Geometric Rank and Birch-Swinnerton-Dyer

Adam B. Durrett^{∗}

January 20, 2002

Mathematics Department Princeton University Princeton, NJ 08544

Abstract

The Birch and Swinnerton-Dyer conjecture tells us that the geometric rank is equal to the analytic rank of a given curve. In conjunction with those investigating analytic rank, we are checking a weak form of B-SD for several families of curves: if the Mordell-Weil group is generated by at least r points, do we find that the analytic rank is at least r?

To do this, I wrote a program to find if points are linearly independent. Many points are forced on curves–we know where these are. Other points are simply found by searching–we will be using points found by the programs of Ellen Eischen and Salman Butt. Once these other points are found, we compare them with the forced points to come up with a lower bound for the number of generators r.

∗E-mail: adurrett@math.princeton.edu

1 Definitions

To begin our exploration of Birch and Swinnerton-Dyer via the Mordell-Weil group, we will first need to understand the notion of the height of a point on an elliptic curve.

Definition 1.1 For a rational point P = (x, y) on an elliptic curve, suppose that x = \frac{m}{n} is a rational number in lowest terms. Then the naive height of this point, h_{0}(P) = log max(|m|, |n|) ≥ 0.

It is an interesting note that there are only a finite number of points below a certain naive height K. For, in order for h_{0}(P) ≤ K, log |m| ≤ K (the same goes for n). Thus |m| ≤ e^{K} and |n| ≤ e^{K}. When counting the possibility that m and n are negative (or 0, in m’s case, this leaves 2K+1 choices for the numerator and 2K choices for the denominator. So there are 2e^{K} ∗(2e^{K} +1) possible points with naive height at most K. More useful to us, though, will be the canonical height of a point.

Definition 1.2 For a rational point P on an elliptic curve, where h_{0}(P) is the naive height of P, the canonical height h(P) can be defined as follows:

h(P) = lim_{n→∞} \frac{h_{0}(2^{n}P)}{4^{n}}  (1.1)

Remark: the canonical height h(P) satisfies the following conditions:

(i) h(P) − h_{0}(P) is bounded (ii) h(2P) = 4h(P) (iii) h(P) = 0 if P is a torsion point

From these 3 properties, we can deduce that h(P) > 0 for all infinite points. h(P) = 0 for all torsion points, which can be seen rather obviously by noticing that, while the numerator in the definition of h(P) is bounded (since P is of finite order and h_{0}(2^{n}P) can only run through a finite number of values) the denominator is unbounded, and thus the limit goes to 0.

Proposition 1.3 h(P) > 0 for all points P of infinite order.

Proof: We know that {P | h(P) ≤ 1} is a finite set. So there must be some point 2^{n}P such that h(2^{n}P) > 1. Then 4^{n}h(P) > 1, so h(P) > 4^{−n} > 0. Since P was an arbitrary member of this set, then h(P) > 0 for all P in the infinite group. □

Alternate Proof: Let k = max(|h(P) − h_{0}(P)|) taken over all P. We know this exists by property (i) above. Now suppose there exists some point P with h(P) = 0. Then h(2^{n}P) = 4^{n}h(P) = 0 for all n. But we also know that the set {P | h_{0}(P) < k} is finite, so there exists some n s.t. h_{0}(2^{n}P) ≥ k +1. But this is impossible, since k was maximal and h(2^{n}P) = 0. So h(P) ≠ 0. □

The following property of h(P), while presented here unproven, will also come in handy. The statement of the theorem (and a proof) can be found in Knapp.

Proposition 1.4 (Parallelogram law)

h(P + Q) + h(P − Q) = 2h(P) + 2h(Q)  (1.2)

With this new law, we can even improve upon property (ii) of h(P).

