The Birch and Swinnerton-Dyer Conjecture for Individual Elliptic Curves

William Stein

University of California, San Diego

http://modular.fas.harvard.edu/

Oberwolfach: July 2005 This talk reports on a long-term collaborative project to verify the Birch and Swinnerton-Dyer conjecture for specific elliptic curves.

Collaborators: Stephen Donnelly, Grigor Grigorov, Andrei Jorza, Stefan Patrikis, Michael Stoll, Corina Tarnita-Patrascu.

Thanks: John Cremona, Noam Elkies, Ralph Greenberg, Barry Mazur, Robert Pollack, Nick Ramsey, and Tony Scholl.

Manin Constant Assumption

For the rest of this talk I will officially assume that the Manin constant of every elliptic curve of conductor ≤ 1000 is 1. This is likely an easy computation, but it seems to not have been done carefully yet.

Main Theorem

Thereom. Suppose E is a non-CM elliptic curve of conductor ≤ 1000 and rank ≤ 1 and p is a prime that does not divide any Tamagawa number of E and that E has no p-isogeny. Then the p-part of the full BSD conjectural formula is true for E.

Once upon a time...

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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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Absolute Value of L-series on Complex

Plane for y^{2} + y = x^{3} − x

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 BSD rank conjecture is true for E.

Refined 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} – order of torsion

• 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

The Shafarevich-Tate Group

� (E) = Ker (H^{1}(ℚ, E) → _{v}^{⨁} H^{1}(ℚ_{v}, E))

The elements of � (E) correspond to (classes of) genus one curves X with Jacobian E that have a point over each p-adic field and ℝ. For example, the curve 3x^{3} + 4y^{3} + 5z^{3} = 0 is in � (x^{3} + y^{3} + 60z^{3} = 0).

Computing � (E) in practice is challenging! It took decades until the first example was computed (by Karl Rubin).

John Cremona’s Book

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Main Theorem

Thereom Suppose E is a non-CM elliptic curve of conductor ≤ 1000 and rank ≤ 1 and p is a prime that does not divide any Tamagawa number of E and that E has no p-isogeny. Then the p-part of the full BSD conjectural formula is true for E.

The rest of this talk is about the proof.

Tools

• SAGE: I did much of this computation using

SAGE: System for Algebra and Geometry Computation

              http://modular.fas.harvard.edu/sage

which is a new computer algebra system that incorporates mwrank, PARI, etc., under one hood.

• MAGMA: I used MAGMA for some 3 and 4-descents.

BSD Conjecture at p

Conjecture 1 (BSD(E, p)). Let (E, p) denote a pair consisting of an elliptic curve E over ℚ and a prime p. Then E(ℚ) has the predicted rank, � (E)[p^{∞}] is finite and

ord_{p}(#� (E)[p^{∞}]) = ord_{p} (\frac{L^{(r)}(E, 1) · (#E(ℚ)_{tor} )^{2}}{r! · Ω_{E} · Reg _{E} · ∏_{p} c_{p}} ) .

Theorem (Tate): The truth of BSD (E, p) is invariant under isogeny.

Computational Evidence for BSD

All of the quantities in the BSD conjecture, except for #� (E/ℚ), have been computed by Cremona for conductor ≤ 80000.

• Cremona (Ch. 4, pg. 106): In Cremona’s book, exactly four optimal curves with conjecturally nontrivial � (E): 571A, 681B, 960D, 960N

• Cremona verified BSD (E, 2) for all curves in his book, except 571A, 960D, and 960N.

The Four Nontrivial � ’s

Conclusion: BSD for the curves in Cremona’s book is the assertion 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

As we will see, we can deal with these four curves...

Victor Kolyvagin

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Victor Kolyvagin

Kolyvagin: When r_{an} ≤ 1, get computable multiple of #� (E).

Let K be a quadratic imaginary field in which all primes dividing the conductor of E split. Let y_{K} ∈ E(K) be the corresponding Heegner point.

Theorem 2 (Kolyvagin). Suppose E is a non-CM elliptic curve and p is an odd prime such that \overline{ρ}_{E,p} is surjective and E(K) has rank 1. Then

ord_{p}(#� (E_{K})) ≤ 2 · ord_{p}([E(K) : ℤy_{K}]).

Kato

Kato: When r_{an} = 0, get bound on #� (E).

Theorem 3 (Kato). Let E be an optimal elliptic curve over ℚ of conductor N, and let p be a prime such that p ∤ 6N and \overline{ρ}_{E,p} is surjective. If L(E, 1) ≠ 0, then � (E) is finite and

ord_{p}(#� (E)) ≤ ord_{p} (\frac{L(E, 1)}{Ω_{E}} ) .

This theorem follows from the existence of an “optimal” Kato Euler system (recent work of Matsuno).

Divisor of Order

Back to our four curves...

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.

The Eighteen Optimal Curves of Rank > 1

There are 18 optimal 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 perhaps nobody currently knows how to show that � (E) is finite, let alone trivial. (But p-adic L-functions, Iwasawa theory, Schneider’s theorem, etc., would give a finite interesting list of p for a given curve.)

Summary

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

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

• Of these, 2441 have conjecturally trivial � .

• Of these, 44 have CM. We prove BSD (E, p) for the remaining 2397 curves at primes p that do not divide Tamagawa numbers and for which \overline{ρ}_{E,p} is irreducible.

