RATIONAL POINTS on ELLIPTIC CURVES Consider the Following

RATIONAL POINTS ON ELLIPTIC CURVES

SERAPHINA LEE

ABSTRACT. Elliptic curves are a special kind of curve that can be given as solutions to 2 3 2 equations of the form y + a1xy + a3y = x + a2x + a4x + a6, with an added point at infinity. I will begin with an introduction to basic definitions, followed by a discussion of their significance, plus what we know (and don’t know) about the group structure on their Q- and Fp- points.

1. INTRODUCTIONAND DEFINITIONS Consider the following example: 2 2 2 Example 1.1. Take the circle, given by C : x + y = r for some r ∈ R. It’s not hard to see that we can classify all rational points on this curve, namely: • If there exists some Q-point P ∈ C, then there are infinitely many Q-points on C. • Otherwise, there are zero Q-points on C. So for example, if r = 1, then we can find an infinite family of rational solutions by: Fix a known rational point, like (1, 0). Then take any q ∈ Q. Then with the line y = q(x − 1) goes through the point (1, 0), and intersects the circle at a second point. This point is rational, since

x2 + q2(x − 1)2 = 1 =⇒ (q2 + 1)x2 − 2q2x + (q2 − 1) = 0 and we compute that the discriminant is

4q4 − 4(q2 − 1)(q2 + 1) = 4q4 − 4(q4 − 1) = 4 i.e. x is rational, and therefore y is rational. Now consider the following curve: Example 1.2. Take C : x3 + y3 = 1. Then if x, y are rational points on C, we have

3 3 3 x0 + y0 = z for some x0, y0, z ∈ Z. Then Fermat’s last theorem (to bring out a sledgehammer of a tool) tells us that this has no nontrivial solutions over Z, meaning the only solutions are those where x0y0z = 0.

Date: November 16, 2018. 1 2 SERAPHINA LEE

What makes some polynomials of two variables have infinitely many rational points, while some others only have finitely many? It turns out, there is a topological/geometric invariant called genus that help us classify which case we are in. For curves with genus 0, there are either infinitely many rational points, or zero rational points. For curves with genus ≥ 2, we have the following:

Theorem 1.1 (Faltings). If C is a nonsingular algebraic curve of genus > 1 over Q, then C has ﬁnitely many rational points. (The ﬁrst example had genus 0, while the second example had genus 3.) So what about genus 1? This is where elliptic curves come into play.

Deﬁnition 1.1. An elliptic curve over a ﬁeld F is (for the purposes of this talk) the zero set of a polynomial of the form 2 3 2 y + a1xy + a3y = x + a2x + a4x + a6

with ai ∈ F, with a point at inﬁnity, which we denote O. More formally, an elliptic curve is really the following object:

Definition 1.2. An elliptic curve over a field F is a smooth projective curve of genus 1 with a prescribed point O. It is a theorem that all curves of genus 1 can be written in the cubic form given above. It is also a theorem that over fields k of characteristic 6= 2, 3, we can find a nicer model for elliptic curves through change of variables, etc, than above, i.e. by

y2 = x3 + Ax + B for some A, B ∈ k. Since Q has characteristic 0, all of our elliptic curves can be written in this way. An interesting and natural question about these curves is the following: “how many” rational points are there on a given elliptic curves? As opposed to the genus 0 case and the genus ≥ 2 case, there are examples of those with no rational points (other than the point at infinity), finitely many, and infinitely many. In this talk, we’ll discuss how genus 1 curves are special in the way that classifying their rational points is unusually difficult, compared to higher or lower genus. Most of the material here is pulled from [1], and more information about the last two sections can be found in [2] and [3].

