jsMath
Worksheet: bsdintro

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

2. We have
r!L(r)(E;1)=#E(Q)2tor#Sha(E)ÁÊEÁRegEÁYpjNcp;
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+x2À12x+2  y2+y=x3+x2À12x+2
The conductor is a product of two primes: 141=3Á47 .
 3Á47  3Á47
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=(sÀ1)  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 À1 .
 0 0
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.
 1 1
 1 1
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 ÊE  is about 2:9765 .
 2.976504024814575615165574 2.976504024814575615165574
The curve has trivial torsion subgroup, so #E(Q)=1 .
 1 1

Now we put it all together:
r!L(r)(E;1)=#E(Q)2tor#Sha(E)ÁÊEÁRegEÁYpjNcp;
 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)Ñ:

## 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.

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

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