14.1. Recapping the GL(1) picture
Throughout, p is a fixed prime. Write
\Gamma^+ := \Gal(\Q(\mu_{p^\infty})^+/\Q) \cong \Zpx/\{\pm 1\} for the Galois
group of the maximal real subfield of the cyclotomic tower, \Lam(\Gamma^+)
for its Iwasawa algebra, and Q(\Gamma^+) for the ring of fractions of
\Lam(\Gamma^+). The notes give three constructions of the Kubota–Leopoldt
p-adic L-function, and the comparison theorems are exactly what is being
generalised to \GL(2).
The three \GL(1) p-adic L-functions are:
-
Analytic. A
p-adic pseudo-measure\zeta_p^{\mathrm{an}} \in Q(\Gamma^+)interpolating special values of the Riemann zeta function, built in Part I. -
Arithmetic. The element
\zeta_p^{\mathrm{arith}} := \Col(c_\infty), the image under the Coleman map\Colof the norm-coherent family of cyclotomic unitsc_\infty. -
Algebraic. The characteristic ideal
\zeta_p^{\mathrm{alg}} := \Char_{\Lam(\Gamma^+)}(\sX_\infty^+) \subset \Lam(\Gamma^+)of the torsion\Lam(\Gamma^+)-module\sX_\infty^+(an unramified Iwasawa module).
This recapitulates Definition 4.3.2 and the Definition 3.6.1 of Part I, the Definition 9.5.1 applied to the Definition 9.3.1, and the Definition 12.1.4 of the Main Conjecture chapter.
The \GL(1) comparison statements read:
-
(Coleman-map computation)
\zeta_p^{\mathrm{an}} = \zeta_p^{\mathrm{arith}}, identifying the analytic and arithmetic constructions. -
(Iwasawa Main Conjecture)
\zeta_p^{\mathrm{alg}} = I(\Gamma^+)\,\zeta_p^{\mathrm{an}}, whereI(\Gamma^+)is the augmentation ideal, identifying the algebraic and analytic constructions.
These rest on Definition 14.1.1, Theorem 9.2.2, Definition 9.5.1, Definition 9.3.1, Theorem 12.3.1 and Definition 12.1.4.
The first equality is the content of the Coleman-map computation of Part II: the
Coleman map sends the cyclotomic units, whose Mellin transform recovers the
Kubota–Leopoldt interpolation property, to the analytic pseudo-measure, so the
arithmetic object equals the analytic one on the nose. The second equality is
the Iwasawa Main Conjecture, equating the characteristic ideal of \sX_\infty^+
with the ideal generated by \zeta_p^{\mathrm{an}} (up to the augmentation
ideal accounting for the pseudo-measure versus measure normalisation). Both are
recorded in detail in the preceding chapters; here they serve as the templates
for the \GL(2) analogues.