, H^1_{\text{et}}(\overline{Y_1(N)}, \mathbb{Q}_p(1))) \xrightarrow{(b)} \, H^1(\mathbb{Q}, V_p(J_1(N))).$
The next step is to use a spectral sequence to get a map from $H^2 \to H^1(\mathbf{Q}, H^1(\cdot\cdot\cdot))$. Precisely, consider the spectral sequence
$E_2^{i,j}=H^i(\mathbb{Q}, H^j_{\text{et}}(\overline{Y_1(N)}, - ) \implies H^{i+j}(Y_1(N), -).$
And use a fact from etale cohomology: $H^i_{\text{et}}(C, -) = 0$ for $i \geq 2$ whenever $C$ is an affine curve over an algebraically closed field. Combining these two facts^{1}, I am told that one obtains a map:
$H^2(Y_1(N), \mathbb{Q}_p(1)) \xrightarrow{(c)} H^1(\mathbb{Q}, H^1_{\text{et}}(\overline{Y_1(N)}, \mathbb{Q}_p(1) ).$
Crucially, in the domain of the map $(c)$, we consider the curve $Y_1(N)$ over $\mathbb{Q}$, whereas in the target, we consider $Y_1(N)$ base-changed to the algebraic closure $\overline{\mathbb{Q}}$. So the group $H^2(Y_1(N), \mathbb{Q}_p(1))$ does not have a Galois action. The benefit, therefore, of working over $\mathbb{Q}$ is that we can use tools from geometry (namely, the Kummer map) to produce classes in $H^2(Y_1(N), \mathbb{Q}_p(1))$.
Then we are going to apply a Tate twist to get a map $(d)$:
$H^2(Y_1(N), \mathbb{Q}_p(2)) \xrightarrow{(d)} H^2(Y_1(N), \mathbb{Q}_p(1))$
There is a cup-product map in etale cohomology:
$H^1(Y_1(N), \mathbb{Q}_p(1)) \times H^1(Y_1(N), \mathbb{Q}_p(1)) \xrightarrow{(e)} H^2(Y_1(N), \mathbb{Q}_p(2))$.
A useful mnemonic: when you apply the cup product, you add the twists (so $\mathbb{Q}_p(1)$ and $\mathbb{Q}_p(1)$ becomes $\mathbb{Q}_p(2)$) and add the degrees (so $H^1 \times H^1$ becomes $H^2$).
The Kummer Map is a map $\kappa: \mathcal{O}(Y_1(N))^{\times} \to H^1(Y_1(N), \mathbb{Q}_p(1))$. Applying the map $\kappa \times \kappa$, we obtain a map $(f)$:
$\left( \mathcal{O}(Y_1(N))^{\times} \right)^{2} \xrightarrow{(f)} H^2(Y_1(N), \mathbb{Q}_p(1))^2$.
Composing the maps $(a)$ through $(f)$, we have:
This is confusing! To recap, the maps are given as follows:
To produce an element of $H^1(\mathbf{Q}, V_pE)$, Kato defines an explicit element of $\left( \mathcal{O}(Y_1(N))^{\times} \right)^{2}$ and lets $z^{\text{Kato}}$ be its image under the maps $(a)$ through $(f)$.
Elements of $\mathcal{O}(Y_1(N))^{\times}$ are called modular units, and we can write some of them down using nothing but classical 19-th century elliptic function theory.
Definition. Let $N \geq 1$ and $0 \geq a,b < N$ be integers. For each pair $\left( \dfrac{a}{M}, \dfrac{b}{M} \right) \in \mathbb{Q}^2 \setminus (0,0)$, define the function $g_{\frac{a}{M}, \frac{b}{M}}: \mathbb{H} \to \mathbb{C}$ as follows:
where $q = e^{2\pi i z}$ and $w = \frac{1}{12} - \frac{a}{N} + \frac{a^2}{2N^2}$.
This is well-defined (independent of the choice of common denominator $N$). We would like to say that it is modular of level $N$, but this is not quite true. Acting on it by an element of $\Gamma(N)$ multiplies it by a root of unity so it defines an element of $\mathcal{O}(Y(N)) \otimes_{\Z} \mathbf{Q}$. We can kill the denominator by making a very simple modification:
Definition. Let $c>1$ be an integer coprime to $6N$. Put
Then ${}_cg_{\frac{a}{N}, \frac{b}{N}}(z)$ is $\Gamma(N)$-invariant so it belongs to $\mathcal{O}(Y(N)_{\mathbf{C}})^{\times}$. However, we can do better: it actually descends to the number field $\mathbf{Q}(\zeta_N)$. Since the affine curve $Y(N)$ is defined over $\mathbf{Q}(\zeta_N)$, we have:
Proposition. The units ${}_cg_{\frac{a}{N}, \frac{b}{N}}(z)$ belong to $\mathcal{O}(Y(N))^{\times}$.
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 $M,N$ appropriately and gets a pair of modular units ${}_cg_{\frac{1}{M}, 0} , {}_dg_{0, \frac{1}{N}}$ in $\left( \mathcal{O}(Y_1(N))^{\times} \right)^{2}$. Then Kato lets $z^{\text{Kato}}$ be its image under the maps $(a)$ through $(f)$.
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 $c,d > 1$ such that $(cd, 6N)=1$. He considers a pair of Seigel units $\left({}_cg_{\frac{1}{M}, 0} , {}_dg_{0, \frac{1}{N}} \right)$ where $M,N$ are chosen appropriately. After doing this, Kato obtains an element which he denotes ${}_{c,d}z_{M,N} = \left({}_c g_{1/M, 0}, {}_d g_{0, 1/N} \right)$. This element ${}_{c,d}z_{M,N}$ belongs to $\left(\mathcal{O}(Y(M,N))^{\times} \right)^2$, where $Y(M,N)$ is a certain "two-level" modular curve and $M,N$ are parameters to be chosen later.
Then for any matrix $\xi \in \textrm{SL}_2(\mathbf{Z})$, Kato "twists" this pair by $\xi$ and applies a norm map down to $Y_1(N)$ to get an element which Kato denotes by
(See page 153 of Kato's paper for the precise definition).
The image of the above element ${}_{c,d}z_{1,N,1}(2,1,1,\xi,S)$ under the maps $(a)$ through $(f)$ is denoted (see Sections 8.9, 8.11 of Kato's paper)
Then in 13.9 of Kato's paper, the zeta element $z^{\text{Kato}}$ (denoted $z_{\gamma}^{(p)}$ in Kato's paper for some auxiliary parameter $\gamma$) is defined as quotients of elements of the form ${}_{c,d} z(f,\xi)$ by certain elements $\mu(c,d)$. It is this element $z_{\gamma}^{(p)}$ which is related to $L$-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.
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. ↩