Analytic continuation of the Hasse–Weil L-series to an entire function (Wiles, Taylor–Wiles, Breuil–Conrad–Diamond–Taylor)

Read-back

Binders and hypotheses. The statement quantifies over a single object $W$, a Weierstrass curve over $\mathbb{Q}$: literally a tuple of five rational numbers $(a_1, a_2, a_3, a_4, a_6)$, thought of as the coefficients of the equation $y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$. There is one typeclass hypothesis, that $W$ is elliptic, which unfolds to: the discriminant $\Delta(W)$ is a unit of $\mathbb{Q}$, i.e. $\Delta(W) \ne 0$, where

$$\Delta = -b_2^2 b_8 - 8 b_4^3 - 27 b_6^2 + 9 b_2 b_4 b_6,$$

with $b_2 = a_1^2 + 4a_2$, $b_4 = 2a_4 + a_1 a_3$, $b_6 = a_3^2 + 4a_6$, $b_8 = a_1^2 a_6 + 4a_2 a_6 - a_1 a_3 a_4 + a_2 a_3^2 - a_4^2$. No other hypothesis is imposed on $W$ (no integrality, no global minimality, no conductor condition).

The claim. For every such $W$, there exists (not "exists uniquely") a function $\Lambda : \mathbb{C} \to \mathbb{C}$ satisfying the two-part predicate "$\Lambda$ is an L-function for $W$" from the definition bundle, namely:

1. (differentiable) $\Lambda$ is complex-differentiable at every point $z \in \mathbb{C}$: for each $z$ there is a $\mathbb{C}$-linear map $\ell_z : \mathbb{C} \to \mathbb{C}$ with $\Lambda(z + h) = \Lambda(z) + \ell_z(h) + o(h)$ as $h \to 0$. In other words $\Lambda$ is an entire function on all of $\mathbb{C}$.

2. (agreement on a half-plane) for every $s \in \mathbb{C}$ with $\operatorname{Re}(s) > \tfrac{3}{2}$ (strict inequality),

$$\Lambda(s) = L_W(s),$$

where $L_W(s)$ is the specific complex number defined below (Mathlib's WeierstrassCurve.LSeries of $W$ at $s$).

Nothing in the statement constrains the values of $\Lambda$ at points with $\operatorname{Re}(s) \le \tfrac32$ other than condition 1. No functional equation, no growth condition, no relation to any other analytic object, and no non-vanishing or non-triviality of $\Lambda$ is asserted.

Unfolding $L_W(s)$. Write $c : \mathbb{N} \to \mathbb{Z}$ for the integer-valued "arithmetic function" $c = \mathrm{LFunction}(W)$ described below (an arithmetic function is a function on $\mathbb{N} = \{0,1,2,\dots\}$ with $c(0) = 0$). Then

$$L_W(s) \;=\; {\sum_{n=0}^{\infty}}' \ t_n(s), \qquad t_n(s) = \begin{cases} 0 & n = 0,\\[2pt] \dfrac{c(n)}{n^{s}} & n \ge 1,\end{cases}$$

where $n^s$ is the principal complex power and $c(n)$ is cast from $\mathbb{Z}$ into $\mathbb{C}$. The primed sum $\sum'$ is Mathlib's total "tsum": if the family $(t_n(s))_n$ is (unconditionally, equivalently absolutely) summable, it is the sum; if the family is not summable at $s$, the value is $0$ by definition. Consequently, at any $s$ with $\operatorname{Re}(s) > \frac32$ where the Dirichlet series $\sum c(n) n^{-s}$ fails to converge absolutely, condition 2 reads $\Lambda(s) = 0$; if the series were not summable anywhere on that half-plane, condition 2 would read "$\Lambda$ vanishes identically on $\{\operatorname{Re} s > \frac32\}$", and the constant function $\Lambda \equiv 0$ would satisfy both conditions. The statement does not assert summability at any point.

