P-Adic Methods for Rational Points on Curves MA841 at BU Fall 2019

p-adic methods for rational points on curves MA841 at BU Fall 2019

Jennifer Balakrishnan December 21, 2019

These are notes for Jennifer Balakrishnan’s course MA841 at BU, Fall 2019. The course webpage is http://math.bu.edu/people/jbala/841.html.

1 Rational points on curves

Lecture 1 5/9/2019 Main Question: How do we determine X Q for X smooth projective of genus 2? What computational tools are involved?( ) Topics:≥ 1. Chabauty-Coleman method 2. Coleman integration (p-adic integration) 3. p-adic heights 4. quadratic Chabauty Evaluation (if you need a grade), TeX 3-4 classes worth of lecture notes. Detailed list of topics: • Chabauty-Coleman • Explicit Coleman integration • p-adic cohomology, based point counting (Kedlaya + Tuitman) • Iterated Coleman integration • Chabauty-Coleman in practice + other tools • Étale descent • Covering collections • Elliptic curve Chabauty • p-adic heights on elliptic curves • p-adic heights on Jacobians on curves • Local heights • Quadratic Chabauty for integral points on aﬃne hyperelliptic curves • Kim’s nonabelian Chabauty program • Nekovář’s p-adic height • Quadratic Chabauty for Q-points on curves • Quadratic Chabauty in practice References for ﬁrst two weeks: • McCallum-Poonen • Stoll: Arithmetic of Hyperelliptic Curves • Kedlaya: p-adic cohomology from theory to practice (notes from 2007 AWS) • Besser: Heidelberg lectures on Coleman integration For computations • Sage • MAGMA

2 The Chabauty-Coleman method

2.1 A question about triangles Does there exist a rational right triangle and a rational isosceles triangle with with same perimeter and same area? (rational means all side lengths are rational) Suppose there does exist such a pair, then introducing parameters, k, t for the right triangle, and l, u for the isosceles we can rescale to

k, t, u Q ∈ 0 < t, u < 1, k > 0 an equate areas and perimeters. Areas:

1 1 2kt k 1 t2 4u 1 u2 2( )( )( − ) 2( )( − ) k2t2 1 t2 2u 1 u2 . ⇒ ( − ) ( − ) Perimeters:

k 1 t2 + k 1 + t2 + 2kt 1 + u2 + 1 + u2 + 4u ( − ) ( ) k + kt 1 + 2u + u2 1 + u 2 ⇒ ( ) so letting x 1 + u, after some algebra we have 1 < x < 2 in Q s.t.

2xk2 + 3x3 2x2 + 6x 4 k + x5 0 (− − − ) this is a quadratic in k, and the discriminant is a square in Q. so

X : y2 3x3 2x2 + 6x 4 2 4 2x x5 (− − − ) − ( ) x6 + 12x5 32x4 + 52x2 48x + 16 − − so this is a genus 2 hyperelliptic curve. We need the Q-points of this.

2 Facts:. Jac X has Mordell-Weil rank 1. The Chabauty-Coleman bound on the size of X( Q) for this curve gives #X Q 10. But we ﬁnd points ( ) ( ) ≤ 12 868 0 : 4 : 1 , , 0 : 4 : 1 , 1 : 1 : 1 , 1 : 1 : 1 , : : 1 , ( − ) ∞± ( ) ( − ) ( ) 11 −1331 12 868 : : 1 , 2 : 8 : 1 , 2 : 8 : 1 11 1331 ( − ) ( ) so this set is X Q . Back in the( original) problem we speciﬁed x < 1 < 2, so there is a unique such pair of triangles: Theorem 2.1 Hirakawa-Matsumura ’18. Up to similitude there exists a unique pair of a rational right triangle and a rational isosceles triangle that have the same perimeters and areas. The unique pair consists of a right triangle with sides

