These are expanded notes prepared for a talk in a learning seminar on Kato's Euler systems, Fall 2016 at Columbia. We motivate the statement of Kato's explicit reciprocity laws and sketch his proof using $q$-expansions. Our main references are [1] and [2].

TopHilbert symbols

Recall the classical quadratic reciprocity law: if $a$,$b$ are odd positive coprime integers, then the quadratic residue symbols satisfies $$\legendre{a}{b}\legendre{b}{a}=(-1)^{\frac{a-1}{2}\cdot\frac{b-1}{2}}.$$ An equivalent formulation in terms of Hilbert symbols (using the product formula) is that for $a,b\in \mathbb{Z}_2^\times$, the Hilbert symbol $$(a,b)_{\mathbb{Q}_2}=(-1)^{\frac{a-1}{2}\cdot\frac{b-1}{2}}.$$

More generally for $K$ a $p$-adic field containing $n$-th roots of unity, Kummer theory/class field theory provide the Hilbert symbol $$(a,b)_K: K^\times/(K^\times)^n\times K^\times/(K^\times)^n\rightarrow \mu_n,\quad (a,b)=\frac{\Art(a)(\sqrt[n]{b})}{\sqrt[n]{b}}.$$ The quadratic reciprocity law can be viewed an explicit formula for $(a,b)_K$ in the case $K=\mathbb{Q}_2$ and $n=2$. So the key to explicating higher reciprocity laws is to give explicit formulas for $(a,b)_K$ in the wild case $p\mid n$. This is more difficult and such a prototype dates back to Kummer [3].

Theorem 1 (Kummer) Let $K=\mathbb{Q}_p(\zeta_p)$ with uniformizer $\pi=1-\zeta_p$. For $a,b\in 1+ \pi\mathcal{O}_K$, we have $$(a,b)_K=\zeta_p^{\tr(\zeta_p \log a\cdot \dlog b)/p}.$$ Here $\dlog b$ means the logarithmic derivative $\dlog f(\pi)=f'(\pi)/f(\pi)$ with respect to $\pi$ of any representation $b=f(\pi)$ ($f\in \mathbb{Z}_p[ [T]]$) and $\tr=\tr_{K/\mathbb{Q}_p}$.
Remark 1 The ride-hand-side of the formula does not depend on the choice of $f\in \mathbb{Z}_p[ [ T]]$ (though $\dlog b$ itself does).
Remark 2 Since $$\frac{1}{p}\tr(\zeta_p\pi^i)\equiv
\begin{cases}
  1 \pmod{p}, &  i=p-1, \\
  0 \pmod{p}, & i\ne p-1,
\end{cases}$$ we know that the exponent appearing above is the same as the coefficient of $\pi^{p-1}$ in $\log a\dlog b$ (= the formal residue of $\frac{\log a \dlog b}{\pi^p}$).

TopClassical explicit reciprocity laws

To facilitate generalization, let us reinterpret Kummer's formula as an explicit formula for the Hilbert symbol using a "dual exponential map". Consider the $n$-th layer of the cyclotomic tower $K_n=\mathbb{Q}_p(\zeta_{p^n})$ (so $K=K_1$). We have the $p^n$-th Hilbert symbol associated to $K_n$, $$K_n^\times \times K_n^\times\rightarrow \mu_{p^n}.$$ Fix a compatible system of $p^n$-roots of unity $(\zeta_{p^n})_{n\ge1}$. Sending $\zeta_{p^n}\mapsto1$ induces a (no longer Galois equivariant) pairing $$K_n^\times\times K_n^\times\rightarrow \mathbb{Z}/p^n.$$ Now fix the first factor to be $m$ and take inverse limit over $n$ (with respect to norm maps) for the second factor, we obtain a pairing $$K_m^\times\times \varprojlim_n K_n^\times\rightarrow \mathbb{Z}_p.$$ This gives a map $$\varprojlim_n K_n^\times\rightarrow \Hom_\mathrm{cont}(K_m^\times, \mathbb{Z}_p).$$ By precomposing with the exponential map $\exp: K_m\rightarrow K_m^\times \otimes \mathbb{Q}$, we obtain a map $$\varprojlim_n K_n^\times\rightarrow\Hom_\mathrm{cont}(K_m, \mathbb{Q}_p).$$ The trace pairing $$K_m\times K_m\rightarrow \mathbb{Q}_p,\quad (x,y)\mapsto\tr_{K_m/\mathbb{Q}_p}(xy)$$ identifies $K_m\cong \Hom_\mathrm{cont}(K_m, \mathbb{Q}_p)$, so at last we obtain a map $$\lambda_m: \varprojlim_n K_n^\times\xrightarrow{\text{Hilbert symbol}} \Hom(K_m^\times, \mathbb{Z}_p)\xrightarrow{\exp^*}\Hom(K_m,\mathbb{Q}_p)\cong K_m.$$

