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 ∈ 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 := {y 2 = x3 + ax + b} ∪ {∞}

(a, b ∈ Z).

Then (x, y) ∈ X (Q)tor ⇒ x, y are integral and

y 2|(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 ∈ 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 ∈ 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 ))\GLn(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 ∈ GK , z ∈ P, r ∈ R,

g(zr) = g(z)g(r).

Note that P is trivial exactly when there is a fixed point z ∈ 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) = {(ζn)n}, 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 ∈ K , get principal Zp(1)-bundle

pn pm P(x) := {(yn)n | yn = x, ynm = yn.}

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)

∞ 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. A main goal of BSD is to remove the conditional aspect. III. Diophantine principal bundles II: The non-abelian case Diophantine principal bundles II: The non-abelian case

To generalise, focus on the sequence of maps

p p ··· - E[p3] - E[p2] - E[p]

of which we take the inverse limit to get the p-adic Tate module of E: T E := lim E[pn]. p ←− This is a free Zp-module of rank 2. The previous finite boundary maps can be packaged into

j : E( ) - lim H1( , E[pn]) = H1( , T E). Q ←− Q Q p Diophantine principal bundles II: The non-abelian case The key point is that p TpE ' π1 (E, O), p where π1 (X , b) refers to the pro-p completion of the fundamental group of a variety X and the map j can be thought of as x 7→ πp(E; O, x).

The pro-p completion of a group A is Ap := lim A/N ←− where N ⊂ A runs over normal subgroups of index a power of p. Given a principal A-bundle P, can form its pro-p completion as a pushout Pp := [P × Ap]/A, of which basic examples are p π1 (X ; b, x). Diophantine principal bundles II: The non-abelian case

Fundamental fact of arithmetic homotopy:

If X is a variety defined over Q and b, x ∈ X (Q), then

p p π1 (X , b), π1 (X ; b, x)

admit compatible actions of G = Gal(Q¯ /Q).

The triples p p (GQ, π1 (X , b), π1 (X ; b, x))

are important concrete examples of (GK , R, P) from the general definitions. Diophantine principal bundles II: The non-abelian case

This action is given via the construction of a pro−p universal cover

p - X˜ := (Xi X )i∈I ,

- where Xi X runs over a cofinal system of finite covering spaces of X of p−power degree. Then the choice of a basepoint lift b˜ ∈ X˜p determine isomorphisms

p ˜p p ˜p π1 (X , b) ' Xb ; π1 (X ; b, x) ' Xx

Each - Xi X is algebraic, and the rationality of b allows us to define the whole system over Q, which gives us the G-action. Diophantine principal bundles II: The non-abelian case

This formulation then extends to general X , whereby we get a map

- 1 p j : X (Q) H (G, π1 (X , b))

given by p x 7→ [π1 (X ; b, x)]

For each prime v, have local versions

- 1 p jv : X (Qv ) H (Gv , π1 (X , b))

given by p x 7→ [π1 (X ; b, x)] which turn out to be far more computable than the global map. Diophantine principal bundles II: The non-abelian case

Localization diagram:

- Y X (Q) X (Qv ) v Q j v jv ? ? 1 p loc- Y 1 p H (G, π1 (X , b)) H (Gv , π1 (X , b)) v As in the elliptic curve case, our interest is in the interaction Q between the images of loc and v jv . Diophantine principal bundles II: The non-abelian case

Actual applications use

- Y X (Q) X (Qv ) v Q j v jv ? ? 1 - Y 1 H (G, U(X , b)) H (Gv , U(X , b)) v where p U(X , b) = ‘π1 (X , b) ⊗ Qp‘ p is the Qp-pro-unipotent completion of π1 (X , b). The effect is that the moduli spaces become pro-algebraic varieties over Qp and the lower row of this diagram an algebraic map. Diophantine principal bundles II: The non-abelian case

That is, the key object of study is

1 Hf (G, U(X , b))

the Selmer variety of X , defined to be the subfunctor of H1(G, U(X , b)) satisfying local conditions at all v. These are conditions like ‘unramified at most primes’, ‘crystalline at p’, and often a few extra conditions. Diophantine principal bundles II: The non-abelian case

- Y X (Q) X (Qv ) v Q j v jv ? ? 1 - Y 1 α- Hf (G, U(X , b)) Hf (Gv , U(X , b)) Qp v If α is an algebraic function vanishing on the image, then Y α ◦ jv v Q gives a defining equation for X (Q) inside v X (Qv ). Diophantine principal bundles II: The non-abelian case

To make this concretely computable, we take the projection

Y - prp : X (Qv ) X (Qp) v and try to compute Y ∩αprp(Z(α ◦ jv )) ⊂ X (Qp). v

Non-Archimedean effective Mordell Conjecture:

Q I. ∩αprp(Z(α ◦ v jv )) = X (Q)

II. This set is effectively computable. Diophantine geometry: remark on non-abelian reciprocity

There is a non-abelian class field theory with coefficients in a fairly general variety X over a number field F generalising CFT with coefficients in Gm. This consists (with some simplications) of a filtration

X (AF ) = X (AF )1 ⊃ X (AF )2 ⊃ X (AF )3 ⊃ · · ·

and a sequence of maps

- recn : X (AF )n Gn(X )

to a sequence of groups such that

−1 X (AF )n+1 = recn (0). Diophantine geometry: remark on non-abelian reciprocity

−1 −1 −1 ··· rec3 (0) ⊂ rec2 (0) ⊂ rec1 (0) ⊂ X (AF )

|| || || ||

··· X (AF )4 ⊂ X (AF )3 ⊂ X (AF )2 ⊂ X (AF )1

rec4 rec3 rec2 rec1 ? ? ? ? 3 ··· G2(X ) G2(X ) G2(X ) G1(X ) Diophantine geometry: remark on non-abelian reciprocity

Put ∞ X (AF )∞ = ∩n=1X (AF )n.