377, 135, 352 ( ) and the isosceles triangle with sides

366, 366, 132 . ( ) 2.2 Why care about X(Q) for X of genus 2? Curves of genus 0: have no Q-points or infinitely many, they satisfy a local to global principle so there exists an algorithm to determine the Q-points in finite time. Curves of genus 1: If we have 1 smooth rational point then we have an elliptic curve, Mordell’s theorem implies that E Q is a finitely generated abelian group, ( ) E Q Zr T ( )' ⊕ where the possible torsion parts T have been determined by Mazur’s theorem. To understand T, and the distribution of T there is work of Harron and Snowden, this often comes down to understanding rational points on X1 N . Upshot: to understand E Q we want to understand r: ( ) Q1: is there an algorithm( to) compute r? Q2: what values of r can occur? Q3: what is the distribution of r? A1: n-descent, the obstacle is III, proving finiteness, it is conjectured that r ords1 L E, s (BSD). A2: record( due) to Noam Elkies an example of E with r 28. A3: minimalist conjecture: 50% of all curves have rank 0,≥ 50% rank 1. Theorem 2.2 Bhargava-Shankar. The average rank is < 1. Baur Bektemirov, Barry Mazur, William Stein, and Mark Watkins, Average ranks of elliptic curves: tension between data and conjecture, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 2, 233–254. MR 2009e:11107 gave average rank graphs, which kept increasing. Sarnak said there would “obviously be a turn around”. Jennifer S. Balakrishnan, Wei Ho, Nathan Kaplan, Simon Spicer, William Stein, and James Weigandt, Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks, LMS J. Comput. Math. 19 (2016), supp. A, pp. 351-370. MR 3540965

3 2.3 Coleman’s bound

Lecture 2 10/9/2019 Goal today: prove Coleman’s refinement of Chabauty’s theorem. Theorem 2.3 Coleman 1985. Let X Q be a curve of genus g 2. Suppose the Mordell-Weil rank of J Q is less than /g. Then if p > 2g is a good≥ prime for X we have ( ) #X Q #XF Fp + 2g 2. ( ) ≤ p ( ) − Definition 2.4 Differentials. Let X be a curve over a field k. The space of Ω1 differentials on X over k is a 1-dimensional k X -vector space X k . There is a nontrivial k-linear derivation ( ) ( )

d: k X Ω1 k ( ) → X( ) i.e. d is k-linear and satisﬁes the Leibniz rule

d f g g d f + f dg ( ) · for all f , g k X and there is some f k X s.t. d f , 0. A general∈ (differential) can be written∈ as( )ω f dg where g K X with Ω1 ∈ ( ) dg , 0. If we fix g this representation is unique. If ω, ω0 X k with ω0 , 0 then there’s a unique f K X s.t. ω f ω . We may write∈ ω ω( ) f . ♦ ∈ ( ) 0 / 0 Definition 2.5 Differentials of the first second and third kinds. Let 0 , 1 ω Ω k and P X k . Let t k X be a uniformizer at P. Then vP ω ∈ X( ) ∈ ( ) ∈ ( ) ( ) vP ω dt is the valuation of ω at P. This valuation is nonzero for only finitely ( / ) many points P X k . The divisor ∈ ( ) Õ div ω vP ω P DivX k ( ) ( ) ∈ ( ) P X k ∈ ( ) is the divisor of ω. If vP ω then ω is regular at P and ω is said to be regular if it is regular ( ) ≥ at all points P X K . Also called∈ differentials( ) of the first kind. A differential of the second kind has residue zero at all points P X K . A differential of the third kind has at most a simple pole at all∈ points( ) P X K (and integer residues there in some references). ♦ ∈ ( ) Since the quotient of any two non-zero differentials is a function

ω1 f1 dg

ω2 f2 dg so ω f1 1 . ω2 f2 The diﬀerence of any two divisors of diﬀerentials is a principal divisor. ω f1 div 1 div ω2 f2

div ω1 div ω2. − So the divisors of diﬀerentials form one linear equivalence class of divisors, the canonical class.

4 Recall. Let X k be a curve and D DivX k . The Riemann-Roch space of D is the k-vector space/ ∈ ( ) L D φ k X × : div φ + D 0 0 ( ) ∈ ( ) ≥ ∪ { } where we write D D if vP D vP D for all P. ≥ 0 ( ) ≥ ( 0) Theorem 2.6 Riemann-Roch. Let X k be a curve of genus g then there is a divisor / W DivX k s.t. for every D DivX k w we have dimk L D is ﬁnite and ∈ ( ) ∈ ( ) ( )

dimk L D deg D g + 1 diml L W D . ( ) − ( − )

In particular, dimk L W g, deg W 2g 2. The canonical class is( exactly) the class of the divisor− W in Riemann-Roch. The k-vector space of regular diﬀerentials has dim L W g, and is denoted as H0 X, Ω1 . ( ) ( X) Example 2.7 Let X : y2 f x be a hyperelliptic curve of genus g over k. Then H0 X, Ω1 has basis ( ) ( X) dx x g 1 dx ,..., − 2y 2y so every regular diﬀerential can be written uniquely as

p x dx ( ) 2y with a polynomial p of degree g 1. ≤ − We want to integrate diﬀerentials in some p-adic sense, Q: What does a p-adic line integral look like?