Proposition 1.5 If P is a point of infinite order, then

h(nP) = n^{2}h(P)  (1.3)

Proof: By induction. The base case, h(2P) = 4h(P) we already know to be true from the above properties. Taking the assumption to be true for h(mP), look at h((m + 1)P):

Applying (1.2) from above, we have:

h(mP + P) = 2h(mP) + 2h(P) − h(mP − P)
= 2m^{2}h(P) + 2h(P) − (m − 1)^{2}h(P)
= m^{2}h(P) + 2mh(P) + h(P)
= (m + 1)^{2}h(P)
□

Before we move on to our discussion of the bilinear form, we need to develop one important lemma. While appearing totally unrelated to our discussion of height that we’ve already begun, we would not be able to find our generators for the Mordell-Weil group without it. While no proof is provided, an excellent proof may be found in Knapp’s textbook on elliptic curves. It also provides the statement of the theorem that we will use:

Theorem 1.6 (Minkowski) If E is a compact set in R^{r} containing 0, closed under negatives, and with volume > 4^{r}, then E contains a point in Z^{r}.

2 The Bilinear Form on E(Q)/T

Given the definitions and tools developed in section 1, we are now ready to proceed with our discussion of the bilinear height function. The statement of the following is taken largely from Knapp.

Theorem 2.1 For an elliptic curve E, there exists a unique Z-bilinear form ⟨P, Q⟩ on E(Q) with the property ⟨P, P ⟩ = h(P) as defined above. The form descends to E(Q)/T (where T represents the Torsion group) and is positive definite there.

Proof: First, we shall show uniqueness. Once we know the form that ⟨P, Q⟩ must take, then we can prove that it is bilinear, positive definite, and modulo the Torsion points. First, assuming that we have a bilinear form associated with a quadratic form q, then it is a fundamental theorem of analysis that ⟨P, P ⟩ = \frac{1}{2}(q(2P) − 2q(P)) so, if we are basing our bilinear form on h(P), by simply substituting into the above equation and using the fact that h(2P) = 4h(P) we see that ⟨P, P ⟩ = h(P). From here we can generalize from ⟨P, P ⟩ to ⟨P, Q⟩ by simply need to looking at h(P + Q, P + Q):

h(P + Q) = ⟨P + Q, P + Q⟩
= ⟨P, P ⟩ + 2⟨P, Q⟩ + ⟨Q, Q⟩
= h(P) + 2⟨P, Q⟩ + h(Q)

Thus, ⟨P, Q⟩ = \frac{1}{2}(h(P + Q) − h(P) − h(Q)).  (2.1)

So if ⟨P, Q⟩ is, in fact, bilinear, it must take the form of (2.1). Now to prove existence. All we have to do is prove that ⟨P +P′, Q⟩ = ⟨P, Q⟩+⟨P′, Q⟩ since we are working with a Z-bilinear form, ⟨aP, Q⟩ = a⟨P, Q⟩ follows immediately from that (for a ∈ Z). Again, the parallelogram law (1.2) will be used several times.

We shall first show that ⟨−P, Q⟩ = −⟨P, Q⟩, noting along the way that since h is quadratic, h(−P) = (−1)^{2}h(P) = h(p).

⟨−P, Q⟩ = \frac{1}{2}(h(−P + Q) − h(P) − h(Q))
= \frac{1}{2}(2h(P) + 2h(Q) − h(P + Q) − h(P) − h(Q))
= −\frac{1}{2}(h(P + Q) − h(P) − h(Q)
= −⟨P, Q⟩.  (2.2)

Next we apply (2.1) and (1.2) to see that:

⟨P + P′, Q⟩ + ⟨P − P′, Q⟩ = \frac{1}{2}(h(P + P′ + Q) + h(P − P′ + Q) − h(P + P′) −
h(P − P′) − 2h(Q))
= \frac{1}{2}(2h(P + Q) + 2h(P′) − 2h(P) − 2h(P′) − 2h(Q)
= 2⟨P, Q⟩.  (2.3)

Using the same strategy but with P and P′ flipped, applying (2.2) we see that:

⟨P + P′, Q⟩ − ⟨P − P′, Q⟩ = ⟨P + P′, Q⟩ + ⟨P′ − P, Q⟩
= 2⟨P′, Q⟩.  (2.4)

Combining (2.3) and (2.4) we have that:

⟨P + P′, Q⟩ = ⟨P, Q⟩ + ⟨P′, Q⟩  (2.5)

and thus ⟨P, Q⟩ is bilinear. And from (2.1) we can see that it is symmetric, so additivity in the first variable implies additivity in the second (and thus ⟨P, aQ⟩ = a⟨P, Q⟩).

