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  and .
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 .
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 , Iwasawa ) 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
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 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
Kato's explicit reciprocity law ([1, Prop. 10.10]) says
To prove Kato's explicit reciprocity law, one uses the -expansion principle (a modular form is determined by its -expansion). More precisely:
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:
Using this theorem one can finally finish the proof of Theorem 3 by choosing specific norm compatible functions:
$p$-adic Hodge theory and values of zeta functions of modular forms, Astérisque (2004), no.295, ix, 117--290.
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Ueber die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math. 56 (1859), 270--279.
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.
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