Theorem 2.8 Let X Qp be a curve with good reduction then there is a p-adic integral / ¹ Q ω Qp P ∈

0 Ω1 defined for each pair of points P, Q X Qp and regular differential ω H X, X Qp that satisfies the following properties:∈ ( ) ∈ ( ( ))

1. The integral is Qp linear in ω ¯ 2. If P, Q both reduce to the same point P XFp Fp then the integral can be evaluated by writing ∈ ( ) ω ω t dt ( ) with t a uniformizer at P reducing to a uniformizer at P¯ and ω a power series. Then integrating formally obtaining a power series l s.t.

dl t w t dt ( ) ( ) and l 0 0 and ﬁnally evaluating ( ) l t Q ( ( )) ∫ P which converges. This implies that P ω 0. 3. ¹ Q ¹ Q0 ¹ Q0 ¹ Q ω + ω ω + ω P P0 P P0

5 so it makes sense to deﬁne: ¹ ω D for n Õ 0 Q j Pj Div Q − ∈ X( p) j1 as n ¹ Õ ¹ Q j ω ω D j1 Pj

∫ 4. If D is principal then D ω 0.

5. The integral commutes with the action of Gal Qp Qp . ( / ) 0 1 6. Fix P0 X Q . If 0 , ω H X, Ω , then the set of points P X Q ∈ ( p) ∈ ( X) ∈ ( p) reducing to a fixed point P0 XF Fp . and s.t. ∈ p ( ) ¹ P ω 0 P0 is finite. Remark 2.9 The statement that the curve has good reduction is not necessary but simplifies the statement of 2. Remark 2.10 This integral is the Coleman integral [31], other works on p-adic integration include Berkovich [12]. Also there is work of Zarhin, Colmez, Vologodsky, Besser, ... Remark 2.11 Theory of Coleman integration of forms the second or third kind developed by Coleman-de Shalit [33]. (additivity in endpoints, linearity, change of variables, FTC). Corollary 2.12 Given the hypotheses of the previous theorem

P0 X Qp ∈ ( ) and J the Jacobian of X let ι : X J → be the embedding P P P0 7→ [ − ] there is a map 0 1 J Qp H X, Ω Qp ( ) × ( X) → P, ω P, ω ( ) 7→ h i that is additive in P and Qp linear in ω which is given by ¹ D , ω ω h[ ] i D in particular for P X Qp ∈ ( ) we have ¹ P ι P , ω ω. h ( ) i P0

6 Remark 2.13 If P J Qp has ﬁnite order, then ∈ ( ) P, ω 0, ω H0 X, Ω1 h i ∀ ∈ ( X) to see this, if nP 0 then 1 1 P, ω nP, ω 0 0. h i n h i n One can show that torsion points are the only points with this property. On the other hand, if ω has the property that P, ω 0 for all P J Qp then ω 0. h i ∈ ( ) Corollary 2.14 Let X Q be a curve of genus g with Mordell-Weil rank less than g. Then #X Q is ﬁnite. Note/ we don’t need g 2, in g 1 this applies to rank 0. ( ) ≥ Proof. Pick a prime of good reduction for X let

V ω H0 X, Ω1 : P J Q : P, ω 0 { ∈ ( X) ∀ ∈ ( ) h i } by additivity in the first argument this condition is equivalent to requiring r that Pj , ω 0 for a basis Pj of the free part of J Q so it leads to at most { }j1 ( ) r linear constraints, so dim V g r > 0. So there is some 0 , ω V pick ≥ − ∈ P0 X Q , if X Q we are done. To define ι : X , J. Since ι P J Q for ∈ ( ) ( ) ∅ ∫ P → ( ) ∈ ( ) all P X Q so it follows that ω 0 for all P X Q . By the theorem the ∈ ( ) P0 ∈ ( ) number of such P is finite in each residue disk of X Q . Since the number of ( ) residue classes is #X Fp which is finite. The total number of points in X Q is finite also. ( ) ( ) To get an actual bound we have to bound the number of zeroes of

¹ z ω P0 as a p-adic power series. We can think of X Qp set theoretically as a finite union of residue disks. Within each residue disk( ) ¹ z ω P0 has finitely many p-adic zeroes. Lecture 3 10/9/2019 We want to give a more refined version of this result which uses results about zeroes of p-adic power series. Theorem 2.15 Let Õ∞ n 0 , l t an t Qp t ( ) ∈ [[ ]] n0 such that an 0 as n in the p-adic topology. Let → → ∞

v0 min vp an : n 0 { ( ) ≥ } and N max n 0, vp an v0 { ≥ ( ) } then there is a constant c Q× ∈ p

