2016-03-04-bsd.sagews

Lecture March 4, 2016

The Birch and Swinnerton-Dyer Conjecture

(part 1 -- elliptic curves over $\QQ$)

William Stein

Overview

BSD is a conjecture about the arithmetic invariants of an abelian variety $A$ over a global field $K$.

BSD Rank: $\text{ord}_{s=1} L(A,s) = \text{rank}(A(K))$

The special case for $E/\QQ$ of the above conjecture is a Clay Math millenium prize problem, which emphasizes its importance.

BSD Formula: write down every number you can think of that is associated to the isomorphism class of $A$ and put them together in a formula that looks a lot like the analytic class number.

Here's what BSD looks like for $E=A$ an elliptic curve defined over $\QQ$, where $r=\text{rank}(E(\QQ))$.

$$\frac{L^{(r)}(E,1)}{r!} = \frac{\Omega_E \cdot \prod c_p \cdot \text{Reg}_E \cdot \#\text{Sha}_E}{\#E(\QQ)_{\rm tor}^2}.$$

"This conjecture relates the behavior of vanishing of a function at a point at which it is not known to be defined to the order of a group that is not known to be finite." -- John Tate, before we knew that $L(E,s)$ is defined everywhere.

Discuss the definition of each quantity...

• $L(E,s) = \sum a_n n^s$, where $f_E = \sum a_n q^n$ is the corresponding modular form.

• $\Omega_E = \int_{\RR} E(\RR) d\omega_E$ is the value of the "elliptic integral" that gave elliptic curves their name; it can be very efficiently computed using the Gauss Arithmetic-Geometric mean (AGM).

• $c_p$ -- the Tamagawa numbers, which are only possibly not 1 for primes of bad reduction. They are $\leq 4$, except when $p$ is a prime of split multiplicative reduction, in which case $c_p = \text{ord}_p(\Delta)$, where $\Delta$ is the discriminant of a minimal Weierstrass equation of $E$.

• $\text{Reg}_E$ -- the regulator of $E$, which is the absolute value of the discriminant of the height quadratic form $h:E(\QQ)\to \RR$, which measures the complexity of a point.

• $E(\QQ)_{\rm tor}$ -- order of the torsion subgroup.

• $\#\text{Sha}_E$ -- the Shafarevich-Tate group of $E$. This is the cardinality of the set (actually group) of equivalence classes of genus one curves $X$ equipped with an action $E\times X \to X$ satisfying the axioms of a simply transitive group action such that $X$ has a point over $\QQ_v$ for all places $v$ of $\QQ$ (i.e., $p$-adics and $\RR$). Equivalently, is $$\#\text{Sha}_E =\ker H^1(\QQ,E) \to \prod H^1(\QQ_v, E).$$

Exercise right now: Compute as much as you can using Sage (and tab completion) about the two conjectures above for the elliptic curve "11a1" of conductor 11.

(Make table on the blackboard with everybody contributing)

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## Computational status of each quantity

- $r_{\rm an} = \text{ord}_{s=1} L(E,s)$ -- if $r_{\rm an} \leq 3$ then there is an algorithm to compute it.  It is an open problem to prove that there exists an elliptic curve with $r_{\rm an} \geq 4$!
- $L(E,s)$ - can compute it at any point to any number of digits of precision.
- $\Omega_E$  -- trivial to compute quickly.
- $c_p$ -- trivial to compute quickly using "Tate's algorithm" (in general)
- $\text{Reg}_E$ -- requires computing $E(\QQ)$; this is very hard in practice, and the only known "algorithm" to do this assumes either that (1) BSD rank is true, or that (2) $\#\text{Sha}_E$ is finite.  Both assumptions are unknown in general, but both are known when $r_{\rm an} \leq 1$.
- $E(\QQ)_{\rm tor}$ -- trivial; Mazur a priori upper bound; use $E(\QQ) \hookrightarrow E(\FF_p)$ and formulas.
- $\#\text{Sha}_E$ -- computable when $r_{\rm an} \leq 1$.  It is an open problem to show that there exists an elliptic curve of rank $\geq 2$ such that $\text{Sha}_E$ is finite.

Computational status of each quantity

• $r_{\rm an} = \text{ord}_{s=1} L(E,s)$ -- if $r_{\rm an} \leq 3$ then there is an algorithm to compute it. It is an open problem to prove that there exists an elliptic curve with $r_{\rm an} \geq 4$!

• $L(E,s)$ - can compute it at any point to any number of digits of precision.

• $\Omega_E$ -- trivial to compute quickly.

• $c_p$ -- trivial to compute quickly using "Tate's algorithm" (in general)

• $\text{Reg}_E$ -- requires computing $E(\QQ)$; this is very hard in practice, and the only known "algorithm" to do this assumes either that (1) BSD rank is true, or that (2) $\#\text{Sha}_E$ is finite. Both assumptions are unknown in general, but both are known when $r_{\rm an} \leq 1$.

• $E(\QQ)_{\rm tor}$ -- trivial; Mazur a priori upper bound; use $E(\QQ) \hookrightarrow E(\FF_p)$ and formulas.

• $\#\text{Sha}_E$ -- computable when $r_{\rm an} \leq 1$. It is an open problem to show that there exists an elliptic curve of rank $\geq 2$ such that $\text{Sha}_E$ is finite.

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**Exercise:** Choose a curve of rank $2$ and compute what you can about the BSD formula for it.

Hint: use `elliptic_curves.rank(2)` to get at some curves of algebraic rank $2$.

Exercise: Choose a curve of rank $2$ and compute what you can about the BSD formula for it.

Hint: use elliptic_curves.rank(2) to get at some curves of algebraic rank $2$.

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**Exercise:** Choose a curve of rank $4$ and compute what you can about the BSD formula for it.

Hint: use `elliptic_curves.rank(4)` to get at some curves of algebraic rank $4$.

Exercise: Choose a curve of rank $4$ and compute what you can about the BSD formula for it.

Hint: use elliptic_curves.rank(4) to get at some curves of algebraic rank $4$.

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If time remains, say more about "$r_{\rm an} = \text{ord}_{s=1} L(E,s)$ -- if $r_{\rm an} \leq 3$ then there is an algorithm to compute it."

If time remains, say more about "$r_{\rm an} = \text{ord}_{s=1} L(E,s)$ -- if $r_{\rm an} \leq 3$ then there is an algorithm to compute it."

Next time: BSD for abelian varieties of $\text{GL}_2$-type.

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