Diophantine Geometry and Principal Bundles Minhyong Kim Perimeter Institute, March, 2019 I. Background: Arithmetic of Algebraic Curves Arithmetic of algebraic curves X : a smooth algebraic curve of genus g defined over Q. For example, given by a polynomial equation f (x; y) = 0 of degree d with rational coefficients, where g = (d − 1)(d − 2)=2: Diophantine geometry is concerned with the set X (Q) of rational solutions. Structure is quite different in the three cases: g = 0, spherical geometry (positive curvature); g = 1, flat geometry (zero curvature); g ≥ 2, hyperbolic geometry (negative curvature). Arithmetic of algebraic curves: g = 0; d ≤ 2 Even now (after millennia of studying these problems), g = 0 is the only case that is completely understood. For g = 0, techniques reduce to class field theory and algebraic geometry: local-to-global methods, generation of solutions via sweeping lines, etc. Idea is to study Q-solutions by considering the geometry of solutions in various completions, the local fields R; Q2; Q3;:::; Q691;:::; Arithmetic of algebraic curves: g = 0 Local-to-global methods sometimes allow us to ‘globalise’. For example, 37x2 + 59y 2 − 67 = 0 has a Q-solution if and only if it has a solution in each of R; Q2; Q37; Q59; Q67; (Hasse principle) a criterion that can be effectively implemented. In other words, there is a successful study of the inclusion: 0 Y X (Q) ⊂ X (AQ) = X (Qp) These are called reciprocity laws. Arithmetic of algebraic curves: g = 1 (d = 3) X (Q) = φ, non-empty finite, infinite, all are possible. Hasse principle fails: 3x3 + 4y 3 + 5 = 0 has points in Qv for all v, but no rational points. If X (Q) 6= φ, then fixing an origin O 2 X (Q) gives X (Q) the structure of a finitely-generated abelian group via the chord-and-tangent method: r X (Q) ' X (Q)tor × Z : Here, r is called the rank of the curve and X (Q)tor is a finite effectively computable abelian group. Arithmetic of algebraic curves: g = 1 To compute X (Q)tor , write X := fy 2 = x3 + ax + bg [ f1g (a; b 2 Z). Then (x; y) 2 X (Q)tor ) x; y are integral and y 2j(4a3 + 27b2): Arithmetic of algebraic curves: g = 1 However, the algorithmic computation of the rank and a full set of generators for X (Q) is very difficult, and is the subject of the conjecture of Birch and Swinnerton-Dyer. In practice, it is often possible to compute these. For example, for y 2 = x3 − 2; Sage will give you r = 1 and the point (3; 5) as generator. Note that 2(3; 5) = (129=100; −383=1000) 3(3; 5) = (164323=29241; −66234835=5000211) 4(3; 5) = (2340922881=58675600; 113259286337279=449455096000) Arithmetic of algebraic curves: g = 1 Figure: Denominators of N(3; 5) Arithmetic of algebraic curves: g ≥ 2 (d ≥ 4) X (Q) is always finite (Mordell conjecture, as proved by Faltings). However, *very* difficult to compute: consider xn + y n = 1 for n ≥ 4. Sometime easy, such as x4 + y 4 = −1: However, when there isn’t an obvious reason for non-existence, e.g., there already is one solution, then it’s hard to know when you have the full list. For example, y 3 = x6 + 23x5 + 37x4 + 691x3 − 631204x2 + 5168743489 obviously has the solution (0; 1729), but are there any other? Arithmetic of algebraic curves: g ≥ 2 (d ≥ 4) Effective Mordell problem: Find a terminating algorithm: X 7! X (Q) The Effective Mordell conjecture (Szpiro, Vojta, ABC, ...) makes this precise using height inequalities. Will describe an approach to this problem using the arithmetic geometry of principal bundles. Ia. A little history Principal bundles in Diophantine geometry: some history X : a smooth projective curve of genus ≥ 2 defined over Q. Andre Weil in 1929 constructed an embedding ⊂ - j : X JX ; where JX is an abelian variety of dimension g. That is, over C, J ( ) = g =Λ = H0(X ( ); Ω1 )∗=H (X ; ): X C C C X (C)) 1 Z The map j is defined over C by fixing a basepoint b and Z x j(x)(α) = α mod H1(X ; Z); b for α 2 H0(X ( ); Ω1 ): C X (C) Principal bundles in Diophantine geometry: some history But the key point is that JX is also a projective algebraic variety defined over Q, and if b 2 X (Q), then the map j is also defined over Q. The reason is that JX is a moduli space of line bundles of degree 0 on X and j(x) = O(x) ⊗ O(−b): The main application is that j : X (Q) ⊂ - J(Q): Weil also proved that J(Q) is a finitely-generated abelian group, and