Now to prove that it descends to E(Q)/T:

From (1.3), (2.1), and (2.2) we have that:

−mn⟨P, Q⟩ = ⟨mP, −nQ⟩
= \frac{1}{2}(h(mP − nQ) − h(mP) − h(nQ))
= \frac{1}{2}(h(mP − nQ) − m^{2}h(P) − n^{2}h(Q))

Multiplying both sides by 2, rearranging, and observing that h(P) ≥ 0 achieves:

0 ≤ h(mP − nQ) = n^{2}h(Q) − 2nm⟨P, Q⟩ + m^{2}h(P).  (2.6)

So, let λ = \frac{n}{m} rational in (2.6), then we know that

0 ≤ λ^{2}h(Q) − 2λ⟨P, Q⟩ + h(P)  (2.7)

And by taking the limit as λ approaches any real number, we see that (2.7) holds for any λ real. Suppose, instead, that for some λ real that (2.7) did not hold. Then, because the rational numbrs are dense in R, we could pick at least one λ′ s.t. 0 ≥ λ^{′2}h(Q) − 2λ′⟨P, Q⟩ + h(P). But this is impossible, for we know that (2.7) holds for any rational λ. So (2.7) must hold for all real λ as well, and thus it is a quadratic in λ with at most one real root (otherwise (2.7) would be negative at some point), so the discriminant cannot be positive, and hence we have the result that:

|⟨P, Q⟩|^{2} ≤ h(P)h(Q).  (2.8)

From (2.8) we see that if either P or Q is a torsion point, since h(P) = 0 for P ∈ T, ⟨P, Q⟩ must be zero. Using this fact, we see that with P a point of infinite order and Q a torsion point,

0 = ⟨P, Q⟩
= \frac{1}{2}(h(P + Q) − h(P) − h(Q))
= \frac{1}{2}(h(P + Q) − h(P))

And thus,

h(P + Q) = h(P)  (2.9)

for all P of infinite order, Q ∈ T. So the bilinear form descends to E(Q)/T.

Now, before we are able to prove that ⟨P, Q⟩ is positive definite, we must first show that this is positive semi-definite. We will proceed as follows:

Suppose that {P_{1}, …, P_{r}} is a basis for the infinite group. Then define c_{ij} = ⟨P_{i}, P_{j}⟩, so that if P = ∑_{i} m_{i}P_{i}, then

0 ≤ h(P) = ⟨P, P ⟩ = ⟨∑_{i} m_{i}P_{i}, ∑_{j} m_{j}P_{j}⟩ = ∑_{i,j} m_{i}c_{ij}m_{j}  (2.10)

Now, we proceed as we did above when we proved (2.8). Divide both sides of (2.10) by N^{2}, N a rational number. Then clearly the equation still holds.

We may then choose that \frac{m_{i}}{N} → λ_{i} given λ_{i} real. So, given an r-tuple (λ_{1}, …, λ_{r}), we may choose an r-tuple (\frac{m_{1}}{N} , …, \frac{m_{r}}{N}) of rational numbers that approaches the real r-tuple, and thus (2.10) must hold for real coefficients as well. Otherwise, if we were able to find some real r-tuple which did not satisfy (2.10), since the rational numbers are dense in R, we could find an r-tuple of rational numbers arbitrarily close to the real r-tuple which also did not satisfy (2.10), a contradiction. So (2.10) holds for any real coefficients as well, and hence (c_{ij}) is symmetric and positive semidefinite.

It may be interesting to note here that it the ≥ is extremely necessary in (2.10). At first, it may seem that, since h(P) > 0 for all infinite points, we need not worry about equality, especially since we can get rational coefficients as close as we want to real values. This would allow us, then, to skip over the following Minkowski argument–a much more attractive alternative. However, the problem is this: if we assume that (2.10) doesn’t hold for all real vectors, we are able to come up with the contradiction that it also must not hold for some rational vectors. However, it is possible to find a vector of real numbers which forces equality in (2.10), and from this we could still have that all rational vectors still force (2.10) to be strictly greater than 0, and hence there would be no contradiction to the strictly greater than condition. A parabola p(x) provides the best analogy: if p(x_{0}) is negative for some real x_{0}, the density of rationals allows us to find some rational x_{1} such that p(x_{1}) is also negative. However, if p(x_{0}) equals zero but is the only root, then we cannot find a rational with p(x_{1}) also equal to zero, despite our best efforts. Thus, we are forced to be content with the ≥ condition when (2.10) is extended to all real numbers.

