THE GRADUATE STUDENT SECTION WHAT IS… an Elliptic Curve? Harris B. Daniels and Álvaro Lozano-Robledo Communicated by Steven J. Miller and Cesar E. Silva Elliptic curves are ubiquitous in number theory, algebraic geometry, complex analysis, cryptography, physics, and beyond. They lie at the forefront of arithmetic geometry, as shown in the feature on Andrew Wiles and his proof of Fermat’s Last Theorem that appears in this issue of the Notices. The goal of arithmetic geometry, in general, is to determine the set of 퐾-rational points on an algebraic variety 퐶 (e.g., a curve given by polynomial equations) defined over 퐾, where 퐾 is a field, and the 퐾-rational points, denoted by 퐶(퐾), are those points on 퐶 with coordinates in 퐾. For instance, Fermat’s Last Theorem states that the algebraic variety 푋푛 + 푌푛 = 푍푛 has only trivial solutions (one with 푋, 푌, or 푍 = 0) over Figure 1. A curve of genus 1 over the complex ℚ when 푛 ≥ 3. Here we will concentrate on the case numbers is a Riemann surface with one hole. of a 1-dimensional algebraic variety, that is, a curve 퐶, and a number field 퐾 (such as the rationals ℚ or the For example, the curve Gaussian rationals ℚ(푖)). Curves are classified by their 퐶 ∶ 3푋3 + 4푌3 = 5 geometric genus as complex Riemann surfaces. When the genus of 퐶 is 0, as for lines and conics, the classical has no ℚ-rational points, but the local methods we use methods of Euclid, Diophantus, Brahmagupta, Legendre, in the genus 0 case to Gauss, Hasse, and Minkowski, among others, completely rule out global points determine the 퐾-rational points on 퐶. For example, fail here.1 A goal of The study of the theory of elliptic 퐶 ∶ 37푋 + 39푌 = 1 and 퐶 ∶ 푋2 − 13푌2 = 1 elliptic curves 1 2 curves is to find all have infinitely many rational points that can be completely the 퐾-rational points grew in the 1980s. determined via elementary methods. However, when the on curves of genus genus of 퐶 is 1, we are in general not even able to decide one. 2 whether 퐶 has 퐾-rational points, much less determine all An elliptic curve 퐸 is a smooth projective curve of the points that belong to 퐶(퐾). genus 1 defined over a field 퐾, with at least one 퐾- rational point (i.e., there is at least one point 푃 on 퐸 with coordinates in 퐾). If the field 퐾 is of characteristic 0 (e.g., Harris B. Daniels is assistant professor at Amherst College. His e-mail address is [email protected]. 1퐶 ∶ 3푋3 + 4푌3 = 5 is an example of Selmer where the local-to- Álvaro Lozano-Robledo is associate professor at the University global principle fails. This means that there are points on 퐶 over of Connecticut. His e-mail address is alvaro.lozano-robledo@ every completion of ℚ—i.e., over ℝ and the 푝-adics ℚ for every uconn.edu. 푝 prime 푝—but not over ℚ itself. For permission to reprint this article, please contact: 2Curves are considered in projective space ℙ2(퐾), where, in addi- [email protected]. tion to the affine points, there may be some points of the curve at DOI: http://dx.doi.org/10.1090/noti1490 infinity. March 2017 Notices of the AMS 241 THE GRADUATE STUDENT SECTION number fields) or characteristic 푝 > 3, then every elliptic The open questions about the rank of an elliptic curve can be given by a nice choice of coordinates, called curve are central to what makes the 퐾-rational points on a short Weierstrass model, of the form elliptic curves so hard to determine. The difficulty arises 퐸 ∶ 푦2 = 푥3 + 퐴푥 + 퐵, from the failure of the local-to-global principle (or Hasse principle) on curves of genus greater than or equal to 1 3 2 with 퐴 and 퐵 in 퐾 (and 4퐴 + 27퐵 ≠ 0 for smoothness). (see footnote 1). For an elliptic curve 퐸/퐾, one defines In this model there is only one 퐾-rational point at infinity, the Tate–Shafarevich group X = X(퐸/퐾) to measure the denoted by 풪. One aspect that makes the theory of failure of the Hasse principle on 퐸. In a sense, X plays the elliptic curves so rich is that the set 퐸(퐾) can be equipped role of the ideal class group of a number field. However, with an Abelian group structure, geometric in nature (see we do not know that X(퐸/퐾) is always a finite group.3 If Figure 2), where 풪 is the zero element (in other words, we knew that X is always finite, then a method Fermat