7 a monic polynomial q Zp t ∈ [ ] of degree N, a power series

Õ∞ n h t bn t 1 + ptZp t ( ) ∈ [[ ]] n0 with bn 0 as n → → ∞ and l t cq t h t . 1 ( ) ( ) ( ) Proof. After rescaling by a0− can assume v0 0 and aN 1 so this in particular l t Zp t the condition an 0 means that the image lm t of l t in Z( )pm ∈Z t]][[ is]] actually a polynomial→ for all m 1. ( ) ( ) / [[ ≥ m The idea is to construct inductively constants cm Z p ×, monic polyno- m ∈ ( / ) m mials qm t Z p t of degree N and polynomials hm t Z p t with ( ) ∈ ( / )[ ] ( ) ∈ ( / )[ ] hm 1 mod pt satisfying ≡ ( )

lm t cm qm t hm t ( ) ( ) ( ) and such that cm+1, qm+1, hm+1 ( ) reduces mod pm to ( ) cm , qm , hm . ( ) Then there is a unique c, q, h as above such that

c, q, h ( ) reduces mod pm to ( ) cm , qm , hm ( ) for all m. To start the induction set c1 1

q1 t l1 t ( ) ( )

h1 t 1 ( ) this is possible since l1 t is monic of degree N. ( ) Assume we’ve constructed cm , qm , hm, let

c˜m+1, q˜m+1, h˜ m+1

m+1 be arbitrary lifts of cm , qm , hm to objects over Z p with /

q˜m+1 monic of degree N

h˜ m+1 t 1 mod pt ( ) ≡ ( ) then m lm+1 t c˜m+1q˜m+1h˜ m+1 p d t ( ) − ( ) with d t Z p t ( ) ∈ ( / )[ ] then we must have m cm+1 c˜m+1 + p γ

8 m qm+1 q˜m+1 + p k t ( ) m hm+1 h˜ m+1 + p η t ( ) with γ Z p, k Z p t of degree < N. and η Z p t with η 0 0. For ∈γ /Z pZ∈, k ( t/ )[ Z] pZ t of degree less than∈ ( /N)[, and] η t ( )Z pZ t ∈ / ( ) ∈ ( / )[ ] ( ) ∈ ( / )[ ] with η 0 0. So the relation lm+1 t cm+1qm+1 t hm+1 t is equivalent ( ) ( ) ( ) ( ) to d t γ + η t l1 t + k t , and γ, k t , and η t are uniquely deter- ( ) ( ( )) ( ) ( ) ( ) ( ) mined through division by l1 t with remainder d t , and this determines ( ) ( ) cm+1, qm+1 t , and hm+1 t . ( ) ( ) Now we apply this to study zeroes of p-adic power series coming from Coleman integrals.

Lemma 2.16 Let l t Qp t with formal derivative w t Zp t . Such that the ( ) ∈ [[ ]] ν + ( ) ∈ [[ ]] image w¯ t Fp t has the form ut with u F×p . Then l converges on pZp. If p > ν (+)2 ∈then[[ ]] ··· ∈ # τ pZp : l t 0 ν + 1. { ∈ ( ) } ≤ Proof. Let w t w0 + w1t + ( ) ··· l t l0 + l1t + ( ) ··· then wn 1 ln+1 Zp n + 1 ∈ n + 1 since vp n + 1 O log n the assumption that ( ) ( ) wn Zp ∈ implies that vp ln vp wn n + 1 c log n for some constant c. If τ pZp ( ) ( /( )) ≥ − ∈ so vp τ 1, then ( ) ≥ n vp ln τ n c log n ( ) ≥ − → ∞ 2 2 as n , hence l τ converges. Now consider l pt l0 + pl1t + p l2t + . The claim→ ∞ is that in( the) notation of the previous theorem( ) we have N ν +···1. ≤ ν+1 vp p l +1 ν + 1 + vp l +1 ( ν ) ( ν ) w ν + 1 + v ν p ν + 1 ν + 1 + vp w vp ν + 1 ν + 1 ( ν) − ( ) ≤ ν as by assumption w¯ t Fp t has the form ut + so that vp wν 0. For n > ν we have( ) ∈ [[ ]] ··· ( )

n+1 vp p ln+1 n + 1 + vp ln+1 ( ) ( ) n + 1 + vp wn vp n + 1 ( ) − ( ) n + 1 vp n + 1 ≥ − ( ) since vp wn 0 ( ) ≥ for n > ν. So it suﬃces to show that

n vp n + 1 > ν − ( ) e This is clear for vp n + 1 0. Otherwise suppose e vp n + 1 then p n + 1 . ( ) ( ) |( ) So n+1 p3 > ν+e+1, where the second inequality can be shown by induction. + For e ≥1 this is our hypothesis that p > ν + 2, then use pe 1 pe + 1. The previous corollary now gives the result. ≥