To prove that it is positive definite, we will have to use the result of Minkowski, theorem 1.6.

Since we now have extended the bilinear form to make sense for real numbers, not just those points with integer coefficients, we will proceed with an orthonormal basis of eigenvectors for (c_{ij}). Label this basis as v^{1}, …, v^{r}, with respective eigenvalues λ^{1} ≥ … ≥ λ^{r} ≥ 0, which we know is possible since (c_{ij}) is positive semi-definite. Our aim will be to prove, then, that our λ^{r} is in fact positive, and hence (c_{ij}) is actually positive definite. We do so by contradiction, so assume that our λ^{r} = 0.

Proceed by calling e^{i} the standard basis with a 1 in the i^{th} row and zeros everywhere else. Then we can define our eigenvectors as v^{i} = ∑_{j} v_{j}^{i}e^{j} and each point P = ∑_{i} m_{i}P_{i} with P_{i} a Z-basis for the infinite group as P = ∑_{i} m_{i}e^{i}. So we let our (c_{ij}) = ⟨e^{i}, e^{j}⟩ and proceed to look at h(P) for some point P. The reason for the extension to real coefficients and the move towards a basis of eigenvectors is to pull out the eigenvalues in our height function, as can be seen below. Note that we can write our point P as a linear combination of the eigenvectors now since we can have non-rational coefficients. The eigenvalues will end up coming out in the second to last line since the v^{i} have been normalized so that v_{k}^{i} ∗ v_{k}^{j} = 1 if i = j and 0 otherwise. Also, note that while we are taking ⟨P*,P*⟩ below, P* is not necessarily a rational point; rather, P* is a linear combination with real coefficients of the eigenvectors. The bilinear form makes sense since we’ve extended it to handle irrational vectors, but when taking ⟨P*,P*⟩ with P* irrational, it no longer makes sense to think of this as the height of P*.

⟨P*,P*⟩ = ⟨∑_{i} m_{i}v^{i}, ∑_{j} m_{j}v^{j}⟩

= ⟨∑_{i,k} m_{i}v_{k}^{i}e^{k}, ∑_{j,l} m_{j}v_{l}^{j}e^{l}⟩
= ∑_{i,j} m_{i}m_{j} ∑_{k,l} v_{k}^{i}(c_{kl})v_{l}^{j}
= ∑_{i,j} m_{i}m_{j} ∑_{k} v_{k}^{i}λ^{k}v_{k}^{j}
= ∑_{i}(m_{i})^{2}λ^{i}.  (2.11)

Using this fact, we can finally see where Minkowski comes into play. First, we must look at the minimum value of h(P) over all points P of infinite order. We know that this minimum must be > 0 since h(P) > 0 for all points P in the infinite group, so call it ϵ > 0. Suppose, instead, that such a minimum didn’t exist. Then since h(P) is positive, there must be some sequence of points P_{n} such that the limit of h(P_{n}) → 0 as n → ∞ But this is impossible, since there are only a finite number of points below any fixed height K. So our minimum positive height must exist.

Now, examine the set of vectors:

E = \{∑_{i} m_{i}v^{i} | max_{1≤i<r} |m_{i}| ≤ (\frac{ϵ}{2(r − 1)λ^{1}} )^{\frac{1}{2}} and |m_{r}| ≤ M\} (2.12)

It is important to note that λ^{1} could not be zero, or else we’d have all eigenvalues being zero, and by equation 2.11, this would imply that h(P) = 0, a contradiction. So we are free to divide by λ^{1}. Also, we can choose our M here to be as large as we want, so we simply need to make M large enough so that vol(E) > 4^{r}. That way, by Minkowski’s theorem, we have that E must contain a rational point P on E of infinite order. Then we look at h(P): By (2.11), if P = ∑_{i} m_{i}v^{i},

h(P) = ∑_{i=1}^{r}(m_{i})^{2}λ^{i}  (2.13)