hoped, without success, that this could be somehow used to study X (Q). Principal bundles in Diophantine geometry: some history In 1938, Weil studied Y BunX (GLn) = GLn(K(X ))nGLn(AK(X ))=[ GLn(Ocx )] x as a ‘non-abelian Jacobian’, hoping to use it to prove the Mordell conjecture, but failed again. However, proved a number of foundational theorems, including the fact that vector bundles of degree zero admit flat connections, beginning non-abelian Hodge theory. This paper was very influential in geometry, leading to paper of Narsimhan and Seshadri: st 1 irr BunX (GLn)0 ' H (X ; U(n)) : Principal bundles in Diophantine geometry: some history This was extended by Donaldson, influencing this work on smooth manifolds and gauge theory, and by Simpson to 1 Higgs(GLn) ' H (X ; GLn): But the Mordell conjecture was proved in the early 1980s by Faltings using the arithmetic geometry of the moduli stack Ag of principally polarised abelian varieties. He deduced the Mordell conjecture from the finiteness of integral points on Ag . II. Arithmetic Principal Bundles Arithmetic principal bundles: (GK ; R; P) K: field of characteristic zero. GK = Gal(K¯=K): absolute Galois group of K. A group over K is a topological group R with a continuous action of GK by group automorphisms: - GK × R R: A principal R-bundle over K is a topological space P with compatible continuous actions of GK (left) and R (right, simply transitive): For g 2 GK , z 2 P, r 2 R, g(zr) = g(z)g(r): Note that P is trivial exactly when there is a fixed point z 2 PGK . Arithmetic principal bundles Example: R = (1) := lim µ n ; Zp − p n where µpn ⊂ K¯ is the group of p -th roots of 1. Thus, Zp(1) = f(ζn)ng; where pn pm ζn = 1; ζnm = ζn: As a group, (1) ' = lim =pn; Zp Zp − Z n but there is a continuous action of GK . Arithmetic principal bundles ∗ Given any x 2 K , get principal Zp(1)-bundle pn pm P(x) := f(yn)n j yn = x; ynm = yn:g over K. P(x) is trivial iff x admits a pn-th root in K for all n. For example, when K = C, P(x) is always trivial. When K = Q, P(x) is trivial iff x = 1. For K = R, and p odd, P(x) is trivial for all x. For K = R and p = 2, P(x) is trivial iff x > 0. Arithmetic principal bundles: localisation Denote by 1 1 H (K; R) = H (GK ; R) the isomorphism classes of principal R-bundles over K, which can also be given an explicit description using continuous group cohomology of GK . When K is a number field, there are completions Kv and injections ⊂ - Gv = Gal(K¯v =Kv ) G = Gal(K¯=K): giving rise to the localisation map 1 - Y 1 loc : H (K; R) H (Kv ; R): v A wide range of problems in number theory rely on the study of its image. The general principle is that the local-to-global problem is easier to study for principal bundles than for points. Diophantine principal bundles: elliptic curves E: elliptic curve over Q. We let G = Gal(Q¯ =Q) act on the exact sequence p 0 - E[p](Q¯ ) - E(Q¯ ) - E(Q¯ ) - 0 to generate the long exact sequence p 0 - E(Q)[p] - E(Q) - E(Q) 1 1 p 1 - H (Q; E[p]) - H (Q; E) - H (Q; E); from which we get the inclusion (Kummer map) 1 0 - E(Q)=pE(Q) ⊂ - H (Q; E[p]) Diophantine principal bundles: elliptic curves The central problem in the theory of elliptic curves is the identification of the image 1 Im(E(Q)=pE(Q)) ⊂ H (G; E[p]): To this end, define the p-Selmer group 1 Sel(Q; E[p]) ⊂ H (Q; E[p]) 1 to be the classes in H (Q; E[p]) that locally come from points. This is useful because the local version of this problem can be solved. Diophantine principal bundles: elliptic curves 1 0 - E(Q)=pE(Q) ⊂ - H (Q; E[p]) locv locv ? ? - ⊂- 1 0 E(Qv )=pE(Qv ) H (Qv ; E[p]) Then −1 Sel(Q; E[p]) := \v locv (Im(E(Qv )=pE(Qv ))): Diophantine principal bundles: elliptic curves The key point is that the p-Selmer group is a finite-dimensional Fp-vector space that is effectively computable and this already gives us a bound on the Mordell-Weil group of E: E(Q)=pE(Q) ⊂ Sel(Q; E[p]): This is then refined by way of the diagram n n 0 - E(Q)=p E(Q) - Sel(Q; E[p ]) ? ? 0 - E(Q)=pE(Q) - Sel(Q; E[p]) for increasing values of n. Diophantine principal bundles: elliptic curves Conjecture: (BSD, Tate-Shafarevich) 1 n Im(E(Q)=pE(Q)) = \n=1Im[Sel(Q; E[p ]] ⊂ Sel(Q; E[p]): Of course this implies that N Im(E(Q)=pE(Q)) = Im[Sel(Q; E[p ]] ⊂ Sel(Q; E[p]) at some finite level pN , and there is a conditional algorithm for verifying this.