May 13, 2024: The Birch–Swinnerton-Dyer conjecture

The Conjecture of Birch and Swinnerton-Dyer (“BSD” for short), originally formulated by Birch and Swinnerton-Dyer in the 1960s for elliptic curves over $\mathbf{Q}$ , is one of the most important open conjectures in number theory. For example, it is one of the seven “Millennium Problems”, for whose solution the Clay Foundation is offering a million dollars each. It relates in a surprising way analytic invariants of an elliptic curve $E$ , which are obtained via its $L$ -series from its local properties (essentially the number of points modulo $p$ on $E$ , for all prime numbers $p$ ), to global arithmetic invariants like the rank of the Mordell–Weil group $E(\mathbf{Q})$ , its regulator, and the rather mysterious Tate–Shafarevich group $\mathrm{Ш}(E)$ . The conjecture has been generalized to cover all abelian varieties over all algebraic number fields. It consists of two parts, which we will explain for the case of an abelian variety $A$ of dimension $g$ over $\mathbf{Q}$ .

The $L$ -function. One attaches to $A$ its $L$ -function $L(A,s)$ , which is defined by an Euler product over all prime numbers $p$ . If $A$ is the Jacobian variety of a curve $X$ of genus $g$ , the Euler factor at $p$ for a prime $p$ of good reduction is determined by the number of $\mathbf{F}_{p^n}$ -points on the mod $p$ reduction of $X$ for $n \leq g$ . It follows from the Weil conjectures for varieties over finite fields that the Euler product converges for $\operatorname{Re}(s) > \frac{3}{2}$ to a holomorphic function. A standard conjecture predicts that $L(A,s)$ extends to an entire function; this is known when $A$ is modular, i.e., occurs as an isogeny factor of the Jacobian $J_0(N)$ of one of the modular curves $X_0(N)$ . By the Modularity Theorem of Wiles and others, this is always the case when $A$ is an elliptic curve over $\mathbf{Q}$ (this is now a special case of Serre’s Modularity Conjecture).

The global BSD invariants. We now introduce the relevant global invariants of $A$ . By the Mordell–Weil Theorem, the abelian group $A(\mathbf{Q})$ of rational points on $A$ is finitely generated, so it splits as $A(\mathbf{Q}) \cong A(\mathbf{Q})_\mathrm{tors}\oplus \mathbf{Z}^r$ , where $A(\mathbf{Q})_\mathrm{tors}$ is the finite torsion subgroup and $r$ is a nonnegative integer, the rank of $A(\mathbf{Q})$ . There is a natural positive definite quadratic form $\hat{h}$ on $A(\mathbf{Q}) \otimes_\mathbf{Z}\mathrm{R}\cong \mathrm{R}^r$ , the canonical height, turning $A(\mathbf{Q})/A(\mathbf{Q})_\mathrm{tors}$ into a lattice in a euclidean vector space. The squared covolume of this lattice (equivalently, the determinant of the Gram matrix of $\hat{h}$ with respect to a lattice basis) is the regulator $\mathrm{Reg}_A$ . The final global arithmetic invariant of $A$ that we need is the Tate–Shafarevich group $\mathrm{Ш}(A)$ . It can be defined as the localization kernel $\mathrm{Ш}(A) = \ker\Big({\mathrm{H}}^1(\mathbf{Q},A) \to \bigoplus_{v} {\mathrm{H}}^1(\mathbf{Q}_v,A)\Big)$ in Galois cohomology; here $\mathbf{Q}_v$ denotes the completion of $\mathbf{Q}$ with respect to a place $v$ and the direct sum is over all places of $\mathbf{Q}$ . Geometrically, $\mathrm{Ш}(A)$ is the group of equivalence classes of everywhere locally trivial $A/\mathbf{Q}$ -torsors. This group is conjectured to be finite, but this is not known in general.

The local BSD invariants. We also need some local invariants. To each prime $p$ , one associates the Tamagawa number $c_p(A)$ ; this is the number of connected components of the special fiber at $p$ of the Néron model $\mathcal{A}/\mathbf{Z}$ of $A$ and equals $1$ for all primes of good reduction. Let $(\omega_1, \ldots, \omega_g)$ be the pull-back to ${\mathrm{H}}^0(A, \Omega^1)$ of a basis of the free $\mathbf{Z}$ -module ${\mathrm{H}}^0(\mathcal{A}, \Omega^1)$ of rank $g$ . Then the real period of $A$ is the volume of $A(\mathrm{R})$ measured using $|\omega_1 \wedge \dots \wedge \omega_g|$ : $\Omega_A = \int_{A(\mathrm{R})} |\omega_1 \wedge \dots \wedge \omega_g|$ .

The weak and the strong BSD conjecture. The weak BSD or BSD rank conjecture says that $L(A,s)$ has an analytic continuation to a neighborhood of $s = 1$ and $r_\mathrm{an}:= \operatorname{ord}_{s=1} L(A, s) = r \,.$ The order of vanishing of $L(A, s)$ at $s = 1$ is also called the analytic rank of $A$ .

The strong BSD conjecture says that in addition $\mathrm{Ш}(A)$ is finite and $L^*(A,1) := \lim_{s \to 1} (s-1)^{-r} L(A, s) = \frac{\Omega_A \prod_p c_p(A) \cdot \mathrm{Reg}_A \#\mathrm{Ш}(A)}{\#A(\mathbf{Q})_\mathrm{tors}\#A^\vee(\mathbf{Q})_\mathrm{tors}} \,.$ Here $A^\vee$ is the dual abelian variety; it is isomorphic to $A$ when $A$ is a Jacobian, or, more generally, when $A$ is principally polarized.

Since all the other invariants of $A$ can (usually) be computed at least numerically, we define the analytic order of Sha to be $\#\mathrm{Ш}(A)_\mathrm{an}:= \frac{L^*(A,1)}{\Omega_A \mathrm{Reg}_A} \cdot \frac{\#A(\mathbf{Q})_\mathrm{tors}\#A^\vee(\mathbf{Q})_\mathrm{tors}}{\prod_p c_p(A)} \,.$ Assuming the BSD rank conjecture, strong BSD can then be phrased as “ $\mathrm{Ш}(A)$ is finite and $\#\mathrm{Ш}(A) = \#\mathrm{Ш}(A)_\mathrm{an}$ .”

Even the weak BSD conjecture for elliptic curves over $\mathbf{Q}$ is wide open in general (this is the Clay Millennium Problem mentioned above). However, the strong BSD conjecture has been verified for many “small” elliptic curves; see below. The goal of the project [K–Stoll 2023] was to verify the strong BSD conjecture for the first time for a number of abelian surfaces $A$ , in a situation where it cannot be reduced to BSD for some elliptic curves. Concretely, this means that $A$ is absolutely simple.