elliptic curves are 1-dimensional Abelian varieties). inaugurated, called descent, would presumably yield an algorithm to determine all the 퐾-rational points on 퐸. In the 1960s, Birch and Swinnerton-Dyer conjectured an analytic approach to computing the rank of an elliptic 푃 + 푄 curve. Later, their conjecture was refined in terms of the Hasse–Weil 퐿-function of an elliptic curve 퐸 (over ℚ for 푃 simplicity), which is defined by an Euler product: −푠 −1 퐿(퐸, 푠) = ∏ 퐿푝(퐸, 푝 ) , 푝 prime 2 푄 where 퐿푝(퐸, 푇) = 1 − 푎푝푇 + 푝푇 for all but finitely many primes, 푎푝 = 푝 + 1 − #퐸(픽푝), and #퐸(픽푝) is the number of 푅 points on 퐸 considered as a curve over 픽푝. Thus defined, 퐿(퐸, 푠) converges as long as Re(푠) > 3/2. In fact, Hasse conjectured more: any 퐿-function of an elliptic curve over Figure 2. The addition law on an elliptic curve. ℚ has an analytic continuation to the whole complex plane. This has now been proved as a consequence of the The Abelian group 퐸(퐾) was conjectured to be finitely modularity theorem that we discuss below. The Birch and generated by Poincaré in the early 1900s and proved to Swinnerton-Dyer conjecture (BSD) claims that the order be so by Mordell for 퐾 = ℚ in 1922. The result was of vanishing of generalized to Abelian varieties over number fields by 퐿(퐸, 푠) at 푠 = 1 Weil in 1928 (a result widely known as the Mordell–Weil An elliptic curve can is equal to 푅퐸/ℚ, Theorem). The classification of finitely generated Abelian the rank of 퐸(ℚ). groups tells us that 퐸(퐾) is the direct sum of two groups: be equipped with an In fact, the conjec- its torsion subgroup and a free Abelian group of rank Abelian group ture also predicts 푅 ≥ 0, i.e., the residue at 푠 = 푅 퐸(퐾) ≅ 퐸(퐾)tors ⊕ ℤ . structure. 1 in terms of in- We then call 푅 = 푅퐸/퐾 the rank of the elliptic curve 퐸/퐾. variants of 퐸/ℚ. For instance, for For instance, the 2 3 3 퐸 ∶ 푦2 + 푦 = 푥3 + 푥2 − 10푥 + 10, curve 퐸 ∶ 푦 +푦 = 푥 −7푥+6 is of rank 3, with 퐸(ℚ) ≅ ℤ , and the graph of 퐿(퐸, 푥) for 0 ≤ 푥 ≤ 3 is displayed in the group 퐸(ℚ) is generated by 푃 = (2, −2) and 푄 = Figure 3. The BSD conjecture is known to hold only in (−4, 1). Here 푃 is a point of order 5 and 푄 is of infinite certain cases of elliptic curves of rank 0 and 1, by work order, and so 퐸(ℚ) ≅ ℤ/5ℤ ⊕ ℤ. of Coates and Wiles, Gross and Zagier, Kolyvagin, Rubin, Which finitely generated Abelian groups can arise as Skinner and Urban, among others. However, Bhargava, the group structure of an elliptic curve over a fixed field Skinner, and Zhang have shown that BSD is true for at 퐾? The possible torsion subgroups 퐸(퐾)tors that can occur least 66 percent of all elliptic curves over the rationals. have been determined only when 퐾 = ℚ, or when 퐾 is The study of elliptic curves grew in popularity in the a quadratic or cubic number field (e.g., 퐾 = ℚ(푖), or 3 1980s when Hellegouarch, Frey, and Serre outlined a road 퐾 = ℚ( √2)). For 퐾 = ℚ, the list of torsion subgroups map to prove Fermat’s Last Theorem by proposing that was conjectured by Levi in 1908, later reconjectured by a certain elliptic curve cannot exist. Roughly speaking, Ogg in 1970, and finally proved in 1976 by Mazur: if 푝 ≥ 11 and 푎푝 + 푏푝 = 푐푝 is a nontrivial solution ℤ/푁ℤ for 1≤푁≤10, or 푁=12, of Fermat’s equation 푋푝 + 푌푝 = 푍푝, then the so-called 퐸(ℚ) ≅ { 2 푝 푝 tors ℤ/2ℤ⊕ℤ/2푀ℤ for 1≤1푀≤4. Frey–Hellegouarch curve 푦 = 푥(푥 − 푎 )(푥 + 푏 ) would have two properties thought to be contradictory. First, In contrast, the list of possible ranks 푅퐸/퐾 is completely the curve would be semistable, which is a mild technical unknown, even over ℚ. We do not even know if this list is finite or infinite for any fixed number field. The largest 3The finiteness of X is known only in certain cases with rank ≤ 1, rank known over ℚ is 28, for a curve found by Elkies. by work of Kolyvagin and Rubin. 242 Notices of the AMS Volume 64, Number 3 THE GRADUATE STUDENT SECTION condition about the type of curves 퐸/픽푝 that we get Freitas, Le Hung, and Siksek extended the modularity the- by reducing the coefficients of E modulo 푝. Second, orem to real quadratic fields. Modularity fits into a much the curve would be modular, a property we explain larger context, together with the Langlands program and in the next paragraph.