Uniqueness of the analytic continuation of the Hasse–Weil L-series

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BSD.isLFunction_unique. Let $W$ be a Weierstrass curve over $\mathbb{Q}$, i.e. an ordered quintuple of rational numbers $(a_1, a_2, a_3, a_4, a_6)$ (thought of as the coefficients of $y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$; the structure itself is just the five numbers). Assume the typeclass hypothesis that $W$ is elliptic, which unfolds to: the discriminant

is a unit of $\mathbb{Q}$, i.e. $\Delta_W \neq 0$. Let $\Lambda_1, \Lambda_2 : \mathbb{C} \to \mathbb{C}$ be two arbitrary functions, and assume that each $\Lambda_i$ satisfies the bundle's predicate $\mathrm{IsLFunction}(W, \Lambda_i)$, which is the conjunction of exactly two conditions:

1. Differentiability everywhere. $\Lambda_i$ is complex-differentiable at every point $s \in \mathbb{C}$ (in the sense of having a complex Fréchet derivative at $s$); there is no exceptional set. In other words $\Lambda_i$ is an entire function.

2. Agreement on the open half-plane $\operatorname{Re} s > 3/2$. For every $s \in \mathbb{C}$ with $\operatorname{Re}(s) > \tfrac{3}{2}$ (strict inequality; the line $\operatorname{Re} s = 3/2$ is excluded, and nothing is required for $\operatorname{Re} s \le 3/2$),

where $L_W(s)$ is Mathlib's WeierstrassCurve.LSeries of $W$ at $s$, defined as follows.

• Mathlib's WeierstrassCurve.LFunction W is not a function of $s$: it is an arithmetic function $a_W : \mathbb{N} \to \mathbb{Z}$ with $a_W(0) = 0$, obtained as an Euler product. Concretely, for each height-one prime ideal $\mathfrak p$ of the ring of integers of the number field $\mathbb{Q}$ (these are the ideals $(p)$ for rational primes $p$), one base-changes $W$ to the $\mathfrak p$-adic completion $K_{\mathfrak p} \cong \mathbb{Q}_p$ with valuation ring $R_{\mathfrak p} \cong \mathbb{Z}_p$, and forms a local Euler factor $E_{\mathfrak p} : \mathbb{N} \to \mathbb{Z}$:

  • Take the minimal model $W'$ of the base-changed curve: $W' = C \cdot W$ for some (non-canonically chosen) admissible change of variables $C$ over $K_{\mathfrak p}$ making the equation integral over $R_{\mathfrak p}$ with the discriminant's valuation maximal among integral models.
  • Let $q$ be the cardinality of the residue field of $R_{\mathfrak p}$ (here $q = p$; the definition uses a cardinality that would take the junk value $0$ for an infinite residue field), and let $a = q + 1 - N$, where $N$ is the cardinality of the set of points of the reduction of $W'$ modulo the maximal ideal, taken in the projective (affine-chart-plus-point-at-infinity) sense, as an integer.
  • The local polynomial $f_{\mathfrak p}(T) \in \mathbb{Z}[T]$ is: $1 - aT + qT^2$ if $W'$ has good reduction (the valuation of $\Delta_{W'}$ equals $1$, i.e. $\Delta_{W'}$ is a unit of $R_{\mathfrak p}$); else $1 - T$ if $W'$ has split multiplicative reduction (valuation of $\Delta_{W'} < 1$, valuation of $c_4$ equal to $1$, and a certain explicit quadratic $c_4 T^2 + a_1 c_4 T - (54 b_6 - 3 b_2 b_4 + a_2 c_4)$ of the integral model splits over the residue field); else $1 + T$ if $W'$ has multiplicative reduction (valuation of $\Delta_{W'} < 1$, valuation of $c_4$ equal to $1$) that is not split; else $1$ (this last case is what remains, namely additive reduction: valuation of $\Delta_{W'} < 1$ and of $c_4$ strictly less than $1$).
  • The local power series is $f_{\mathfrak p}(T)^{-1} \in \mathbb{Z}[[T]]$ (formal inverse; the constant term of $f_{\mathfrak p}$ is $1$ in every branch, so this is the genuine inverse).
  • $E_{\mathfrak p}$ is the arithmetic function obtained by substituting $T = q^{-s}$ formally: if $q > 1$ then $E_{\mathfrak p}(q^k)$ is the coefficient of $T^k$ in $f_{\mathfrak p}(T)^{-1}$ for each $k \ge 0$, and $E_{\mathfrak p}(n) = 0$ for every $n$ not a power of $q$ (if $q \le 1$ the definition falls back to a junk value, namely the constant coefficient times the identity arithmetic function).

  Then $a_W = \prod'_{\mathfrak p} E_{\mathfrak p}$ is the infinite product of these arithmetic functions under Dirichlet convolution, where the infinite product is taken in the topology on arithmetic functions in which convergence means: for each fixed $n$, the $n$-th coefficient of the finite partial products is eventually constant along the cofinite filter on the index set of primes. If this family is not "multipliable" in that sense, the infinite product is assigned the default value $1$, the multiplicative identity arithmetic function ($n \mapsto [n = 1]$).

• With $a_W$ so defined, and writing $\tilde a_W(n) := a_W(n) \in \mathbb{C}$ for the image under $\mathbb{Z} \hookrightarrow \mathbb{C}$,

where $n^s$ is the principal complex power of the positive integer $n$, and $\sum'$ is Mathlib's unconditional sum: it equals the limit of the finite partial sums over finite subsets of $\mathbb{N}$ (directed by inclusion) when the family $(t_n(s))_n$ is unconditionally summable in $\mathbb{C}$, and equals $0$ otherwise. Thus at any $s$ with $\operatorname{Re} s > 3/2$ at which $\sum_n t_n(s)$ fails to be (absolutely) summable, condition 2 requires $\Lambda_i(s) = 0$.

Conclusion. Under these hypotheses,

as functions $\mathbb{C} \to \mathbb{C}$, i.e. $\Lambda_1(s) = \Lambda_2(s)$ for every $s \in \mathbb{C}$ — including all $s$ with $\operatorname{Re}(s) \le \tfrac{3}{2}$, where the hypotheses impose no pointwise constraint beyond differentiability.

Remarks on scope and edge cases (literal content only).

• The theorem does not assert that any $\Lambda$ satisfying $\mathrm{IsLFunction}(W, \Lambda)$ exists; if for some $W$ no such function exists, the statement is vacuously true for that $W$. It also does not assert that $L_W$ itself (or any extension of it) is entire or satisfies the predicate.
• The ellipticity hypothesis $\Delta_W \neq 0$ is carried as an assumption, but none of the notions appearing in the statement (IsLFunction, WeierstrassCurve.LSeries, WeierstrassCurve.LFunction) require it in order to be defined; they are defined for every Weierstrass curve over $\mathbb{Q}$.
• The bundle's BSD.rank (the $\mathbb{Z}$-module rank of the group of rational points of $W$, with the finrank convention that non-finitely-generated or infinite-rank modules get rank $0$) is not referenced by this theorem.
• The Dirichlet coefficients $a_W(n)$ depend on choices made inside Mathlib's definitions (a chosen minimal model at each prime); the theorem quantifies over whatever those choices produce, and the predicate compares $\Lambda_i$ to that specific $L_W$.