Verifying the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves

William Stein Harvard University Math 129: May 5, 2005 This talk reports on a project to verify the Birch and Swinnerton-Dyer conjecture for many specific elliptic curves over ℚ.

Joint Work: Grigor Grigorov, Andrei Jorza, Corina Patrascu, Stefan Patrikis

Thanks: John Cremona, Stephen Donnelly, Ralph Greenberg, Grigor Grigorov, Barry Mazur, Robert Pollack, Nick Ramsey, Tony Scholl, Micahel Stoll.

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

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

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}} )·_{p|Δ}^{∏} L_{p}(E, s)

Here a_{p} = p + 1 − #E(𝔽_{p}) for all p ∤ Δ, where Δ is divisible by the primes of bad reduction for E. We do not include the factors L_{p}(E, s) at bad primes here.

Real Graph of the L-Series of

y^{2} + y = x^{3} − x

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

8

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.

What about Taylor series of L(E, s)

around s = 1?

Taylor Series

For y^{2} + y = x^{3} − x, the Taylor series about 1 is

L(E, s) = 0 + (s − 1)0.3059997 …

+(s − 1)^{2}0.18636 … + ⋯

In general, it’s

L(E, s) = c_{0} + c_{1}s + c_{2}s^{2} + · · · .

Big Mystery: Do these Taylor coefficients c_{n} have any deep arithmetic meaning?

BSD Formula Conjecture

Let r = ord_{s=1} L(E, s). Then Birch and Swinnerton-Dyer made

a famous guess for the first nonzero coefficient c_{r}:

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

• #E(ℚ)_{tor} – torsion order

• t_{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

What about c_{r+1} , c_{r+2} , etc?

I think nobody has even a wild and crazy guess for an interpretation of these.

They are probably not “periods” like c_{r} is, so perhaps should not have any nice interpretation...

Motivating Problem 1

Motivating Problem 1. For specific curves, compute every quantity appearing in the BSD formula conjecture in practice. NOTE: This is not meant as a theoretical problem about computability, though by compute we mean “compute with proof.”

Status

1. When r_{an} = ord_{s=1} L(E, s) ≤ 3, then we can compute r_{an}. Open Problem: Show that r_{an} ≥ 4 for some elliptic curve.

2. “Relatively easy” to compute #E(ℚ)_{tor} , c_{p}, Ω_{E}.

3. Computing Reg _{E} essentially same as computing E(ℚ); interesting and sometimes very difficult.

4. Computing #� (E) is currently very very difficult.

Theorem (Kolyvagin):

r_{an} ≤ 1 ⟹ � (E) is finite (with bounds)

Open Problem:

Prove that � (E) is finite for some E with r_{an} ≥ 2.

Victor Kolyvagin

Kolyvagin’s work on Euler systems is crucial to our project.

Motivating Problem 2: Cremona’s Book

Motivating Problem 2. Prove BSD for every elliptic curve over ℚ of conductor at most 1000, i.e., in Cremona’s book.

1. By Tate’s isogeny invariance of BSD, it suffices to prove BSD for each optimal elliptic curve of conductor N ≤ 1000.

2. Rank part of the conjecture has been verified by Cremona for all curves with N ≤ 40000.

3. All of the quantities in the conjecture, except for #� (E/ℚ), have been computed by Cremona for conductor ≤ 40000.

4. Cremona (Ch. 4, pg. 106): We have 2 ∤ #� (E) for all optimal curves with conductor ≤ 1000 except 571A, 960D, and 960N. So we can mostly ignore 2 henceforth.

John Cremona

John Cremona’s software and book are crucial to our project.

The Four Nontrivial � ’s

Conclusion: In light of Cremona’s book and the above results, the problem is to show that � (E) is trivial for all but the following four optimal elliptic curves with conductor at most 1000:

Curve a-invariants  � (E)_{?}
571A [0,-1,1,-929,-105954]  4
681B [1,1,0,-1154,-15345]  9
960D [0,-1,0,-900,-10098]  4
960N [0,1,0,-20,-42]  4

We first deal with these four.

Divisor of Order:

1. Using a 2-descent we see that 4 | #� (E) for 571A, 960D, 960N.

2. For E = 681B: Using visibility (or a 3-descent) we see that 9 | #� (E).

Multiple of Order:

1. For E = 681B, the mod 3 representation is surjective, and 3 || [E(K) : y_{K}] for K = ℚ(√−8), so Kolyvagin’s theorem implies that #� (E) = 9, as required.

2. Kolyvagin’s theorem and computation ⟹ #� (E) = 4^{?} for 571A, 960D, 960N.

3. Using MAGMA’s FourDescent command, we compute Sel^{(4)} (E/ℚ) for 571A, 960D, 960N and deduce that #� (E) = 4. (Note: MAGMA Documentation currently misleading.)

The Eighteen Optimal Curves of Rank

> 1

There are 18 curves with conductor ≤ 1000 and rank > 1 (all have rank 2):

389A, 433A, 446D, 563A, 571B, 643A, 655A, 664A, 681C, 707A, 709A, 718B, 794A, 817A, 916C, 944E, 997B, 997C

For these E nobody currently knows how to show that � (E) is finite, let alone trivial. (But mention, e.g., p-adic L-functions.)

Motivating Problem 3: Prove the BSD Conjecture for all elliptic curve over ℚ of conductor at most 1000 and rank ≤ 1.

SECRET MOTIVATION: Our actual motivation is to unify and extend results about BSD and algorithms for elliptic curves. Also, the computations give rise to many surprising and interesting examples.

Our Goal

• There are 2463 optimal curves of conductor at most 1000.

• Of these, 18 have rank 2, which leaves 2445 curves.

• Of these, 2441 are conjectured to have trivial � .

Thus our goal is to prove that #� (E) = 1 for these 2441 elliptic curves.

Our Strategy

1. [Find an Algorithm] Based on deep work of Kolyvagin,

Kato, et al. Input: An elliptic curve over ℚ with r_{an} ≤ 1. Output: B ≥ 1 such that if p ∤ B, then p ∤ #� (E).

2. [Compute] Run the algorithm on our 2441 curves.

3. [Reducible] If E[p] is reducible say nothing.

Kolyvagin Bound on #� (E)

INPUT: An elliptic curve E over ℚ with r_{an} ≤ 1. OUTPUT: Odd B ≥ 1 such that if p ∤ 2B, then p ∤ #� (E/ℚ).

1. [Choose K] Choose a quadratic imaginary field K = ℚ(√D) with certain properties, such that E/K has analytic rank 1. Assume ℚ(E[p]) has degree # GL_{2}(𝔽_{p}).

2. [Compute Mordell-Weil]

(a) If r = 0, compute generator z for E^{D}(ℚ) mod torsion.

(b) If r = 1, compute generator z for E(ℚ) mod torsion.

3. [Index of Heegner point] Compute the “Heegner point” y_{K} ∈ E(K) associated to K. This is a point that comes from the “modularity” map X_{0}(N) → E.

4. [Finished] Output B = I ·A, where A is the product of primes such that ℚ(E[p]) has degree less than # GL_{2}(𝔽_{p}).

Theorem (Kolyvagin): p ∤ 2B ⟹ p ∤ #� (E/ℚ).