Unfolding the coefficients $c = \mathrm{LFunction}(W)$. Let $\mathcal{O}_{\mathbb{Q}}$ be the ring of integers of the number field $\mathbb{Q}$ (isomorphic to $\mathbb{Z}$), and let $\mathfrak{p}$ range over its height-one spectrum: the nonzero prime ideals, i.e. (up to this isomorphism) the prime numbers $p$. For each such $\mathfrak{p}$ let $K_\mathfrak{p}$ be the $\mathfrak{p}$-adic completion of $\mathbb{Q}$ (so $\mathbb{Q}_p$), $R_\mathfrak{p} \subset K_\mathfrak{p}$ its valuation ring (so $\mathbb{Z}_p$), and $\kappa_\mathfrak{p} = R_\mathfrak{p}/\mathfrak{m}$ its residue field (so $\mathbb{F}_p$). Then

$$c \;=\; {\prod_{\mathfrak{p}}}' \ E_\mathfrak{p},$$

an infinite product of arithmetic functions under Dirichlet convolution, taken as a "tprod" in the topology of coefficientwise eventual equality (each coefficient of a finite partial product must stabilise). If the family $(E_\mathfrak{p})$ is not multipliable in that sense, the product is by definition the multiplicative identity, the arithmetic function $\varepsilon$ with $\varepsilon(1) = 1$ and $\varepsilon(n) = 0$ for $n \ne 1$; in that junk case $L_W(s) = 1$ for every $s$, and $\Lambda \equiv 1$ would satisfy the theorem. The statement does not assert multipliability.

The local factor $E_\mathfrak{p}$. Let $W_\mathfrak{p}$ be $W$ with coefficients mapped into $K_\mathfrak{p}$. Let $W'_\mathfrak{p} = C_\mathfrak{p} \cdot W_\mathfrak{p}$ be a minimal model: $C_\mathfrak{p}$ is a change of variables over $K_\mathfrak{p}$, chosen (by the axiom of choice, from a proof that one exists) such that $C_\mathfrak{p}\cdot W_\mathfrak{p}$ is integral (its five coefficients lie in $R_\mathfrak{p}$) and, among all changes of variables $C'$ with $C' \cdot W_\mathfrak{p}$ integral, the multiplicative valuation $v(\Delta(C'\cdot W_\mathfrak{p})) \in \{1, \pi, \pi^2,\dots\}\cup\{0\}$ is maximal at $C' = C_\mathfrak{p}$ (maximal in the sense: any competitor whose value is $\ge$ has value $\le$). Multiplicative valuation "$= 1$" means "a unit"; "$< 1$" means "in the maximal ideal". Let $I_\mathfrak{p}$ be a chosen integral model of $W'_\mathfrak{p}$ over $R_\mathfrak{p}$ (five elements of $R_\mathfrak{p}$ mapping to the coefficients of $W'_\mathfrak{p}$), and let $\widetilde{W}_\mathfrak{p}$ be its reduction: $I_\mathfrak{p}$ with coefficients reduced modulo $\mathfrak{m}$ into $\kappa_\mathfrak{p}$.

Set

$$q_\mathfrak{p} = \#\kappa_\mathfrak{p} \in \mathbb{Z}, \qquad a_\mathfrak{p} = q_\mathfrak{p} + 1 - \#\,\widetilde{W}_\mathfrak{p}(\kappa_\mathfrak{p}) \in \mathbb{Z},$$

where both cardinalities are Nat.card (which returns the junk value $0$ for an infinite type; here $\kappa_\mathfrak{p}$ is finite), the subtraction is in $\mathbb{Z}$ (no truncation), and $\widetilde{W}_\mathfrak{p}(\kappa_\mathfrak{p})$ is the type of affine points of $\widetilde W_\mathfrak{p}$: one point at infinity together with all pairs $(x, y) \in \kappa_\mathfrak{p}^2$ that are nonsingular on $\widetilde W_\mathfrak{p}$, meaning $y^2 + a_1xy + a_3y - (x^3 + a_2x^2 + a_4x + a_6) = 0$ and at least one of the two partial derivatives of this expression is nonzero at $(x,y)$ (coefficients those of $\widetilde W_\mathfrak{p}$). Singular points of the reduced curve, if any, are not counted.