The classical explicit reciprocity law (Artin—Hasse [4], Iwasawa [5]) gives an explicit formula for this map $\lambda_m$ (encoding the Hilbert symbol on the $m$-th layer). To state their formulas, let $A=\mathbb{Z}_p[ [t-1]]$ be the ring of formal power series in the variable $t-1$. The ring homomorphism $t\mapsto t^p$ presents $A$ as a free $A$-module of rank $p$ and induces a norm map $$N: A^\times\rightarrow A^\times: f(t)\mapsto\prod_{i=1}^p f(\zeta_p^i t^{1/p}).$$

Theorem 2 (Artin—Hasse, Iwasawa) Let $(\theta_n(t))_{n\ge1}\in\varprojlim_n A^\times$ be a norm compatible sequence of $A^\times$. Then $u_n=\theta_n(\zeta_{p^n})$ form a norm compatible sequence in $\varprojlim_n K_n^\times$, and $$\lambda_m(u)=p^{-m}\cdot \zeta_{p^m} (\dlog \theta_m)(\zeta_{p^m}).$$
Remark 3 When $m=1$, we recover Kummer's formula by taking $u_1=\theta_1(\zeta_p)$ to be $b=f(\pi)$ and obtain that $(a,b)_K=\zeta_p^{\tr\log a\cdot \lambda_1(u)}=\zeta_p^{\tr(\zeta_p \log a\cdot \dlog b)/p}$. Notice this formula is valid for general units and does not involve any particular cyclotomic units yet.
Remark 4 To obtain cyclotomic units (needed to relate to $L$-functions — in this case — partial Riemann zeta functions $\zeta_{\equiv a\pmod{p^m}}(s)$), we choose the norm compatible elements $$\theta_n(t)=(1-t)(1-t^{-1}).$$ So that $\theta_n(\zeta_{p^n})=(1-\zeta_{p^n})(1-\zeta_{p^n}^{-1})$. Since $\dlog ((1-t)(1-t^{-1}))=\frac{1}{t}\cdot\frac{t+1}{t-1}$, we obtain $$\lambda_m(u)= p^{-m}\cdot \frac{\zeta_{p^m}+1}{\zeta_{p^m}-1},$$ where the cyclotomic units show up!
Remark 5 Notice that $\lambda_m$ can be essentially viewed as (after identifying the terms with their duals using the Hilbert symbol) 
\begin{equation*}
  \tag{1}
 \footnotesize \varprojlim_n K_n^\times\rightarrow\varprojlim_n H^1(K_n, \mu_{p^n})\rightarrow H^1(K_m, T_p\mathbb{G}_m)=K_m^\times \otimes \mathbb{Z}_p\xrightarrow{\exp^*} \Lie \mathbb{G}_m(K_m)=K_m.
\end{equation*}
Notice the first map is nothing but the connecting homomorphism in Kummer theory and the last map is nothing but the Block—Kato dual exponential map for the $p$-adic Galois representation $T_p\mathbb{G}_m$. % The miracle is this composition map has an explicit description and hence gives an explicit description of the Hilbert symbol.
Remark 6 Two lessons we learned:
  • Differential forms are more explicit than Galois cohomology. To switch to the former, one composes the etale Abel—Jacobi map with the dual exponential map.
  • After choosing an explicit norm compatible sequence, the explicit reciprocity law connects explicit classes in Galois cohomology with special values of $L$-functions.

Needless to say, the first step (though purely local) is by no means easy. The second step (constructing Euler systems) is even harder! (but see a series of recent works of Bertolini—Darmon—Rotger and Kings/Lei—Loeffler—Zerbes on generalized Kato classes).

TopKato's explicit reciprocity laws

Kato's explicit reciprocity law can be viewed as a generalization from the tower of cyclotomic fields $\Spec K_m$ to the tower of open modular curves $Y(m):=Y(Mp^m, Np^m)$. Here we fix two positive integers $M, N$ coprime to $p$ and $Y(Mp^n, Np^n)$ roughly parametrizes elliptic curves together with a marked $Mp^n$-torsion $e_{1,n}$ and a marked $Np^n$-torsion point $e_{2,n}$. Similarly define the tower of compact modular curves $X(m):=X(Mp^m, Np^m)$. One needs $M+N\ge5$ to avoid stacky issues but it instructional to just think hypothetically as if $M=N=1$.

