14.3. The analytic p-adic L-function
The analytic p-adic L-function L_p^{\mathrm{an}}(f) is a p-adic
distribution on \Zpx that interpolates the critical values of
L(f,s). Its existence is governed by the slope v_p(\alpha_p) of the
U_p-eigenvalue \alpha_p of f.
- No associated Lean code or declarations.
Let \alpha_p be the U_p-eigenvalue of f, and suppose
v_p(\alpha_p) < k+1. Then there is a unique locally analytic distribution
L_p^{\mathrm{an}}(f) \in \sD^{\mathrm{la}}(\Zpx) on \Zpx such that:
-
L_p^{\mathrm{an}}(f)has growth of orderv_p(\alpha_p); and -
for every Dirichlet character
\chiof conductorp^nand every0 \le j \le k,L_p^{\mathrm{an}}(f,\overline{\chi},j+1) = \int_{\Zpx} \chi(x)\,x^j \cdot L_p^{\mathrm{an}}(f) = -\alpha_p^{-n}\left(1 - \chi(p)\tfrac{p^j}{\alpha_p}\right) \frac{G(\chi)\,j!\,p^{nj}}{(2\pi i)^{j+1}} \cdot \frac{L(f,\overline{\chi},j+1)}{\Omega_f^{\pm}},whereG(\chi)is the Gauss sum and\Omega_f^{\pm}the canonical complex periods.
This generalises Theorem 5.1.1 and lives in the space of Definition 3.7.3; it depends on Definition 14.2.1.
This is the theorem of Mazur–Swinnerton-Dyer, Amice–Vélu and Višik. One forms
the modular symbol attached to f, an H-valued additive functional on
\Q-paths in the upper half-plane whose values along \{i\infty \to a/p^n\}
encode the twisted L-values L(f,\overline{\chi},j+1). Applying the
U_p-operator and dividing by \alpha_p makes these symbols compatible under
the maps in the cyclotomic tower; the slope bound v_p(\alpha_p) < k+1
guarantees that the resulting compatible system has bounded growth of order
v_p(\alpha_p), so it defines an admissible (locally analytic) distribution on
\Zpx rather than merely a sequence of values. Uniqueness follows because a
distribution of growth < k+1 is determined by its values against the
characters x \mapsto \chi(x)x^j with 0 \le j \le k, which are exactly the
prescribed critical L-values. The Euler factor
1 - \chi(p)p^j/\alpha_p and the power \alpha_p^{-n} arise from the
U_p-stabilisation, exactly as the factor 1-p^{-s} arose for \zeta_p.
If f is p-ordinary, i.e. v_p(\alpha_p) = 0, then the growth condition
forces L_p^{\mathrm{an}}(f) to lie in the subspace of p-adic measures on
\Zpx, that is in the Iwasawa algebra \Lam(\Zpx) \subset \sD(\Zpx). This
uses Theorem 14.3.1 and Definition 3.3.3.
Growth of order 0 is precisely boundedness, and a bounded locally analytic
distribution is a measure; this is the same dichotomy seen for \zeta_p, where
p-ordinarity (here v_p(\alpha_p) = 0) places the distribution in the
Iwasawa algebra rather than the larger distribution space. The measure then
sits in \Lam(\Zpx) under the Mahler/Mellin identification of measures with
the Iwasawa algebra.
Boundary and infinite slope. If \alpha_p \neq 0 then
v_p(\alpha_p) \le k+1, but the theorem above excludes the boundary case
v_p(\alpha_p) = k+1; this is handled by the overconvergent modular-symbol
methods of Pollack–Stevens and Bellaïche, with a slightly modified statement. The
infinite slope case \alpha_p = 0 is harder still: partial p-adic
L-functions with good interpolation are constructed from Kato's Euler system
via Perrin-Riou's big logarithm maps. We do not develop these refinements here.