Birch and Swinnerton-Dyer conjecture: $\operatorname{ord}_{s=1} L(E,s) = \operatorname{rank} E(\mathbb{Q})$ for every elliptic curve $E/\mathbb{Q}$ (Clay Millennium Prize Problem)

Read-back

Declaration BSD.birch_swinnerton_dyer. Let $W$ be a Weierstrass curve over $\mathbb{Q}$, i.e. an arbitrary quintuple of rational numbers $(a_1, a_2, a_3, a_4, a_6) \in \mathbb{Q}^5$, thought of as the equation

$$Y^2 + a_1 XY + a_3 Y = X^3 + a_2 X^2 + a_4 X + a_6 .$$

Assume (as a typeclass hypothesis) that $W$ is elliptic in Mathlib's sense: the discriminant

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

(with $b_2 = a_1^2 + 4a_2$, $b_4 = 2a_4 + a_1 a_3$, $b_6 = a_3^2 + 4a_6$, $b_8 = a_1^2 a_6 + 4a_2 a_6 - a_1 a_3 a_4 + a_2 a_3^2 - a_4^2$) is a unit of $\mathbb{Q}$; since $\mathbb{Q}$ is a field this is exactly the condition $\Delta(W) \neq 0$. No other hypothesis is placed on $W$.

The theorem asserts that there exists (at least one; not claimed unique) a function $\Lambda : \mathbb{C} \to \mathbb{C}$ satisfying the following three conditions simultaneously.

(i) Entire. $\Lambda$ is complex-differentiable at every point of $\mathbb{C}$.

(ii) Agreement with Mathlib's Dirichlet series 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 following specific complex number. Let $\mathcal{L}_W : \mathbb{N} \to \mathbb{Z}$ be the arithmetic function described in (ii-a) below (with $\mathcal{L}_W(0) = 0$). Then

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

where the primed sum is Mathlib's unconditional sum: if the family $(t_n(s))_n$ is summable in $\mathbb{C}$ (for complex numbers this is the same as absolute convergence) it is the value of the sum, and otherwise it is the junk value $0$. Nothing in the statement asserts that the series converges for any $s$; wherever it fails to converge with $\operatorname{Re} s > \tfrac32$, condition (ii) requires $\Lambda(s) = 0$ there. Here $n^s$ is the complex power $\exp(s \log n)$ for $n \ge 1$, and $\mathcal{L}_W(n) \in \mathbb{Z}$ is coerced to $\mathbb{C}$.

(ii-a) The arithmetic function $\mathcal{L}_W$ (the "L-function as a formal Dirichlet series"). Let $\mathcal{O}$ be Mathlib's ring of integers of the number field $\mathbb{Q}$, and let $\mathfrak{p}$ range over the height-one spectrum of $\mathcal{O}$, i.e. over all nonzero prime ideals of $\mathcal{O}$. For each such $\mathfrak{p}$, let $K_\mathfrak{p}$ be the completion of $\mathbb{Q}$ with respect to the $\mathfrak{p}$-adic valuation and let $R_\mathfrak{p} \subseteq K_\mathfrak{p}$ be its valuation ring (the elements of valuation $\le 1$), a discrete valuation ring with maximal ideal $\mathfrak{m}_\mathfrak{p}$ and residue field $\kappa_\mathfrak{p} = R_\mathfrak{p}/\mathfrak{m}_\mathfrak{p}$. Write $W_\mathfrak{p}$ for the base change of $W$ to $K_\mathfrak{p}$ (same five coefficients, viewed in $K_\mathfrak{p}$). Then

$$\mathcal{L}_W = {\prod_{\mathfrak{p}}}' \; E_\mathfrak{p},$$

the primed product being Mathlib's unconditional infinite product $\prod'$ of arithmetic functions under Dirichlet convolution, taken in the topology of pointwise convergence where $\mathbb{Z}$ carries the discrete topology (so the product converges iff for every $n$ the $n$-th coefficient of finite partial products is eventually constant). If the family $(E_\mathfrak{p})_\mathfrak{p}$ is not multipliable in this sense, the product is the junk value $1$, the arithmetic function $\varepsilon$ with $\varepsilon(1) = 1$ and $\varepsilon(n) = 0$ for $n \ne 1$. The local factor $E_\mathfrak{p} : \mathbb{N} \to \mathbb{Z}$ is built as follows.

