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Computing Shafarevich-Tate Groups of Elliptic Curves

1. The Birch and Swinnerton-Dyer Conjecture

Let E be an elliptic curve over Q :

y2+a1xy+a3y=x3+a2x2+a4x+a6:

The BSD conjecture asserts that

The rank r of E(Q) equals ran=ords=1L(E;s) .

We have

r!L(r)(E;1)=#E(Q)2tor#Sha(E)��E�RegE�YpjNcp;

where N is the conductor of E .

This talk is about computing quantities appearing in the above conjecture and in p -adic analogues of it.

Let's give it a shot...

First we look up a curve that will illustrate a range of methods in Cremona's tables:

Here's the equation of the curve:

y2+y=x3+x2�12x+2

y2+y=x3+x2�12x+2

The conductor is a product of two primes: 141=3�47 .

3�47

3�47

The graph of the real points on the curve is connected (and pretty):

We next plot a bunch of rational points on the curve using a 3-d ray tracer (for fun). Lighter points have larger height.

We use Tim Dokchitser's L -functions program to compute with the L -function of this elliptic curve.

Dokchitser L-function associated to Elliptic Curve defined by y^2 + y = x^3 +
x^2 - 12*x + 2 over Rational Field

Dokchitser L-function associated to Elliptic Curve defined by y^2 + y = x^3 + x^2 - 12*x + 2 over Rational Field

For example, here is the Taylor expansion about z=(s�1) of the L -series.

0.718550172498336z+(-0.0426008377591305)z2+(-0.322488300418669)z3+O(z4)

0.718550172498336z+(-0.0426008377591305)z2+(-0.322488300418669)z3+O(z4)

Of course the derivative at 1 is consistent:

0.718550172498336

0.718550172498336

Next we plot the L -series:

And it does in fact vanish at 1 because the sign in the functional equation is �1 .

Not that this is related to BSD, but we can compute the zeros of the L -function on the critical line using Mike Rubinstein's Lcalc program:

Our curve has rank and analytic rank both 1, in accord with the conjecture.

We compute a generator of the Mordell-Weil group:

[(-3 : 4 : 1)]

[(-3 : 4 : 1)]

The regulator is as follows.

0.0344867750175524

0.0344867750175524

The Tamagawa number at 3 is 7 ; this will cause us some difficulty.

[7, 1]

[7, 1]

The regulator �E is about 2:9765 .

2.976504024814575615165574

2.976504024814575615165574

The curve has trivial torsion subgroup, so #E(Q)=1 .

Now we put it all together:

r!L(r)(E;1)=#E(Q)2tor#Sha(E)��E�RegE�YpjNcp;

L1 =  0.718550172498336
rhs = 0.718550172498336 * Sha

L1 =  0.718550172498336
rhs = 0.718550172498336 * Sha

We Conclude

Thus the BSD conjecture asserts in this case that #Sha(E)=1 . We will compute #Sha(E) and hence prove the conjecture. The methods we use are fairly general (assuming that r<2 ), though much interesting work remains to be done.

Sha(E)=ker�H1(Q;E)!MH1(Qv;E)�:

The 2-Selmer Group

We can get some information about Sha(E) by by computing the 2-Selmer group. From this we see that #Sha is odd.

From the first calculation we see that the rank of the elliptic curve is at most one.

Since the curve is known to have rank 1, this means that Sha(E)[2]=0 .

This talk is about how it is becoming possible to in practice to systematically compute much more about Sha(E) .

Next: Theorems of Kolyvagin and Kato