But since we are operating under the assumption that λ^{r} = 0, this reduces to

h(P) = ∑_{i=1}^{r−1}(m_{i})^{2}λ^{i}  (2.14)

Since λ^{1} ≥ λ^{i} for all i, and since |m_{i}| ≤ (\frac{ϵ}{2(r−1)λ^{1}} )^{\frac{1}{2}} , we can reduce again to

h(P) ≤ ∑_{i=1}^{r−1} λ^{1} (\frac{ϵ}{2(r − 1)λ^{1}} ) = (r − 1)λ^{1} (\frac{ϵ}{2(r − 1)λ^{1}} ) = \frac{ϵ}{2}.  (2.15)

But we had that h(P) ≥ ϵ for all P of infinite order, so we have our contradiction. Hence, all the eigenvectors must have been positive, and (c_{ij}) is, in fact, positive definite. □

Corollary 2.2 A set of points {P_{1}, …, P_{r}} on an elliptic curve E are linearly independent ⟺ the matrix (c_{ij}) = ⟨P_{i}, P_{j}⟩ has nonzero determinant.

Proof: ⇒: This is an automatic result of Theorem 2.1. For, we were simply dealing with a set of linearly independent points as a basis for Z^{r}. Since all of the eigenvalues were greater than zero, and the product of eigenvalues is the determinant, the determinant cannot be zero.

⇐: Assume the determinant is nonzero. Then is it possible that any of the points were linearly dependent? If so, we could find a non-trivial linear combination of these points equal to zero. Call this point P. Then h(P) = 0 , by definition. But P must have some projection onto at least one of the eigenvectors which we defined in the proof of Theorem 2.1. So, if we write P = ∑_{i} m_{i}v^{i} as before, with λ^{1}, …, λ^{r} > 0 since 2.1 is true, then we have that h(P) = ∑_{i}(m_{i})^{2}λ^{i} with at least one m_{i} nonzero, and it doesn’t matter if the projection onto that eigenvector is negative because it will end up being squared. So we get the whole sum must be positive, which is a contradiction since h(P) was zero. So the points must have been linearly independent. □

3 Results

Using corollary 2.2 as a basis for calculation, I took all of this information on heights to find independent points in the Mordell-Weil group (modulo Torsion) of different curves of expected rank at least 2. I will first explain how such curves may be constructed, then discuss the two “nice” curves with which I worked most and what I observed about them.

3.1 Construction of curves of high rank

It is possible to construct curves using an algorithm developed by Mestre. It generally uses the following steps:

Step 1: Take a polynomial P (x) = ∏_{i=1}^{4}(x − α_{i}). For our purposes P(x) has 4 roots.

Step 2: Construct the polynomial G(x) = P (x + t)P (x − t) for some fixed t. Notice, now, that G(x) will have 8 roots: α_{i} ± t for i = 1 to 4.

Step 3: Write G(x) as S(x)^{2} − R(x), where R(x) is a cubic polynomial (if possible…).

Step 4: Check for independence of points on your new curve. For any of these 8 points, G(x) = 0, so S(x)^{2} = R(x). Consider now the equation y^{2} = R(x), where a cubic polynomial dumps out a perfect square, and we have a new elliptic curve to work with. Plus, we already know of at least 8 points on this curve! (Exactly where G(x) is zero) While the ±t should produce some dependence among points, we will hopefully force some rank on this new curve.

3.2 y^{2} = x^{3} − t^{2}x + t^{4}

The curve E_{t} : y^{2} = x^{3} − t^{2}x + t^{4} is a nice curve to work with, so I did. Why is it nice? First of all, when looking at the family of curves E/Q(t), we see that its A_{ϵ}(P) comes out to 2P, so the family has rank 2 over Q(t). Then we can use the results from Silverman’s specialization theorem (for further mention of this theorem, see Silverman’s text on the arithmetic of elliptic curves):

Theorem 3.1 (Silverman’s Specialization Theorem) If you have a 1parameter family E over Q(t) such that the rank of E/Q(t) is r, then there exists some t_{0} such that for all t > t_{0}, the curve E_{t} over Q has rank at least r.