Theorem (Non-abelian reciprocity)

X (F ) ⊂ X (AF )∞. IV. Computing Rational Points Computing rational points

[Dan-Cohen, Wewers] 1 For X = P \{0, 1, ∞},

X (Z[1/2]) = {2, −1, 1/2} ⊂ {D2(z) = 0} ∩ {D4(z) = 0},

where D2(z) = `2(z) + (1/2) log(z) log(1 − z), 3 D4(z) = ζ(3)`4(z) + (8/7)[log 2/24 + `4(1/2)/ log 2] log(z)`3(z) 3 3 +[(4/21)(log 2/24 + `4(1/2)/ log 2) + ζ(3)/24] log (z) log(1 − z), and ∞ X zn ` (z) = . k nk n=1 Numerically, the inclusion appears to be an equality. Computing rational points Some qualitative results:

(Coates and K.)

axn + by n = c for n ≥ 4 has only finitely many rational points.

Standard structural conjectures on mixed motives (generalised BSD) ⇒ There exist (many) non-zero α as above ⇒ Faltings’s theorem over Q. Recently, Lawrence and Venkatesh have succeeded in giving an unconditional new proof of Faltings’s theorem using the same diagram with U replaced by a reductive completion of π1 with respect to a semi-simple representation - ρ : π1 Aut(V ). Computing rational points

A recent result on modular curves by Balakrishnan, Dogra, Mueller, Tuitmann, Vonk (arXiv 1711.05846, ‘Explicit Chabauty-Kim theory for the split modular curve of level 13’):

+ + Xs (N) = X (N)/Cs (N), where X (N) the the compactification of the moduli space of pairs

2 (E, φ : E[N] ' (Z/N) ),

+ and Cs (N) ⊂ GL2(Z/N) is the normaliser of a split Cartan subgroup. + Bilu-Parent-Rebolledo had shown that Xs (p)(Q) consists entirely of cusps and CM points for all primes p > 7, p 6= 13. They called p = 13 the ‘cursed level’. Computing rational points

Theorem (BDMTV) The modular curve + Xs (13) has exactly 7 rational points, consisting of the cusp and 6 CM points.

This concludes an important chapter of a conjecture of Serre: There is an absolute constant A such that

G - Aut(E[p])

is surjective for all non-CM elliptic curves E/Q and primes p > A. Computing rational points [Burcu Baran]

y 4 + 5x4 − 6x2y 2 + 6x3z + 26x2yz + 10xy 2z − 10y 3z

−32x2z2 − 40xyz2 + 24y 2z2 + 32xz3 − 16yz3 = 0

Figure: The cursed curve

{(1:1:1), (1:1:2), (0:0:1), (-3:3:2), (1:1:0), (0,2:1), (-1:1:0) } V. Some speculations on rational points and critical points Some speculations on rational points and critical points

Would like to think of

1 - Y 1 H (G, U(X , b)) H (Gv , U(X , b)) v as being like S(M, G) ⊂ A(M, G).

In particular, functions cutting out the image of localisation should be thought of as ‘classical equations of motion’ for gauge fields. Some speculations on rational points and critical points

When X is smooth and projective, X (Q) = X (Z), and we are actually interested in

1 Y 1 Y 1 Im(H (GS , U)) ∩ Hf (Gv , U) ⊂ H (Gv , U), v∈S v∈S where 1 1 Hf (Gv , U) ⊂ H (Gv , U) is a subvariety defined by some integral or Hodge-theoretic conditions. In order to apply symplectic techniques, replace U by

∗ ∗ T (1)U := (LieU) (1) o U. Some speculations on rational points and critical points Then Y 1 ∗ H (Gv , T (1)U) v∈S is a symplectic variety and

1 ∗ Y 1 ∗ Im(H (GS , T (1)U)), Hf (Gv , T (1)U) v∈S

are Lagrangian subvarieties. Thus, the (derived) intersection

1 ∗ Y 1 ∗ DS (X ) := Im(H (GS , T (1)U)) ∩ Hf (Gv , T (1)U) v∈S

has a [−1]-shifted symplectic structure. Zariski-locally the critical set of a function. [Ben-Basset, Brav, Bussi, Joyce] Some speculations on rational points and critical points

- −1 ⊂ - Y X (Z) jS (DS (X )) X (Qv ) v∈S g j jS jS ? ? loc ? 1 ∗ S - ⊂ - Y 1 ∗ Hf (GS , T (1)U) DS (X ) H (Gv , T (1)Un) v∈S From this view, the global points can be obtained by pulling back ‘Euler-Lagrange equations’ via a period map. Some speculations on rational points and critical points

Figure: Pierre de Fermat (1607-1665)