14.5. The algebraic p-adic L-function and the Main Conjecture
For a p-ordinary eigenform the analytic/arithmetic p-adic L-function is
a measure, and we can ask for its algebraic counterpart: a characteristic ideal
of a Selmer-type Iwasawa module, exactly as in the \GL(1) Main Conjecture.
Suppose f is p-ordinary (v_p(\alpha_p)=0), so V_f is an ordinary
Galois representation. As in the Greenberg–Selmer formalism this yields a Selmer
group, an Iwasawa module \sX_{p^\infty}(V_f) over the Iwasawa algebra
\Lam(\Gamma) of \Gamma := \Gal(\Q(\mu_{p^\infty})/\Q) \cong \Zpx. Kato
proved this module is \Lam-torsion, so it has a characteristic ideal, and the
algebraic p-adic L-function of f is
L_p^{\mathrm{alg}}(f) := \ch_{\Lam(\Zpx)}\bigl(\sX_{p^\infty}(V_f)\bigr).
This is the \GL(2) analogue of \zeta_p^{\mathrm{alg}}; it uses
Definition 14.4.1, Definition 12.1.4 and
Theorem 12.1.3.
That \sX_{p^\infty}(V_f) is \Lam-torsion is a consequence of Kato's Euler
system: a non-trivial Euler system bounds the Selmer group, forcing it to be
torsion over \Lam. A finitely generated torsion \Lam-module has a
well-defined characteristic ideal by the structure theory of \Lam-modules,
and L_p^{\mathrm{alg}}(f) is by definition this ideal. The construction
mirrors \zeta_p^{\mathrm{alg}} = \Char(\sX_\infty^+) from the \GL(1) case,
with \sX_\infty^+ replaced by the Selmer module of V_f.
(Iwasawa Main Conjecture for f.) Let f be a p-ordinary cuspidal
Hecke eigenform. Under mild technical hypotheses, the analytic/arithmetic
p-adic L-function is a measure in \Lam(\Zpx), and the algebraic
p-adic L-function equals the ideal it generates:
L_p^{\mathrm{alg}}(f) = \bigl(L_p^{\mathrm{an}}(f)\bigr)
\subset \Lam(\Zpx).
This is the \GL(2) analogue of Theorem 12.3.1; it depends
on Definition 14.5.1, Proposition 14.3.2,
Theorem 14.4.4 and Theorem 14.4.2.
This is a theorem of Kato together with Skinner–Urban. Kato's Euler system gives
one divisibility: the existence of the bounding Euler system forces
L_p^{\mathrm{alg}}(f) \mid (L_p^{\mathrm{an}}(f)), i.e. the characteristic
ideal divides the analytic p-adic L-function — the same mechanism by which
an Euler system yields the divisibility half of any Main Conjecture. The reverse
divisibility is supplied by Skinner–Urban, who construct congruences between
f and Eisenstein series on a larger unitary (or \GL(2)) group; the
Eisenstein congruence manufactures enough elements of the Selmer group to bound
it from the other side, exactly as Ribet's method underlies the \GL(1) Main
Conjecture. Combining the two divisibilities gives the equality of ideals. Much
subsequent work weakens the hypotheses and treats non-ordinary f.
Applications. The cyclotomic Iwasawa theory of modular forms has deep applications to elliptic curves and the Birch–Swinnerton-Dyer conjecture. There is also a parallel anticyclotomic theory, working over an auxiliary imaginary quadratic field, with its own striking applications to BSD. These directions lie beyond the scope of these notes.