9 Theorem 2.17 Coleman ’85. Let X Q be a curve of genus g, with Mordell-Weil rank of J less than g. Then /

#X Q #X Fp + 2g 2. ( ) ≤ ( ) − [32]. Proof. We assume P0 X Q , now arguing as in the proof of today’s first corollary there is a non-zero∈ ( differential) ω H0 X, Ω1 such that X Qp ∈ ( / ) ¹ P ω 0 P0 for all P X Q . Now consider a point Q¯ X Fp and lift it to Q in X Qp , ∈ ( ) ∈ ( ) ( ) we can pick a uniformizer t Qp X s.t at Q t reduces to a uniformizer ∈ ( )× t¯ Fp X at Q¯ . We rescale ω s.t. its reduction ω¯ is defined an non-zero. Then ∈ ( )× ω¯ H0 X, Ω1 . Recall that div ω¯ is effective and has degree 2g 2. Let ∈ ( X Fp ) ( ) − ¯ / ¯ ¯ ν Q denote the valuation at Q of ω¯ . ν Q νQ¯ ω¯ . We write ω t w t dt with( ) ( ) ( ) ( ) ( ) w t Zp t ( ) ∈ [[ ]] the coefficients are in Zp since ω¯ is defined. Then

ω¯ w¯ t dt¯ ( ) ν Q¯ w¯ t¯ t¯ u0 + u1t¯ + . ( ) ( )( ···) ¹ P ω l t P P0 ( ( )) for P X Qp such that p¯ q¯ and apply previous lemma. Now summing over residue∈ ( disks) we get

¹ P #X Q # P X Qp : ω 0 ( ) ≤ ∈ ( ) P0 Õ ν Q¯ + 1 ≤ ( ) Q¯ X¯ Fp ∈ ( ) Õ Õ ν Q¯ + 1 ( ) Q¯ X¯ Fp Q¯ X¯ Fp ∈ ( ) ∈ ( )

deg div ω + #X¯ Fp ≤ ( ) ( ) 2g 2 + #X¯ Fp . − ( ) Remark 2.18 Stoll (06) showed that we can choose the best ω in each residue disk, can improve the bound, r < g and p > 2r + 2 is a good prime then

#X Q #X Fp + 2r ( ) ≤ ( ) can also weaken the assumption that

p > 2r + 2.

10 If p > 2 then 2r #X Q #X Fp + 2r + . ( ) ≤ ( ) p 1 − [87]. Katz-Rabinoﬀ-Zuerieck-Brown (12) extend Stoll’s result tot the case of bad reduction, if p > 2g and a proper regular model for X over Zp then X

#X Q # sm Fp + 2r ( ) ≤ X ( ) where Fp is the set of smooth points in the special fiber of minimal proper X( ) regular model of X over Zp. [58]. Lecture 4 17/9/2019 A few results applying Chabauty-Coleman to prove uniform bounds: Theorem 2.19 Stoll ’13. If X Q is hyperelliptic of genus g with Jacobian of Mordell-Weil rank r g 3, then/ ≤ − #X Q 8r g + 33 g 1 + 1 ( ) ≤ ( − ) [88]. Theorem 2.20 Katz-Rabinoff-Zuerieck-Brown ’19. If X Q curves of genus g with r g 3. / ≤ − #X Q 84g2 98g + 28. ( ) ≤ − Ref KRZB and expository paper. Suppose X Q is genus 3, hyperelliptic curve of rank 0, Stoll’s bound gives #X Q 67. Is/ there a curve meeting this bound? Or even #X Q 10? (In) the ≤ LMFDB we find in g 2, r 0 the record seems to( be) #X Q 8. For http://lmfdb.org/Genus2Curve/Q/1116.a.214272.1 we have #X Q ( )8. ( ) J Q Z 39 ( )' / with simple Jacobian (first found by Elkies). It is possible to use constructions of Howe, Leprevost, Poonen, Elkies, others to construct Jacobians with even larger torsion (and possibly curves of low rank with many rational points? Earlier we talked about computing annihilating differentials in the Chabauty-Coleman method. Here is a concrete example, to motivate a discussion of explicit Coleman integration. Example 2.21 Consider

X : y2 x5 4x3 + 3x + 1 − http://lmfdb.org/Genus2Curve/Q/3920.b.62720.1.