The map generalizing (1) is then given by 
\begin{equation*}
  \tag{2}
\scriptsize \varprojlim_n K_2(Y(n))\rightarrow \varprojlim_{n} H^2(Y(m), \mathbb{Z}/p^n(1))\rightarrow H^1(\mathbb{Q}_p, H^1(Y(m)_{\overline{\mathbb{Q}}_p},\mathbb{Q}_p) (1))\xrightarrow{\exp^*}M_2(X(m)) \otimes \mathbb{Q}_p.
\end{equation*}

Here:

  • $K_2(Y(m))$ is the second $K$-group of the open modular curve $Y(m)$. Notice in the classical case $K_m^\times$ is the first $K$-group of $\Spec K_m$.
  • The first map is induced by composing the Chern class map (the etale regulator) together with cup product with $\zeta_{p^n}^{-1}$, $$K_2(Y(n))\rightarrow H^2(Y(n), \mathbb{Z}/p^n(2))\rightarrow H^2(Y(n), \mathbb{Z}/p^n(1)).$$
  • The second map comes from Leray spectral sequence for $Y(m)\rightarrow \Spec \mathbb{Q}_p$ (so the first two maps together essentially gives the etale Abel-Jacobi map).
  • The third map is the Bloch—Kato dual exponential map $$H^1(K, V)\rightarrow \mathrm{Fil}^0D_\mathrm{dR}(V).$$ For $V=H^1(Y(m), \mathbb{Q}_p))$, we have $$\mathrm{Fil}^0D_\mathrm{dR}(V)=M_2(X(m))\otimes \mathbb{Q}_p,$$ where $M_2(X(m))$ is the space of weight 2 modular forms on $X(m)$.

Kato's explicit reciprocity law ([1, Prop. 10.10]) says

Theorem 3 (Kato) The composition map (1) sends $_{c,d}z_{Mp^n, Np^n}$ to $_{c,d}z_{Mp^m,Np^m}(2,1,1)$. Here

  • $_{c,d}z_{Mp^n, Np^n}$ are the norm compatible zeta elements ([1, 2.2]), constructed from a pair of explicit Siegel units $\{_{c}g_{1/Mp^n,0},{}_{d}g_{0,Np^n}\}$ (the integer parameters $c,d$ are needed for technical purposes);
  • $_{c,d}z_{Mp^m,Np^m}(2,1,1)$ is the the zeta modular form ([1, 4.2]), constructed as the product of two explicit weight 1 Eisenstein series $_cE^{(1)}_{1/Mp^m,0}$ and $_dE^{(1)}_{0,1/Np^m}$.

Top$q$-expansions

To prove Kato's explicit reciprocity law, one uses the $q$-expansion principle (a modular form is determined by its $q$-expansion). More precisely:

  • Consider $F $, the $p$-adic completion of the function field of the modular curve of level $(M,N)$ at a cusp. It is a complete discrete valuation field of characteristic 0 with valuation ring $$\mathcal{O}_F=\varprojlim_n (\mathbb{Z}[\zeta_N]_\mathfrak{p}[ [q^{1/M}]][q^{-1}]/p^n$$ and imperfect residue field $$k=\mathbb{F}_p(\zeta_N)((q^{1/M})).$$ One can visualize $F $ as a "2-dimensional local field" (geometrically a puncture disk around the cusp). Imagine if $M=N=1$, then $\mathcal{O}_F$ is just the $p$-adic completion of $\mathbb{Z}_p[ [q]][q^{-1}]$ and $k=\mathbb{F}_p((q))$.
  • Let $\mathcal{E}/\mathcal{O}_F$ be the Tate curve $\mathcal{E}/\mathbb{Z}[ [q]]$ base changed to $\mathcal{O}_F$. Notice that $q$ is invertible, so $\mathcal{E}$ is indeed an elliptic curve.
  • Let $E/F$ be its generic fiber.
  • The triple $(E, \langle q^{1/M}\rangle, \langle\zeta_N\rangle)$ defines an $F$-point of $Y(M,N)$, denoted by $Y_0$.
  • Let $Y_n$ be the pullback $Y_0$ along the natural projection $Y(n)\rightarrow Y(0)$. These cusps $Y_n$ of $Y(n)$ form a Galois covering of the cusp $Y_0$ of $Y(0)$ with Galois group $GL_2(\mathbb{Z}/p^n)$.