Determining im(\overline{ρ}_{E,p}) ⊂ Aut(E[p])

Theorem 4 (Cojocaru, Kani, and Serre). If E is a non-CM elliptic curve of conductor N, and

p ≥ 1 + \frac{4√6}{3} · N · ∏_{prime ℓ|N} (1 + \frac{1}{ℓ} ) ^{1/2} ,

then \overline{ρ}_{E,p} is surjective.

Proposition 5 (WS, Grigorov, Serre (Inv. 1972)). Let E be an elliptic curve over ℚ of conductor N and let p ≥ 5 be a prime. For each prime ℓ ∤ p · N with a_{ℓ} ≢ 0 (mod p), let

s(ℓ) = (\frac{a_{ℓ}^{2} − 4ℓ}{p} ) ∈ \{0, −1, +1\},

where the symbol (\frac{·}{·}) is the Legendre symbol. If −1 and +1 both occur as values of s(ℓ), then \overline{ρ}_{E,p} is surjective. If s(ℓ) ∈ \{0, 1\} for all ℓ, then im(\overline{ρ}_{E,p}) is contained in a Borel subgroup (i.e., reducible), and if s(ℓ) ∈ {0, −1} for all ℓ, then im(\overline{ρ}_{E,p}) is a nonsplit torus.

This + division polynomials ⟹ efficient algorithm to compute image. (Tables now available online.)

Generalizations of Kolyvagin’s Theorem

Theorem 6 (Cha). If p ∤ D_{K}, p^{2} ∤ N, and \overline{ρ}_{E,p} is irreducible, then

ord_{p}(#� (E/K)) ≤ 2 · ord_{p}([E(K) : ℤy_{K}]).

Example 7. Let E be the elliptic curve 608B, which has rank 0. Then BSD (E, 5) is true for E by Cha’s theorem, but not Kato’s since \overline{ρ}_{E,5} is irreducible but not surjective.

Theorem 8 (Donnelly, Jorza, Patrikis, Stoll, WS). If E is a non-CM curve over ℚ, K is a quadratic imaginary field that satisfies the Heegner hypothesis, and p is an odd prime such that p ∤ #E′(K)_{tor} for any curve E′ that is ℚ-isogenous to E, then

ord_{p}(#� (E)) ≤ 2 ord_{p}([E(K) : ℤy_{K}]),

unless disc (K) is divisible by exactly one prime ℓ, in which case the conclusion is only valid if p ≠ ℓ.

Computing Indexes of Heegner Point

Use the Gross-Zagier formula to compute h(y_{K}) from special values of L-functions. When we can compute E(K) we obtain the index using properties of heights. If E(K) is too difficult to compute, can sometimes use the Cremona-Prickett-Siksek height bound and search to bound [E(K) : ℤy_{K}]:

Example 9. Let E be 906E1 which has rank 0:

y^{2} + xy + y = x^{3} + x^{2} − 40466325x + 99063769563.

All \overline{ρ}_{E,p} are surjective. The first few Heegner discriminants are

−23, −71, −119, −143, −263, −335.

The heights are approx. 7705, 20400, 33785, 19284, 39658, 63256. Finding these Heegner points could be difficult. Let F be the quadratic twist of E by −23. The CPS bound for F is B = 13.649 . . .. Search for points on F of naive logarithmic height < 21, and find no points, so

[E(K) : ℤy_{K}] < √7705/(2 · (21 − 13.649)) ∼ 22.89 < 23.

But Kato implies only 2, 3, 151 could divide #� (E)!

Major Obstruction: Tamagawa Numbers

Serious Issue: The Gross-Zagier formula and the BSD conjecture together imply that if an odd prime p divides a Tamagawa number, then p | [E(K) : ℤy_{K}].

• Rank 0: If E has r_{an} = 0, and p ≥ 5, and ρ_{E,p} is surjective,

then Kato’s theorem (and Mazur, Rubin, et al.) imply that

ord_{p}(#� (E)) ≤ ord_{p}(L(E, 1)/Ω_{E}),

so squareness of #� (E) frequently helps.

• Rank 1: In many cases with r_{an} = 1, there is a big Tamagawa number—there are 91 optimal curves up to conductor 1000 with Tamagawa number divisible by a prime p ≥ 7.

Conclusion

Throw in explicit 3 and 4-descents to deal with a handful of reluctant cases. Everything works out so that all our techniques are just enough to complete the proof. If Cremona’s book were larger, this might not have been the case.

For more, see:

        http://modular.fas.harvard.edu/papers/bsdalg/

Future Projects

1. [Manin] Prove that c = 1 for curves of conductor ≤ 80000.

2. [CM] Verify the BSD conjecture for CM curves up to some conductor. About half of rank 0 and half of rank 1. Very extensive theory here, beginning with Rubin—should be relative “easy”, especially for rank 0.

3. [Extend] Consider curves of conductor > 1000. Have to verify nontriviality of big � (E)’s; use visibility and Grigor Grigorov’s thesis.

4. [Big Rank] Verify BSD at all primes p ≤ 100 for some curve of rank 2.

5. [Isogenies] Verify the BSD conjecture at primes p that are the degree of an isogeny from E. Mazur’s “Eisenstein descent” does prime level case; but then p = 2. Perhaps direct p-descent is doable, or use congruences...

6. [Tamagawa] Verify the BSD conjecture at primes p that divide a Tamagawa number. Prove a refinement of Kolyvagin’s theorem and/or develop p-adic methods.

7. [Find a Very Efficient Algorithm!] I.e., prove the full BSD conjecture for elliptic curves of rank 0 and 1.