J Q Z Z 2. ( )' ⊕ / N 3920 24 5 72. · · And X Q , 0, 1 , 1, 1 , 1, 1 ( ) ⊇ {∞ ( ± ) ( ± ) (− ± )} #XF F11 13 11 ( ) #XF F13 14 13 ( ) so the Chabauty-Coleman bound by itself does not prove that we found all the Q-points already. The point 1, 1 [( ) − ∞]

11 is of infinite order in J Q . We use it to construct an annihilating differential. Let p 11. Then a basis( of) H0 X, Ω1 is given by ( ) xi dx ωi 2y i0,1 so the annihilating differential η is some Qp-linear combination of ω0, ω1. We use the values of 1,1 1,1 ¹ ( ) ¹ ( ) ω0, ω1 ∞ ∞ to compute η. We find

¹ 1,1 ( ) 2 3 7 8 9 ω0 8 11 + 7 11 + 7 11 + 4 11 + 9 11 + O 11 α · · · · · ( ) ∞ ¹ 1,1 ( ) 2 3 4 5 6 7 8 9 ω1 3 11+2 11 +4 11 +3 11 +6 11 +6 11 +8 11 +3 11 +O 11 β. · · · · · · · · ( ) ∞ Then 1,1 ¹ ( ) βω0 αω1 0 − ∞ so take η βω0 αω1. − To use η to compute X Q or more precisely, a ﬁnite subset of X Qp containing X Q we need to compute( ) the collection of indeﬁnite Coleman( integrals) ( ) ¹ Pt η ∞ where Pt ranges over all residue disks. And solve for z X Qp such that ∈ ( ) ¹ z η 0. ∞ So to compute α, β and the functions we needed Coleman integrals between points not in the same residue disk. Goal: show how to compute these p-adic integrals. Let X Q be a curve. Let Xan be the associated rigid analytic space. (Let / X be a smooth curve over Zp s.t.

Qp X Qp, X ⊗ ' ⊗ an then X denotes the rigid analytic space over Qp which is the generic fibre of .) X Definition 2.22 A wide open subspace of Xan is the complement in Xan of the union of a finite collection of disjoint closed disks of radius λi < 1. ♦ Example 2.23 Let 5 2 Ö X : y x αi ( − ) i1 take out closed disks of radius λi for each Pi αi , 0 and . ( ) ∞

12 Theorem 2.24 Coleman, Coleman-de Shalit. Let η, ξ be 1-forms on a wide open an V of X and P, Q, R V Qp . Let a, b Qp. The deﬁnite Coleman integral has the following properties∈ ( ) ∈ 1. Linearity ¹ Q ¹ Q ¹ Q αη + bξ α η + b ξ P P P 2. Additivity in endpoints

¹ Q ¹ R ¹ Q η η + η P P R

3. Change of variables, if V0 X0 is a wide open subspace of a rigid analytic space X and φ : V V is a rigid⊆ analytic map then 0 → 0 ¹ Q ¹ φQ φ∗η η. P φP

4. Fundamental theorem of calculus

¹ Q d f f Q f P P ( ) − ( ) for f a rigid analytic function on V. Goal: want to integrate ¹ Q ω P for a 1-form of the second kind, P, Q V Qp . Idea ∈ ( ) 1. Take φ to be a lift of frobenius from the special ﬁbre.

2. Write a basis ωi of 1-forms of the second kind. { } ∫ Q 3. Compute φ∗ωi and use properties of Coleman integral to relate P φ∗ωi ∫ Q to P ωi and other terms we can compute. [59], [37], also Stephanie Chan MMath thesis (is this online?) Setup p , 2 prime n X Fq , q p / hyperelliptic of genus g with aﬃne equation

y2 P x ( ) with P x monic degree 2g + 1, with no repeated roots. ( ) X : X r , y 0 . {∞ }

W ring of Witt vectors over Fq, (the unique unramiﬁed extension of Zp with residue ﬁeld Fq. Choose a lift P˜ of P, to a monic polynomial of degree 2g + 1. Over W 1 2 this gives a lift Xe of X. Let A W x, y, y− y P˜ x Let A† be the weak [ ]/( − ( )) completion of A, explicitly let vp denote the p-adic valuation on W extend it