2. MORDELL–WEIL GROUP We can think about these arithmetic questions about elliptic curves by studying the group structure that we can put on their Q-rational points: Definition 2.1. The Mordell–Weil group structure on the Q-rational points on E is defined: If P,Q ∈ E(Q), then we get the unique point R ∈ E(Q) such that P, Q, R are collinear, and define P ⊕ Q ⊕ R = O. First, we examine the structure of this group in the next few examples: RATIONAL POINTS ON ELLIPTIC CURVES 3

Example 2.1. Take 2 3 2 E1 : y + y = x − x given in this ﬁgure 1

2 3 2 FIGURE 1. y + y = x − x

which has Mordell–Weil group Z/5Z, with generator (0, 0), where the other points are

(0, 0) → (1, −1) → (1, 0) → (0, −1) → O

Example 2.2. Take 2 3 E2 : y = x + x + 1, given in this ﬁgure 2

2 3 FIGURE 2. y = x + x + 1

which has Mordell–Weil group Z, with generator (0, 1). Some of the other points are:

1 9 287 40879 43992 30699397 (0, 1) → , − → (72, 611) → − , → , − → ... 4 8 1296 46656 82369 23639903 where (0, 1) and (72, 611) are the only integral points. 4 SERAPHINA LEE

Example 2.3. Take 2 3 E0 : y = x − x given in this ﬁgure 3

2 3 FIGURE 3. y = x − x

This elliptic curve has Mordell–Weil group Z/2Z × Z/2Z, with torsion generators (0, 0), (1, 0). For example, note that (−1, 0) ⊕ (0, 0) ⊕ (1, 0) = O, which means that since (1, 0) = −(1, 0), we have

(−1, 0) ⊕ (0, 0) = (1, 0). How do we know that we can find such a unique point R? First, if P,Q ∈ E(Q) are collinear, we can find a line y = mx + b or x = b which goes through them, with m, b ∈ Q. Therefore, in the first case, we have the equation

x3 − (mx + b)2 + Ax + b = 0 which has the two roots x(P ), x(Q), the x-coordinates of P and Q. This means that the third root to the equation above is also rational, which gives us a unique rational point R that is collinear with P,Q, and is in E. In the second case, we have the vertical line x = b, which crosses P,Q, and O. There is no other point on E that is collinear with P,Q since it would have to be a root to the equation

y2 − b3 − Ab − B = 0 which has exactly two roots. We can also invoke a giant hammer and note Bezout’s theorem: Theorem 2.1 (Bezout). If X,Y are projective curves of degree m, n respectively, then mn- points in their intersection, up to multiplicity. So a line is a curve of degree 1, and an elliptic curve is one of degree 3. Therefore, a line and an elliptic curve have 3 intersection points, counted up to multiplicity. The next few statements are some things we know about the Modell–Weil group over Q. They’re pretty famous theorems, proven by pretty famous mathematicians, so need- less to say, we won’t be proving any of them in this talk. RATIONAL POINTS ON ELLIPTIC CURVES 5

r Theorem 2.2 (Mordell–Weil). The group E(Q) is ﬁnitely generated, i.e. E(Q) = Z ×Etors, where Etors is torsion and r < ∞. For integral points, however, we can say something even stronger: Theorem 2.3 (Siegel). If E/Q is an elliptic curve, then there are only ﬁnitely many points on E with x, y ∈ Z. So we have a rough handle on “how many” rational points may be on an elliptic curve E. There is a little more that we know about the structure of the Mordell–Weil group, especially the torsion part. Theorem 2.4 (Mazur). The torsion part of E(Q) are one of the following: Z/NZ, with 1 ≤ N ≤ 10 or N = 12, Z/2Z × Z/2NZ, with 1 ≤ N ≤ 4. Do we know anything about the free part? Not really, as we see below: Theorem 2.5 (Elkies). The current known highest rank of an elliptic curve was found by Noam Elkies in 2006, when he found an elliptic curve with rank at least 28. The curve is given by

y2 + xy + y = x3−x2 − 20067762415575526585033208 209338542750930230312178956502x + 3448161179503055646703298569039 07203748559443593191803612 66008296291939448732243429 We don’t know if rank is bounded or unbounded!