So after a certain t, all curves in this family will have at least rank 2. So we can expect to find at least 2 linearly independent points on these curves.

The problem, then, becomes finding any points on the curve, not just independent ones. While we don’t necessarily know the polynomials that may have generated our curve, we can still find some points. For example, observe that t^{4} is already a perfect square. So if we let x = 0, we can isolate the t^{4} and get the point P_{0}(0, t^{2}). Similarly, we can cancel the x terms by having x = ±t, again isolating the t^{4} and yielding the points P_{1} and P_{2} = (±t, t^{2}). We can cancel out the two terms with t coefficients when x = t^{2}, which conveniently makes the x^{3} term into a perfect square, from whence we have the point P_{3} = (t^{2}, t^{3}). To find a fourth point, we can treat the right side as a quadratic in t^{2}. We have y^{2} = (t^{2})^{2} − xt^{2} + x^{3}. We want x^{3} to be square, though, so we let x = b^{2}. Then it becomes y^{2} = (t^{2})^{2} − b^{2}t^{2} + b^{6}, but we want to be able to write it as y^{2} = (t^{2} − b^{3})^{2}. In order to do this, the cross terms must match up, so b = \frac{1}{2}, thus x = \frac{1}{4}. The point P_{4} = (\frac{1}{4}, t^{2} − \frac{1}{8}) is then also forced.

To begin our attempt at finding the generators for the Mordell-Weil group, we used a technique proposed in a paper by Cremona. By simply calculating the determinant of the height matrices that any group of these points generates, we can figure out a lot. For example, we can determine automatically a lower bound for the geometric rank. From this, hopefully we can figure out what t_{0} is in Theorem 3.1. The worry is that some of these points that we’ve found to be forced may be linearly dependent, or are perhaps even torsion points for some t (which we can verify easily since h(P) = 0 for torsion points P). This is especially a worry at the lower values of t, since Silverman’s theorem only tells us that eventually we will have rank 2, not necessarily initially. So we may expect that for lower values of t, we may find torsion points or dependence.

Using the described technique, I discovered that for 1 < t ≤ 43640 any two of the points P_{1}, P_{2}, P_{4} are linearly independent, while any set of three points is always linearly dependent. Therefore, our t_{0} in 3.1 must be 1.

3.3 y^{2} = x^{3} + 5x^{2} − 16t^{2}x + 64t^{2}

This family of curves, unlike the curve above, has rank 3. Naturally, this proves even more interesting when studying independent points in the infinite group. After all, there are more of them!

Finding points on this curve followed the same pattern as last time. It basically relied on the fact that 4 + 5 is a perfect square (3^{2}).

Letting x = 4 gets the right portion of the equation to cancel, leaving just the x^{3}+5x^{2}, which conveniently is the square of 12. So the point P_{1} = (4, 12) is on the curve. Similarly, x = −5 gets the left portion to cancel, leaving only 144t^{2} remaining. So P_{2} = (−5, 12t) must also be on the curve. We can also factor the equation as y^{2} = x(x^{2} − 16t^{2}) + 5x^{2} + 64t^{2}, so x = ±4t leaves only 5x^{2} + 64t^{2}, which again simplifies to 144t^{2} when x is substituted in. So the points P_{3}andP_{4} = (±4t, 12t) must be on the curve. Finally, as with the first curve, we can treat the right side as a quadratic in t if we let x = at^{2}. This yields y^{2} = t^{2}(a^{3}(t^{2})^{2} + (5a^{2} − 16a)t^{2} + 64). Noticing that we want a^{3} to be a square, let a = b^{2} and get y^{2} = t^{2}(b^{6}(t^{2})^{2} + b^{2}(5b^{2} − 16)t^{2} + 8^{2}), but we want to be able to write the right side as t^{2}(b^{3}t^{2} +8)^{2}. This requires that the crossterms match up, and thus 16b^{3} = b^{2}(5b^{2} − 16). We end up having b = 4, so the point P_{5} = (16t^{2}, 8t(8t^{2} + 1)) is also forced.