13 Í i to polynomials. If g x ai x , deﬁne vp g min vp ai . The elements of ( ) ( ) { ( )} A† are series Õ∞ 2n Sn x + Tn x y y ( ( ) ( ) ) −∞ where Sn and Tn are polynomials of degree at most 2g s.t. limits are positive. Lecture 5 19/9/2019 References for Rigid Geometry: [38] [21]. X Fq a hyperelliptic curve of genus g, with odd degree model and monic, no repeated/ roots. X : X r , y 0 {∞ } Xe is a lift of X to Zq the ring of Witt vectors over Fq.

y2 P˜ x . ( )

1 2 A : Zq x, y, y− y˜ P˜ x [ ]/( − ( )) A† the weak completion of A, this is ( ) Õ∞ n n sn x y : sn Zq x , deg sn 2g, ordp sn > c for some c > 0 . ( ) ∈ [ ] ≤ ( ) −∞ Monsky-Washnitzer cohomology is a p-adic cohomology theory for smooth affine varieties, over fields of characteristic p. Theorem 2.25 Special case, Berthelot, (1974, 1997). The algebraic de Rham cohomology of Xe coincides with the Monsky-Washnitzer cohomology of X. Monsky-Washnitzer cohomology is finite dimensional and is equipped with an action of Frobenius. So the theorem tells us that we can compute via a description of de Rham cohomology. Proposition 2.26 The de Rham cohomology of A splits into eigenspaces under the hyperelliptic involution: a positive eigenspace generated by

xi dx , i 0,..., 2g y2 and a negative eigenspace generated by

xi dx , i 0,..., 2g 1. y −

We lift p-power frobenius to an endomorphism of A† by deﬁning it as the canonical Witt vector frobenius on Zq.

p p a0, a1,... a , a ,... ( ) 7→ ( 0 1 ) p 2 for ai Fq, then extend it to Zq x by mapping x x . Then since y P˜ x , we have∈ [ ] 7→ ( ) yσ 2 y2 σ P˜ x σ ( ) ( ) ( ( )) p y2 y2p P˜ x σ P˜ x σ ( ) ( ( )) P˜ x P˜ x p ( ) ( ) 14 1 P˜ x σ 2 y yp ( ) 7→ P˜ x p ( ) 1 P˜ x σ P˜ x p 2 yp 1 + ( ) − ( ) P˜ x p ( ) i Õ∞ 1 2 P˜ x σ P˜ x p yp / ( ) − ( ) i y2p i0 Remark 2.27 Here is why we removed the Weierstrass points (we don’t want to divide by y and have things diverge). Its possible to compute a Frobenius lift without deleting Weierstrass points, but then we need to solve for images of x, y using a 2 variable newton iteration. Further extend to diﬀerentials by sending

p p 1 dx d x px − dx 7→ ( ) q deﬁne F σlogp this is a lift of q-power frobenius. Key reduction∗ lemmas, (to prove prop on eigenspaces).

Lemma 2.28 If A x P˜ x B x + P˜ C x then ( ) ( ) ( ) 0 ( ) A x dx 2C x dx ( ) B x + 0( ) y2 ( ) s 2 ys 2 − − as elements of H1 X . MW ( ) We also have i j i 1 j i j 1 d x y ix − dx y + x j y − dy, ( ) use y2 P˜ x which implies ( ) 2 d y P˜ x 2y dy P˜ 0 x dx ( ( )) ( ) P˜ x dx dy 0( ) ⇒ 2y giving ˜ i j i 1 j i j 1 P0 x dx d x y ix − y dx + x j y − ( ) . ( ) 2y A special case of this: let Q x xm 2g then ( ) −

dx 1 d Q x y Q x P˜ 0 x + 2Q0 x P˜ x 0 in H X . ( ( ) ) ( ( ) ( ) ( ) ( )) y ≡ MW ( )

Goal for Coleman integration: We compute

σ xi dx y reduce using the above reductions to get a cohomologous diﬀerential that’s a linear combination of the basis xi dx . y i0,...,2g 1 − What does this look like?

15 1. The reduction process is essentially subtracting the right linear combi- nations of d xi y j and using y2 P˜ x . ( ) ( ) 2. Precision is lost when we divide by p in the reduction algorithm, so we’ll need to measure the loss of precision at each step to know how many provably correct p-adic digits we have.