Analogous to (2), we obtain the map around the cusps, $$\lambda_m: \varprojlim_n K_2(Y_n)\rightarrow\varprojlim_{n}H^2(Y_n, \mathbb{Z}/p^n(1))\rightarrow H^2(Y_m, \mathbb{Z}_p(1))\xrightarrow{\exp^*}\mathcal{O}(Y_m).$$ Here the construction of $\exp^*$ is not so easy and needs $p$-adic Hodge theory over imperfect residual fields. For this reason it is a nontrivial task to check the compatibility with the usual Bloch—Kato dual exponential map in (2), but this is done in [1, 11]. If we take this compatibility for granted, then it remains to compute the map $\lambda_m$ explicitly.

Now we make obvious changes to $\mathbb{G}_m$ in the classical case:

  • The norm compatible functions will be on the $p$-divisible group $G$ of $E $, instead of $\mathbb{G}_m$, and the multiplication $p: G\rightarrow G$ induces a norm map $N: \mathcal{O}(G)^\times\rightarrow\mathcal{O}(G)^\times$.
  • We evaluate these functions at the marked torsion points $e_{1,n}, e_{2,n}$ of $E $, instead of $\zeta_{p^n}$.

The main theorem of [2] (quoted as Prop. 10.12 in [1]) is the following.

Theorem 4 (Kato) Let $\theta_{1,n},\theta_{2,n}\in \mathcal{O}(G)^\times$ be two norm compatible sequences. Then $$u_n=\{\theta_{1,n}(e_{1,n}), \theta_{2,n}(e_{2,n})\}\in K_2(Y_n)$$ gives a norm compatible sequence $u\in \varprojlim_n K_2(Y_n)$ and $$\lambda_m(u)=p^{-2m}\cdot \dlog \theta_{1,m}(e_{1,m})\dlog \theta_{2,m}(e_{2,m}).$$ Here $\dlog \theta=\frac{d\theta}{\theta \omega}$, where $\omega=\frac{dt}{t}$ is the canonical basis of $\omega_G\cong\omega_{\mu_{p^\infty}}$.

Using this theorem one can finally finish the proof of Theorem 3 by choosing specific norm compatible functions:

  • Take $\theta_{1,n},\theta_{2,n}$ to be the theta functions $_{c}\theta_\mathcal{E},_{d}\theta_\mathcal{E}$ associated to $\mathcal{E}$. These theta functions have product expansion roughly of the form (again imagine $M=N=1$) $$q^{1/12}\prod_{n\ge0}(1-q^nt)\prod_{n\ge1}(1-q^nt^{-1}).$$ This is analogous to the rational function $(1-t)(1-t^{-1})$ in the classical case.
  • Using the product expansion one can explicitly evaluate their $\dlog $ (along the vertical $t$-direction rather than the horizontal $q$-direction) at the two marked torsion points $e_{1,m}$ and $e_{2,m}$. This gives explicit Eisenstein series $_cE^{(1)}_{1/Mp^m,0}$ and $_dE^{(1)}_{0,1/Np^m}$ of weights $k_1=k_2=1$, which appear in the Rankin—Selberg integrals computing $L(f, k-1+s)L(f \otimes \chi, k_1+s)$ at $s=0$ for forms $f$ of weight $k=2$ (so one gets the desired central value $L(f,1)$). They are analogous to cyclotomic units in the classical case.
Last Update: 01/04/2017. Copyright © 2015 - 2017, Chao Li.

References

[1]Kato, Kazuya, $p$-adic Hodge theory and values of zeta functions of modular forms, Astérisque (2004), no.295, ix, 117--290.

[2]Kato, Kazuya, Generalized explicit reciprocity laws, Adv. Stud. Contemp. Math. (Pusan) 1 (1999), 57--126.

[3]Kummer, E. E., Ueber die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math. 56 (1859), 270--279.

[4]Artin, E. and Hasse, H., Die beiden Ergänzungssätze zum reziprozitätsgesetz der $l^n$-ten potenzreste im körper der $l^n$-ten Einheitswurzeln, Abh. Math. Sem. Univ. Hamburg 6 (1928), no.1, 146--162.

[5]Iwasawa, Kenkichi, On explicit formulas for the norm residue symbol, J. Math. Soc. Japan 20 (1968), 151--165.