3. LOCAL-TO-GLOBALAND ELLIPTIC CURVESOVER FINITE FIELDS So consider this idea: what if we knew the structure of the elliptic curve over all ﬁelds Fp for p prime. Then would we know the structure of E over Q by “patching” together all this local data? It is generally false that if a curve C has solutions in Fp for all p, that it will also have solutions in Q, but we still think that the local data gives us some information about the global data. So let’s ﬁrst study the structure of E(Fp). The most useful information we can get is from this Hasse–Weil bound, which states

Theorem 3.1 (Hasse-Weil Bound). If E is an elliptic curve over Fp, then √ |#E(Fp) − (p + 1)| ≤ 2 p. We denote p + 1 − #E(Fp) =: ap. This is not too surprising; naively, one may reason heuristically that #E(Fp) should be on the order of p. A trivial upper bound can be given by #E(Fp) ≤ 2p + 1 , since each x value yields two y values. And since about half of elements in Fp are squares, you may expect #E(Fp) to be around p. Every math talk should have a proof, probably. So here is a very rough sketch of theorem 3.1: 6 SERAPHINA LEE ¯ ¯ Proof. Consider E(Fp). We note that for P ∈ E(Fp), P ∈ E(Fp) ⇐⇒ φ(P ) = P , where φ is the Frobenius endomorphism, which sends φ :(x, y) 7→ (xp, yp). Specigically, this means P ∈ ker(1 − φ). Then note that # ker(1 − φ) = deg(1 − φ), and that deg function satisﬁes the condition: p | deg(ψ1 − ψ2) − deg(ψ1) − deg(ψ2)| ≤ 2 deg(ψ1) deg(ψ2). deg(1) = 1, deg(φ) = p. Then this shows that √ |#E(Fp) − p − 1| ≤ 2 p. Theorem 3.2 (Sato–Tate, sort of). #{a > 0 : p ≤ N} #{a < 0 : p ≤ N} lim p = lim p = 0.5 N→∞ #{p ≤ N} N→∞ #{p ≤ N}

This theorem tells us that for all elliptic curves, the distribution of ap values are more or less the same. However, we think there might still be some info to be gleaned from this local data.

4. BIRCHAND SWINNERTON-DYER The idea is that if the rank of an elliptic curve over Q is low (resp. high), then “for more primes p, the elliptic curve will have fewer (resp. more) points mod p.” This is precisely what Birch and Swinnerton-Dyer studied, i.e. letting

Y Np f (N) = ,N = #E( ) = p + 1 − a , E p p Fp p p≤N they computed

lim fE(N) N→∞ r and saw that the rate of growth was ∼ C log(N) , r = rank(E/Q). They were able to make this conjecture more precise by thinking about L-functions, since formally, we have

Y Np Y p + 1 − ap 1 = = p p L (1) p p E/Q

5. L-FUNCTIONS The L-function is a generating function that encodes all the local data attached to the elliptic curve. Definition 5.1. The L-function of an elliptic curve E over Q is defined as the following: Y 1 LE/Q(s) = −s 1−2s . 1 − app + p p-∆ P −s which, if expanded as a Dirichlet series cnn , satisfies cp = ap for all primes p. Now we can state a version of the Birch and Swinnerton-Dyer conjecture: RATIONAL POINTS ON ELLIPTIC CURVES 7

Conjecture 5.1 (Birch and Swinnerton-Dyer). The Taylor expansion of LE(s) at s = 1 has the form r LE(s) = c(s − 1) + higher order terms where r = rank(E(Q)). Now, we want to make a statement analogous to this claim, such that we can use less local information to make a similar claim about the algebraic rank. Definition 5.2. We define a new L-function, Y 1 LE,sgn(s) = −s . 1 − sgn(ap)p p-∆ Conjecture 5.2. Is there a similar claim we can make, about maybe the Taylor expansion 1 of LE,sgn(s) around s = 2 ? Maybe! For instance, define the renormalized version of this signed L-function,

Deﬁnition 5.3. 0 Y 1 1 LE,sgn(s) = −1 −s 1 + p 1 − sgn(ap)p p-∆ 1 This is at least a reasonable function to study, since at s = 2 , we have (formally) √ Y p + 1 − sgn(ap) p 1 ∼ . p 0 1 p LE,sgn 2

REFERENCES [1] Joseph H. Silverman. The Arithmetic of Elliptic Curves. Springer, 2009. [2] Andrew Wiles. The birch and swinnerton-dyer conjecture. [3] B. J. Birch and H. P. F. Swinnerton-Dyer. Notes on elliptic curves. II. Journal fur die reine und angewandte Mathematik, 1964.