The Birch and Swinnerton-Dyer Conjecture


Picture I took of Birch and Swinnerton-Dyer in 2000 in Utrecht, Holland

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John Coates (2002): "The ancient union between theory and computation
is as potent a force as ever today.  It is my strong personal view
that the best computations on elliptic curves are those that lead to
new insights for attacking the unsolved theoretical problems. Equally,
I firmly believe that no abstract theorem about the arithmetic of
elliptic curves is worth its salt unless illuminating numerical
examples of it can be given."

 
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EDSAC

{{{id=141| /// }}} {{{id=0| /// }}} {{{id=84| /// }}} {{{id=49| /// }}} {{{id=27| /// }}}

Birch and Swinnerton-Dyer's EDSAC Computation

Heuristic Observation: If $E$ has infinitely many rational points, then the numbers $N_p$ will tend to be "large", since you can reduce all those points modulo $p$.  Since the ``zeta function'' $$ \prod_p \frac{p}{N_p}$$ will tend to be small.

 

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Replicating the computation...

{{{id=146| %auto def prods(E, B=10000): a = [float(1)] for p in primes(B): a.append(a[-1]*float(p)/E.Np(p)) return finance.TimeSeries(a)[1:] /// }}}

A Rank 0 Curve:

{{{id=145| prods(EllipticCurve('11a')).plot(ymax=.2) /// }}} {{{id=159| time prods(EllipticCurve('11a'),10^5).plot(ymax=.2) /// Time: CPU 1.51 s, Wall: 1.60 s }}} {{{id=163| time prods(EllipticCurve('11a'),10^6).plot(ymax=.2) /// Time: CPU 10.27 s, Wall: 10.36 s }}} {{{id=164| time prods(EllipticCurve('11a'),10^7).plot(ymax=.2) /// Time: CPU 110.32 s, Wall: 112.23 s }}}

NOTE: Amazingly, the above plot is expected to converge to a limit!  Dorian Goldfeld proved that if the Generalized Riemann Hypothesis (for $L(E,s)$) is true and $L(E,1)\neq 0$, then $\prod_p \frac{p}{N_p} = \frac{L(E,1)}{\sqrt{2}}.$

{{{id=161| EllipticCurve('11a').lseries()(1)/math.sqrt(2) /// 0.179493301160226 }}} {{{id=162| /// }}}

A Rank 1 Curve

{{{id=34| prods(EllipticCurve('37a')).plot(ymax=.2) /// }}}

A Rank 2 Curve

{{{id=150| prods(EllipticCurve('389a')).plot(ymax=.2) /// }}}

A Rank 3 Curve

{{{id=153| prods(EllipticCurve('5077a')).plot(ymax=.2) /// }}}

A Rank 4 Curve

{{{id=156| prods(EllipticCurve([1,-1,0,-79,289])).plot(ymax=.2) /// }}} {{{id=152| prods(EllipticCurve([1,-1,0,-79,289])).plot(ymax=.001) /// }}} {{{id=173| /// }}}

 

The $L$-function

Let $E$ be an elliptic curve. For each prime number $p$, let $N_p=\#E(\mathbf{F}_p)$ be the number of solutions modulo $p$.

Definition: Let $a_p = p+1-N_p$.

Theorem (Hasse): $|a_p| < 2\sqrt{p}$.

Let $\Delta = -16(4a^3+27b^2)$. We have the following very deep theorem:

Theorem (Breuil, Conrad, Diamond, Taylor, Wiles -- 1999): The function $$L^*(E,s) = \prod_{{\rm primes\, } p\nmid \Delta}\left(\frac{1}{1-a_p p^{-s} + p^{1-2s}}\right)$ extends (uniquely) to an entire complex-analytic function on the complex plane $\mathbf{C}$.

 

NOTE: In practice, one usually works with a slightly more complicated function $L(E,s)$.  When $\Delta$ is "minimal", this function is $$L(E,s) = L^*(E,s) \cdot \prod_{{\rm primes\, } p\mid \Delta}\left(\frac{1}{1-a_p p^{-s}}\right),$$ where $a_p=p+1-\#E_{ns}(\mathbf{F}_p)$ and $\#E_{ns}(\mathbf{F}_p)$ is the subgroup of nonsingular points.

 

{{{id=128| # A rank 2 curve: y*(y+1) = x*(x-1)*(x+2) E = EllipticCurve([0,1,1,-2,0]); E /// Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field }}} {{{id=125| L = E.lseries(); L /// Complex L-series of the Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field }}} {{{id=126| L(2) /// 0.360092863578881 }}} {{{id=127| L(1+I) /// -0.638409938588039 + 0.715495239204667*I }}} {{{id=83| time show(line([(i,L(i)) for i in [-1,-.95,..,2]]), figsize=[7,1.5], ymin=-1) /// Time: CPU 0.40 s, Wall: 0.45 s }}} {{{id=158| /// }}}

The Official Conjecture

 

Theorem (Mordell): There is a finite set $P_1, \ldots, P_r$ of rational points on $E$ so that all rational points (modulo points of finite order) can be generated from these using the group law.

We call the smallest $r$ in Mordell's theorem the rank of $E$.

Conjecture (Birch and Swinnerton-Dyer): $$\text{ord}_{s=1} L(E,s) = \text{rank}(E)$$

This problem, exactly as stated above, is the Clay Math Institute Million Dollar prize problem in algebraic number theory.  Its solution would also resolve the 1000-year old congruent number problem.

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An Example

{{{id=85| E = EllipticCurve([0,1,1,-2,0]) L = E.lseries() /// }}} {{{id=36| L.taylor_series(1) /// -2.69129566562797e-23 + (1.52514901968783e-23)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 - 0.193509313829981*z^4 + 0.459971558373642*z^5 + O(z^6) }}} {{{id=22| E.rank() /// 2 }}} {{{id=134| /// }}} {{{id=132| /// }}}

The Kolyvagin -- Gross-Zagier Theorem

Theorem: If $\text{ord}_{s=1} L(E,s) \leq 1$ then the Birch and Swinnerton-Dyer conjecture is true for $E$.


The proof involves Heegner points, modular curves, Euler systems and Galois cohomology.



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We do not know in general how to compute ${\rm ord}_{s=1} L(E,s)$ yet...

Open Problem: Show that the Birch and Swinnerton-Dyer conjecture holds for even a single specific curve with rank $r=4$ (or higher).  For example, for the curve $$y^2 + xy = x^3 - x^2 - 79x + 289.$$

{{{id=50| E = EllipticCurve([1,-1,0,-79,289]) /// }}}

The following command does in fact verify with certainty that the rank is $4$:

{{{id=136| E.rank() /// 4 }}} {{{id=138| E.gens() /// [(-9 : 19 : 1), (-8 : 23 : 1), (-7 : 25 : 1), (4 : -7 : 1)] }}}

The $L$-series looks like it vanishes to order $4$ at 1, but we can't be sure since we get only 0.000000....  It doesn't vanish to order greater than $4$.

{{{id=30| E.lseries().taylor_series(1, 53, 6) /// 5.54631009473167e-24 + (-2.08951550639391e-23)*z + (-4.15704192504384e-22)*z^2 + (1.66720224204167e-21)*z^3 + 8.94384739590089*z^4 - 33.6950287693207*z^5 + O(z^6) }}} {{{id=140| /// }}} {{{id=139| /// }}}
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