I am wondering if there is a multi-dimensional analog of the
Birch/Swinnerton-Dyer (BSD) conjecture.
The recent famous result inching toward resolution of that conjecture is:
Bhargava, Manjul, and Christopher Skinner. "A positive proportion of elliptic curves over $\mathbb{Q}$ have rank one." 2014. arXiv link.
As I understand it: "66.48% of elliptic curves satisfy the (rank part of the) BSD Conjecture."
I see there are books such as
Miyaoka, Joichi, and Thomas Peternell, eds. Geometry of higher dimensional algebraic varieties. Vol. 26. Springer, 1997.
that extend elliptic curves to "higher dimensional algebraic varieties." So,
Q. Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?
Obviously this is a naive question. Thanks for educating me!
Best Answer
Yes, there is a (well-known) analogue for abelian varieties of all dimensions. I was going to suggest looking at the Wikipedia article on the Birch/Swinnerton-Dyer conjecture and was surprised to see that it only talks about the elliptic curve case. (Clearly an opportunity for someone to add a section on generalizations.) There are also further major generalizations due to Tate (concerning the order of vanishing), Beilinson (concerning the transcendental factor in the leading coefficient, analogous to the real period) and Bloch and Kato (concerning the form of the algebraic part of the leading coefficient, analogous to the SHA*Regulator/(torsion)$^2\cdot (c_p$ factors)). [And I'm probably missing some other names here.]
The abelian variety version is undoubtedly described in many places, but since I have a copy handy, I'll mention the statement over $\mathbb Q$ is in my book with Hindry, Diophantine Geometry: An Introduction, Conjecture F.4.1.6 (page 462). It says:
Conjecture Let $A/\mathbb Q$ be an abelian variety, and assume that $L(A/\mathbb Q,s)$ has an analytic continuation to $\mathbb C$. Then $$ \operatorname{ord}_{s=1} L(A/\mathbb Q,s) = \operatorname{rank} A(\mathbb Q)$$ and if the rank is $r$, then $$ \lim_{s\to1} \frac{L(A/\mathbb Q,s)}{(s-1)^r} = \Omega_A \frac{\#\text{SHA}(A/\mathbb Q,s)\cdot\text{Reg}(A/\mathbb Q,s)} {\#A(\mathbb Q)_{\text{tors}}\cdot\#\hat A(\mathbb Q)_{\text{tors}}} \cdot \prod_{p} c_p. $$ Most of the terms have the same meaning as for elliptic curves. Notice the appearance of the torsion in the dual $\hat A$. An elliptic curve is self-dual, that's why $\#(E(\mathbb Q)_{\text{tors}})^2$ appears in the elliptic curve version. Also, the regulator is the pairing between a basis for the free parts of $A(\mathbb Q)$ and $\hat A(\mathbb Q)$ relative to the Poincare bundle on the product $A\times\hat A$. The fudge factors $c_p$ are for the primes of bad reduction, with $c_p=\#A(\mathbb Q_p)/A^0(\mathbb Q_p)$, where $A^0$ is the identity component of the Neron model.
Alternatively, as suggested by user52824, if one lets $\mathcal A_p$ be the reduction of the Neron model of $A$ at $p$ and $\mathcal A_p^0$ its identity component, then $c_p = \#(\mathcal A_p/ \mathcal A^0_p)(\mathbb F_p)$.