• Minimal model. $W'_\mathfrak{p}$ is a chosen (by the axiom of choice) change of variables of $W_\mathfrak{p}$ over $K_\mathfrak{p}$ that is minimal: it is integral (equal to the base change of some Weierstrass curve with coefficients in $R_\mathfrak{p}$), and among all changes of variables $C$ of $W_\mathfrak{p}$ that are integral, the multiplicative valuation $v(\Delta(C \cdot W_\mathfrak{p})) \in \mathbb{Z}^{\mathrm{m}0}$ (which is $\le 1$ for integral curves; larger multiplicative valuation means smaller additive valuation) is maximal at $C = 1$, i.e. at $W'_\mathfrak{p}$ itself.
• Residue-field cardinality. $q_\mathfrak{p} := \operatorname{Nat.card}(\kappa_\mathfrak{p}) \in \mathbb{Z}$, the number of elements of $\kappa_\mathfrak{p}$ if $\kappa_\mathfrak{p}$ is finite, and the junk value $0$ if $\kappa_\mathfrak{p}$ is infinite.
• Reduction and point count. $\widetilde{W}_\mathfrak{p}$ is the Weierstrass curve over $\kappa_\mathfrak{p}$ obtained from a chosen integral model of $W'_\mathfrak{p}$ (a curve over $R_\mathfrak{p}$ whose base change to $K_\mathfrak{p}$ is $W'_\mathfrak{p}$) by reducing each coefficient modulo $\mathfrak{m}_\mathfrak{p}$. Let $N_\mathfrak{p} := \operatorname{Nat.card}\bigl(\widetilde{W}_\mathfrak{p}(\kappa_\mathfrak{p})\bigr)$, where $\widetilde{W}_\mathfrak{p}(\kappa_\mathfrak{p})$ is the type consisting of one point "zero" (at infinity) together with all pairs $(x,y) \in \kappa_\mathfrak{p}^2$ that satisfy the Weierstrass equation of $\widetilde{W}_\mathfrak{p}$ and are nonsingular there (at least one of the two partial derivatives $a_1 y - (3x^2 + 2a_2 x + a_4)$, $2y + a_1 x + a_3$ of the reduced curve is nonzero at $(x,y)$). Again $\operatorname{Nat.card}$ is $0$ if this type is infinite. Set $a_\mathfrak{p} := q_\mathfrak{p} + 1 - N_\mathfrak{p} \in \mathbb{Z}$.
• Local polynomial $f_\mathfrak{p}(T) \in \mathbb{Z}[T]$, decided by the first applicable case (classically):
  1. if $W'_\mathfrak{p}$ has good reduction — meaning $W'_\mathfrak{p}$ is minimal and $v(\Delta(W'_\mathfrak{p})) = 1$ (the discriminant is a unit of $R_\mathfrak{p}$) — then $f_\mathfrak{p}(T) = 1 - a_\mathfrak{p} T + q_\mathfrak{p} T^2$;
  2. else if $W'_\mathfrak{p}$ has split multiplicative reduction — meaning it is minimal, $v(\Delta(W'_\mathfrak{p})) < 1$, $v(c_4(W'_\mathfrak{p})) = 1$ where $c_4 = b_2^2 - 24 b_4$, and, writing $I$ for the chosen integral model, the polynomial $c_4(I)\,T^2 + a_1(I) c_4(I)\, T - \bigl(54\, b_6(I) - 3\, b_2(I) b_4(I) + a_2(I) c_4(I)\bigr)$ with coefficients reduced into $\kappa_\mathfrak{p}$ lies in the multiplicative closure of the constant polynomials and the monic linear polynomials over $\kappa_\mathfrak{p}$ (i.e. splits into linear factors) — then $f_\mathfrak{p}(T) = 1 - T$;
  3. else if $W'_\mathfrak{p}$ has multiplicative reduction (minimal, $v(\Delta) < 1$, $v(c_4) = 1$, but the quadratic above does not split) then $f_\mathfrak{p}(T) = 1 + T$;
  4. otherwise (in particular in the remaining exhaustive case $v(\Delta) < 1$ and $v(c_4) < 1$, "additive reduction") $f_\mathfrak{p}(T) = 1$.
• Local power series. $g_\mathfrak{p}(T) \in \mathbb{Z}[[T]]$ is the formal inverse of $f_\mathfrak{p}(T)$ computed by the recursive inverse-of-a-unit formula with the constant term taken to be the unit $1$; since every $f_\mathfrak{p}$ above has constant coefficient $1$, this is the genuine reciprocal $1/f_\mathfrak{p}(T)$.
• Local Euler factor. If $q_\mathfrak{p} > 1$ then $E_\mathfrak{p}(n) = [T^k]\, g_\mathfrak{p}$ when $n = q_\mathfrak{p}^{\,k}$ for some $k \ge 0$, and $E_\mathfrak{p}(n) = 0$ if $n$ is not a power of $q_\mathfrak{p}$ (in particular $E_\mathfrak{p}(0)=0$). If $q_\mathfrak{p} \le 1$ (which happens exactly when the junk value $q_\mathfrak{p}=0$ occurs, or $q_\mathfrak{p}=1$), then $E_\mathfrak{p}$ is instead the junk value $[T^0] g_\mathfrak{p} \cdot \varepsilon = \varepsilon$, the multiplicative identity.

