14.4. The arithmetic p-adic L-function
The arithmetic construction generalises the Coleman-map route of Part II,
passing through the Galois representation of f, Kato's Euler system and the
Perrin-Riou big logarithm map.
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Attached to f is a two-dimensional p-adic Galois representation V_f,
realised by Deligne inside the étale cohomology of the modular curve. It
contains a Galois-stable integral lattice T_f. The localisation of the
relevant cohomology at p carries an Iwasawa cohomology group
\HIw(\Qp, V_f), the \GL(2) analogue of the module of norm-coherent local
units.
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(Kato.) There exists an Euler system \mathbf{z}_{\mathrm{Kato}}(f) attached
to the lattice T_f. Localising at p yields a canonical class
\mathbf{z}_{\mathrm{Kato}}(f) \in \HIw(\Qp, V_f).
This depends on Definition 14.4.1 and is the \GL(2) analogue of
Definition 9.3.1.
This is the first main theorem of Kato's Hodge–Tate, p-adic and zeta paper.
The classes are constructed from cup products of Siegel units (modular units on
the modular curve) via the Beilinson–Kato map, producing for each integer m
a class in the motivic, hence étale, cohomology \Hone(\Q(\mu_m), T_f(2)). The
norm-compatibility of Siegel units along the cyclotomic tower endows the family
with the Euler-system distribution relations — exactly the role played by the
norm-coherence of cyclotomic units in the \GL(1) Coleman-map construction.
Localising at p and passing to the inverse limit over the tower yields the
single Iwasawa-cohomology class \mathbf{z}_{\mathrm{Kato}}(f).
The Perrin-Riou big logarithm map is a \Lam-morphism
\Log_{V_f} : \HIw(\Qp, V_f) \longrightarrow \sD^{\mathrm{la}}(\Zpx),
the \GL(2) analogue of the Coleman map. The arithmetic p-adic
L-function of f is the image of Kato's class,
L_p^{\mathrm{arith}}(f) := \Log_{V_f}\bigl(\mathbf{z}_{\mathrm{Kato}}(f)\bigr)
\in \sD^{\mathrm{la}}(\Zpx).
This uses Theorem 14.4.2 and Definition 9.5.1.
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(Kato's explicit reciprocity law.) The analytic and arithmetic p-adic
L-functions agree:
L_p^{\mathrm{an}}(f) = L_p^{\mathrm{arith}}(f) \in \sD^{\mathrm{la}}(\Zpx).
This is the \GL(2) analogue of Theorem 14.1.2, and depends on
Theorem 14.3.1 and Definition 14.4.3.
This is the second main theorem of Kato. The Perrin-Riou logarithm map is
characterised by an interpolation property: pairing \Log_{V_f} of a class
against the de Rham data of V_f recovers the dual exponential (or Bloch–Kato
exponential) of the class twisted by characters \chi(x)x^j. Kato computes
these dual exponentials for his Siegel-unit classes explicitly and shows they
reproduce the same critical L-values L(f,\overline{\chi},j+1), with the
identical Euler and Gauss-sum factors, that characterise
L_p^{\mathrm{an}}(f). Since both distributions have growth v_p(\alpha_p)
and agree on the determining set of characters x \mapsto \chi(x)x^j, they are
equal. This is precisely the \GL(2) incarnation of the equality
\zeta_p^{\mathrm{an}} = \zeta_p^{\mathrm{arith}} from \GL(1).