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 -expansions. Our main references are [1] and [2].

## Hilbert symbols

Recall the classical quadratic reciprocity law: if , are odd positive coprime integers, then the quadratic residue symbols satisfies An equivalent formulation in terms of Hilbert symbols (using the product formula) is that for , the Hilbert symbol

More generally for a -adic field containing -th roots of unity, Kummer theory/class field theory provide the Hilbert symbol The quadratic reciprocity law can be viewed an explicit formula for in the case and . So the key to explicating higher reciprocity laws is to give explicit formulas for in the wild case . This is more difficult and such a prototype dates back to Kummer [3].

Theorem 1 (Kummer) Let with uniformizer . For , we have Here means the logarithmic derivative with respect to of any representation () and .
Remark 1 The ride-hand-side of the formula does not depend on the choice of (though itself does).
Remark 2 Since we know that the exponent appearing above is the same as the coefficient of in (= the formal residue of ).

## Classical 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 -th layer of the cyclotomic tower (so ). We have the -th Hilbert symbol associated to , Fix a compatible system of -roots of unity . Sending induces a (no longer Galois equivariant) pairing Now fix the first factor to be and take inverse limit over (with respect to norm maps) for the second factor, we obtain a pairing This gives a map By precomposing with the exponential map , we obtain a map The trace pairing identifies , so at last we obtain a map

The classical explicit reciprocity law (Artin—Hasse [4], Iwasawa [5]) gives an explicit formula for this map (encoding the Hilbert symbol on the -th layer). To state their formulas, let be the ring of formal power series in the variable . The ring homomorphism presents as a free -module of rank and induces a norm map

Theorem 2 (Artin—Hasse, Iwasawa) Let be a norm compatible sequence of . Then form a norm compatible sequence in , and
Remark 3 When , we recover Kummer's formula by taking to be and obtain that . 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 -functions — in this case — partial Riemann zeta functions ), we choose the norm compatible elements So that . Since , we obtain where the cyclotomic units show up!
Remark 5 Notice that can be essentially viewed as (after identifying the terms with their duals using the Hilbert symbol) 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 -adic Galois representation .
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 -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).

## Kato's explicit reciprocity laws

Kato's explicit reciprocity law can be viewed as a generalization from the tower of cyclotomic fields to the tower of open modular curves . Here we fix two positive integers coprime to and roughly parametrizes elliptic curves together with a marked -torsion and a marked -torsion point . Similarly define the tower of compact modular curves . One needs to avoid stacky issues but it is instructional to just think hypothetically as if .

The map generalizing (1) is then given by

Here:

• is the second -group of the open modular curve . Notice in the classical case is the first -group of .
• The first map is induced by composing the Chern class map (the etale regulator) together with cup product with ,
• The second map comes from Leray spectral sequence for (so the first two maps together essentially gives the etale Abel-Jacobi map).
• The third map is the Bloch—Kato dual exponential map For , we have where is the space of weight 2 modular forms on .

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

Theorem 3 (Kato) The composition map (1) sends to . Here

• are the norm compatible zeta elements ([1, 2.2]), constructed from a pair of explicit Siegel units (the integer parameters are needed for technical purposes);
• is the the zeta modular form ([1, 4.2]), constructed as the product of two explicit weight 1 Eisenstein series and .

## -expansions

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

• Consider , the -adic completion of the function field of the modular curve of level at a cusp. It is a complete discrete valuation field of characteristic 0 with valuation ring and imperfect residue field One can visualize as a "2-dimensional local field" (geometrically a puncture disk around the cusp). Imagine if , then is just the -adic completion of and .
• Let be the Tate curve base changed to . Notice that is invertible, so is indeed an elliptic curve.
• Let be its generic fiber.
• The triple defines an -point of , denoted by .
• Let be the pullback along the natural projection . These cusps of form a Galois covering of the cusp of with Galois group .

Analogous to (2), we obtain the map around the cusps, Here the construction of is not so easy and needs -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 explicitly.

Now we make obvious changes to in the classical case:

• The norm compatible functions will be on the -divisible group of , instead of , and the multiplication induces a norm map .
• We evaluate these functions at the marked torsion points of , instead of .

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

Theorem 4 (Kato) Let be two norm compatible sequences. Then gives a norm compatible sequence and Here , where is the canonical basis of .

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

• Take to be the theta functions associated to . These theta functions have product expansion roughly of the form (again imagine ) This is analogous to the rational function in the classical case.
• Using the product expansion one can explicitly evaluate their (along the vertical -direction rather than the horizontal -direction) at the two marked torsion points and . This gives explicit Eisenstein series and of weights , which appear in the Rankin—Selberg integrals computing at for forms of weight (so one gets the desired central value ). They are analogous to cyclotomic units in the classical case.
[1]Kato, Kazuya, $p$-adic Hodge theory and values of zeta functions of modular forms, Astérisque (2004), no.295, ix, 117--290.
[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.