Now, proceeding with calculations similar to before, I found that any pair of the above points is linearly independent for t > 1. So for t > 1, we have that this curve has rank ≥ 2. However, for t > 5, the point P_{5} paired with any of the other 2 points forms a set of linearly independent points, implying that for t > 5, the rank of this curve is at least 3. So again, our t_{0} in 3.1 is most likely 5, and we even have the points to prove it. This was tested up to t = 250.

3.4 Notes about these curves

The rank 2 family was formed explicitly to force the points (±t, t^{2}) and (t^{2}, t^{3}) on each curve. The other points were then found using the methods described, and again checked against each other for independence.

The rank 3 family came from Fermigier, who apparently used Mestre’s method to come up with the curve. However, he did this by taking 2 roots to be zero (and thus forcing only 6 points) to come up with a curve of rank 3. Some simplifying was done to get the curve into the form that it appears; unfortunately, recovering α_{1} and α_{2} was too difficult a task. Instead, we simply stared at the curve to find the points metioned above.

4 What’s Next

Now that, using Pari, I have a way of determining whether or not a set of points on a curve are independent, and for the two curves I’ve worked with I’ve been able to find points that force a lower bound of the geometric rank to be what we expect from Silverman’s theorem 3.1, it would be nice to do even more. One way to proceed is with points of low height. There are algorithms to find points of low height on a curve; once these are found, are they independent from the points we’ve already forced on these curves? If a family of curves has rank r, can we find points that force the rank to be greater than r?

Eventually, once we’ve generated a more accurate lower bound for the geometric rank of these curves by looking at a larger sampling of points (exactly these points of low height that we are generating), we will hopefully be numerically verifying the Birch and Swinnerton Dyer conjecture. I present it here as it appears in Knapp:

Conjecture 4.1 (Birch and Swinnerton-Dyer) For any elliptic curve over Q, the L-series L(s) extends to be entire for s ∈ C, and the order of vanishing of L(s) at s = 1 is equal to the rank r of the curve.

Since I am essentially coming up with a lower bound for the rank r of these curves, we should expect that L(E, 1) vanishes to an order that it is at least as large as the size of the largest set of independent points on the curve that I can find–assuming that 4.1 is true. And since other students in the class are calculating the order of vanishing of L(E, 1), when our work is combined we will hopefully see a significant amount of information supporting 4.1. In fact, knowing the order of vanishing, then, should tell us exactly where to look for these extra linearly independent points: exactly where the order of vanishing is greater than the rank of E/Q(t).

So far, we have Jonathon Hsu’s results to compare with. Jonathon has calculated the values of the derivatives of L(E, 1) to find the order of vanishing for both curves that I have worked with. Surprisingly enough, our results have been consistent with the Birch and Swinnerton-Dyer conjecture; for the rank 2 family, the analytic rank is ≥ 2 for t ≥ 2, and for the rank 3 family, the analytic rank is ≥ 3 for t ≥ 6. My results have discovered at least two independent points for the curve of rank 2 when t ≥ 2, and at least three independent points for the curve of rank 3 when t ≥ 6.

Lastly, it is important to note that we are not actually finding generators of the Mordell-Weil group. To see this, suppose that there is a point P in the infinite group, and at some point in our search we discover the point 2P and, being independent from our other points, we tack it on as one of the “basis” points of the infinite group. However, we will never recover the point P from these generators. Therefore, we are merely looking to find generators for some subset of the Mordell-Weil group with the same rank as the group itself. Eventually, using a method outlined in the Cremona paper, perhaps our program could be refined to take care of this problem.

5 Bibliography

Cremona, John, Algorithms for Modular Elliptic Curves, Cambridge University Press, Cambridge, 1992.

Fermigier, Stefan, “Etude experimentale du rang de familles de courbes elliptiques sur Q”, Experimental Math., Vol 5 No. 2, 1996, p 119.

Knapp, Anthony W., Elliptic Curves, Princeton University Press, Princeton, 1992.

Silverman, Joseph H., and Tate, John, Rational Points on Elliptic Curves, Springer-Verlag, New York, 1992.

Silverman, Jospeh H., The Arithmetic of Elliptic Curves, Springer-Verlag, 1986.