Elliptic Curves and the Birch and

Swinnerton-Dyer Conjecture

William Stein Harvard University

http://modular.fas.harvard.edu/129-05/

Math 129: April 5, 2005 This talk is a first introduction to elliptic curves and the Birch and Swinnerton-Dyer conjecture.

Elliptic Curves over the Rational Numbers ℚ

An elliptic curve is a nonsingular plane cubic curve with a rational point (possibly “at infinity”).

x y_{y^{2} + y = x^{3} − x}  ^{EXAMPLES} y^{2} + y = x^{3} − x^{x3 + y3 = z3 (projective)} y^{2} = x^{3} + ax + b 3x^{3} + 4y^{3} + 5z^{3} = 0

The Group Operation

x y_{y^{2} + y = x^{3} − x}^{∞}^{Point at infinity}^{⊕} ^{=}^{(−1, 0) ⊕ (0, −1) = (2, 2)}_{The set of rational points}_{on E forms an abelian group.}

The First 150 Multiples of (0, 0)

(The bluer the point, the bigger the multiple.)

Fact: The group E(ℚ) is infinite cylic, generated by (0, 0).

In contrast, y^{2} + y = x^{3} − x^{2} has only 5 rational points!

x

y

y^{2} + y = x^{3} − x

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Mordell’s Theorem

Theorem (Mordell). The group E(ℚ) of rational points on an elliptic curve is a finitely generated abelian group, so

E(ℚ) ≅ ℤ^{r} ⊕ T,

with T = E(ℚ)_{tor} finite. Mazur classified the possibilities for T. Folklore conjecture: r can be arbitrary, but the biggest r ever found is (probably) 24.

Conjectures Proliferated

“The subject of this lecture is rather a special one. I want to describe some computations undertaken by myself and Swinnerton-Dyer on EDSAC, by which we have calculated the zeta-functions of certain elliptic curves. As a result of these computations we have found an analogue for an elliptic curve of the Tamagawa number of an algebraic group; and conjectures have proliferated. [...] though the associated theory is both abstract and technically complicated, the objects about which I intend to talk are usually simply defined and often machine computable; experimentally we have detected certain relations between different invariants, but we have been unable to approach proofs of these relations, which must lie very deep.” – Birch 1965

Birch and Swinnerton-Dyer (Utrecht, 2000)

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The L-Function

Theorem (Wiles et al., Hecke) The following function extends to a holomorphic function on the whole complex plane:

L^{∗}(E, s) = _{p∤Δ}^{∏} (\frac{1}{1 − a_{p} · p^{−s} + p · p^{−2s}} ) .

Here a_{p} = p + 1 − #E(𝔽_{p}) for all p ∤ Δ_{E}. Note that formally,

L^{∗}(E, 1) = _{p∤Δ}^{∏} (\frac{1}{1 − a_{p} · p^{−1} + p · p^{−2}} ) = _{p∤Δ}^{∏} (\frac{p}{p − a_{p} + 1} ) = _{p∤Δ}^{∏} \frac{p}{N_{p}}

Standard extension to L(E, s) at bad primes.

Real Graph of the L-Series of y^{2} + y = x^{3} − x

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More Graphs of Elliptic Curve L-functions

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The Birch and Swinnerton-Dyer Conjecture

Conjecture: Let E be any elliptic curve over ℚ. The order of vanishing of L(E, s) as s = 1 equals the rank of E(ℚ).

The Kolyvagin and Gross-Zagier Theorems

Theorem: If the ordering of vanishing ord_{s=1} L(E, s) is ≤ 1, then the conjecture is true for E.

BSD Conjectural Formula

\frac{L^{(r)}(E, 1)}{r!} = \frac{Ω_{E} · Reg _{E} · ∏_{p|N} c_{p}}{#E(ℚ)_{tor}^{2}} · #� (E)

• #E(ℚ)_{tor} – torsion order

• c_{p} – Tamagawa numbers

• Ω_{E} – real volume ∫_{E(ℝ)} ω_{E}

• Reg _{E} – regulator of E

• � (E) = Ker (H^{1}(ℚ, E) → ⨁_{v} H^{1}(ℚ_{v}, E)) – Shafarevich-Tate group

One of My Research Projects

Project. Find ways to compute every quantity appearing in the BSD conjecture in practice.

NOTES:

1. This is not meant as a theoretical problem about computability, though by compute we mean “compute with proof.”

2. I am also very interested in the same question but for modular abelian varieties.

3. Working with Harvard Undergrads: Stephen Patrikas, Andrei Jorza, Corina Patrascu.