(iii) Order of vanishing at $1$ equals the rank. In the extended natural numbers $\mathbb{N}\cup\{\infty\}$,

$$\operatorname{ord}_{z=1} \Lambda = \operatorname{rank}(W),$$

where the right-hand side is the natural number $\operatorname{rank}(W)$ described in (iii-b), coerced into $\mathbb{N}\cup\{\infty\}$, and the left-hand side is Mathlib's analyticOrderAt, defined by cases:

• if $\Lambda$ is not analytic at $1$, the value is the junk value $0$ — but under (i), complex differentiability on all of $\mathbb{C}$ implies analyticity at $1$, so this case cannot occur for a $\Lambda$ satisfying (i);
• if $\Lambda$ is analytic at $1$ and vanishes identically on some neighbourhood of $1$, the value is $\infty$;
• otherwise the value is the unique $n \in \mathbb{N}$ such that, on some neighbourhood of $1$, $\Lambda(z) = (z-1)^n\, g(z)$ with $g$ analytic at $1$ and $g(1) \ne 0$ (so $n = 0$ iff $\Lambda(1) \ne 0$).

Because the right-hand side is a finite natural number, equality in $\mathbb{N}\cup\{\infty\}$ excludes the value $\infty$: any $\Lambda$ witnessing the theorem is not identically zero near $1$ (hence, being entire, not identically zero on $\mathbb{C}$). In particular the constant function $\Lambda \equiv 0$ does not satisfy (iii). Conversely, by (i) and the identity theorem, if a witness exists it is uniquely determined by its values on $\{\operatorname{Re} s > \tfrac32\}$, i.e. by the function $L_W$ there, including any junk zeros of $L_W$ arising from non-summability.

(iii-b) The natural number $\operatorname{rank}(W)$. Let $W(\mathbb{Q})$ denote the type of points of $W$ in affine coordinates: the single point "zero" (at infinity) together with all $(x,y) \in \mathbb{Q}^2$ satisfying $y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$ and nonsingular there (at least one of $a_1 y - (3x^2 + 2a_2 x + a_4)$, $2y + a_1 x + a_3$ nonzero). Mathlib equips $W(\mathbb{Q})$ with the abelian group structure given by the chord–tangent (secant–tangent) law with "zero" as identity, and hence with a $\mathbb{Z}$-module structure. Then

$$\operatorname{rank}(W) := \operatorname{finrank}_{\mathbb{Z}} W(\mathbb{Q}) = \operatorname{toNat}\Bigl(\sup\bigl\{\, |S| \;:\; S \subseteq W(\mathbb{Q}) \text{ is } \mathbb{Z}\text{-linearly independent}\,\bigr\}\Bigr),$$

where the supremum of cardinals is taken over all subsets $S$ such that no nontrivial finite $\mathbb{Z}$-linear combination of distinct elements of $S$ is zero (so torsion points never belong to such an $S$), and $\operatorname{toNat}$ sends a finite cardinal to the corresponding natural number and every infinite cardinal to $0$. The statement does not assume or assert that $W(\mathbb{Q})$ is finitely generated; if the supremum above is infinite, $\operatorname{rank}(W) = 0$ by convention, and condition (iii) then requires $\Lambda(1) \neq 0$.

Summary of the literal claim. For every quintuple $(a_1,a_2,a_3,a_4,a_6) \in \mathbb{Q}^5$ with $\Delta \ne 0$, there is an entire function $\Lambda$ that coincides on the open half-plane $\operatorname{Re} s > \tfrac32$ with the (junk-tolerant) Dirichlet series $L_W$ of the Euler product of the local factors $E_\mathfrak{p}$ described above, and whose order of vanishing at $s = 1$ is the finite natural number $\operatorname{finrank}_{\mathbb{Z}} W(\mathbb{Q})$ (with the conventions: order $\infty$ excluded, infinite $\mathbb{Z}$-rank read as $0$, and non-convergent sums or products, or infinite residue fields, read as $0$ or $\varepsilon$ respectively).