The local polynomial $f_\mathfrak{p}(T) \in \mathbb{Z}[T]$ is chosen by a cascade of four cases on $W'_\mathfrak{p}$:

• if $W'_\mathfrak{p}$ has good reduction ($W'_\mathfrak{p}$ minimal as above and $v(\Delta(W'_\mathfrak{p})) = 1$), then $f_\mathfrak{p} = 1 - a_\mathfrak{p} T + q_\mathfrak{p} T^2$;
• otherwise, if $W'_\mathfrak{p}$ has split multiplicative reduction (minimal, $v(\Delta(W'_\mathfrak{p})) < 1$, $v(c_4(W'_\mathfrak{p})) = 1$ where $c_4 = b_2^2 - 24 b_4$, and the polynomial $c_4 T^2 + a_1 c_4 T - (54 b_6 - 3 b_2 b_4 + a_2 c_4)$, with coefficients taken from $I_\mathfrak{p}$ and reduced into $\kappa_\mathfrak{p}$, is a product of constants and monic linear factors over $\kappa_\mathfrak{p}$), then $f_\mathfrak{p} = 1 - T$;
• otherwise, if $W'_\mathfrak{p}$ has multiplicative reduction (minimal, $v(\Delta(W'_\mathfrak{p})) < 1$, $v(c_4(W'_\mathfrak{p})) = 1$) — i.e. non-split multiplicative — then $f_\mathfrak{p} = 1 + T$;
• otherwise (which, for a minimal model, is the case $v(\Delta) < 1$ and $v(c_4) < 1$, "additive reduction"), $f_\mathfrak{p} = 1$.

In every branch $f_\mathfrak{p}$ has constant term $1$. Let $g_\mathfrak{p}(T) = \sum_k g_{\mathfrak{p},k} T^k \in \mathbb{Z}[[T]]$ be the formal power series obtained from $f_\mathfrak{p}$ by Mathlib's inverse-of-a-unit recursion with the unit $1$ (namely $g_{\mathfrak{p},0} = 1$ and $g_{\mathfrak{p},k} = -\sum_{i=1}^{k} f_{\mathfrak{p},i}\, g_{\mathfrak{p},k-i}$ for $k \ge 1$), so that $f_\mathfrak{p}\, g_\mathfrak{p} = 1$ as formal power series. Finally $E_\mathfrak{p}$ is the arithmetic function "$g_\mathfrak{p}(q_\mathfrak{p}^{-s})$": when $q_\mathfrak{p} > 1$,

$$E_\mathfrak{p}(n) = \begin{cases} g_{\mathfrak{p},k} & \text{if } n = q_\mathfrak{p}^{\,k} \text{ for some } k \ge 0,\\ 0 & \text{if } n \text{ is not a power of } q_\mathfrak{p}, \end{cases}$$

and when $q_\mathfrak{p} \le 1$ (junk case, not occurring for finite residue fields of size $\ge 2$) $E_\mathfrak{p}$ is instead the constant term $g_{\mathfrak{p},0}$ placed at $n = 1$ and $0$ elsewhere.

Items in the bundle not used by this statement. The definition $\operatorname{rank}(W) = \dim_{\mathbb{Z}}$-finrank of the affine point group of $W$ over $\mathbb{Q}$ (the natural number obtained from the $\mathbb{Z}$-module rank, junk $0$ if infinite) appears in the bundle but does not occur in this theorem.

Summary of what is asserted. For every five-tuple of rationals $W$ with $\Delta(W) \ne 0$, there is at least one entire function $\Lambda : \mathbb{C} \to \mathbb{C}$ whose values on the open half-plane $\operatorname{Re}(s) > \frac32$ coincide, point by point, with the total-function value $L_W(s)$ of the Dirichlet series $\sum_{n \ge 1} c(n) n^{-s}$ built from the Euler product of the local factors above (with the value $0$ wherever that series is not absolutely summable, and with $c = \varepsilon$ if the Euler product is not multipliable).