We compute

i σ pi+p 1 L σ p i x dx px − dx Õ 1 2 P˜ x P˜ x − / ( ( ) − ( ) ) y yp i y2pi i0 we need to know how large L must be to get provably correct expansions. If the result of this is N Õ Aj x dx ( ) y2j+1 j M − using the reduction formulas to eliminate the j N term then the N 1 term until no terms with j > 0 remain. Do likewise with the j M, M−+ 1,... terms. − − At the end of the reduction algorithm we are left with

σ 2g 1 xi dx Õ− x j dx d f + M y i ji y j0 the d fi is whats eliminated by the reduction algorithm, we sum the d’s at each step. Do this for each i 0,..., 2g 1. Then M Mij gives the matrix of Frobenius. Its characteristic polynomial− gives you the( numerator) of the zeta function of X. Lemmas on precision:

Lemma 2.29 Let A x Zq x be a polynomial of degree 2g. For some m > 0 consider the reduction( ) of ∈ [ ] ≤ A x dx ω ( ) y2m+1 by Reduction 1 A x dx B x dx ω ( ) ( ) + d f y2m+1 y with B x Qq x with deg B x 2g 1. We have ( ) ∈ [ ] ( ) ≤ − j k logp 2m 1 p ( − ) B x Zq x . ( ) ∈ [ ] m 1 Õ− Fk x f ( ) , deg Fk 2g. y2k+1 ≤ k 1 − Lemma 2.30 Let A x Zq x be a polynomial of degree 2g. For some m > 0 consider the reduction( ) of ∈ [ ] ≤ A x y2m dx ω ( ) y

16 by Reduction 2 A x y2m dx B x dx ω ( ) ( ) + d f y y with B x Qq x with deg B x 2g 1, ( ) ∈ [ ] ( ) ≤ − m 1 2m+1 Õ− 2k+1 f c y + Fk x y ( ) k0 j k + + logp 2g 1 2m 1 c Qq, deg Fk 2g, p ( )( ) B x Zq x . ∈ ≤ ( ) ∈ [ ] Proposition 2.31 To get N correct digits in the expansion after reduction we need to start with precision nj k j ko j k N1 N + max log 2M 3 , log 2g + 1 + 1 + log 2g 1 , p( − ) p( ) p( − ) where M is the smallest integer s.t. nj k j k o M max log 2M + 1 , log 2g + 1 . − p( ) p( ) Example 2.32 Let y2 P˜ x x3 + x + 1 Q ( ) / let p 5 (or take this over F5 and lift to Z5). Let N 2 be the number of correct dx x dx 5-adic digits, so M 3, so N1 3, use the diﬀerentials y , y

σ dx 25x + 50 75x2 + 100x + 25 50x2 + 50x + 100 75x + 50 50x2 + 50x 70x2 + 70x + 25 5x + + + + + + dx mod 53 y y15 y13 y11 y9 y7 y5 y3 ( ) similar for σ x dx 100x2 + 100x + 75 + dx mod 53 y y15 ··· ( ) let Fk be the polynomial in the term

Fk dx y2k+1 starting from k 7, set sk x Fk x , compute a series of polynomials induc- ( ) ( ) tively for k 1, k 2,..., 0. Given Sk+ ﬁnd polynomials Ak+ , Bk+ s.t. − − 1 1 1

Ak+1P˜ + Bk+1P˜ 0 sk+1

2B0k+1 x then set sk x Fk x + Ak+ x + ( ) ( ) ( ) 1( ) 2k+1 σ dx dx 15x mod 52 y y ( )

σ x dx dx 22x + 18 mod 52 y ( ) y ( ) 0 18 M mod 52 . 15 22 ( )

17 I missed a day heere! Lecture 7 1/10/2019 Set-up for Tuitman’s algorithm: X: Smooth projective curve Fq, birational to

dx 1 + dx 1 + + Q x, y y − Qdx 1 y − Q0 0 ( ) − ··· is irreducible, where Qi Fq x for i 0,..., dx 1. This is part II of Tuitman [93] as we have a not necessarily∈ [ ] smooth model.− Tuitman’s idea: 1. Use the (low degree) map x : X P1. → 2. remove the ramification locus of x, call this r x 0, c.f. Kedlaya’s algorithm where we deleted the Weierstrass points.( ) 3. Choose a lift of Frobenius that sends x to xp. Compute y via Hensel lifting. 4. Compute the action of Frobenius on differentials, reduce in cohomology, using Lauders fibration algorithm.

1 Then for a basis ωi of H X Tuitman’s algorithm computes: { } dR( ) Õ φ∗ωi d fi + Mji ωj j and as before this can be used to give an algorithm for Coleman integration [8]. Let S Zq x, 1 r , R Zq x, 1 r, y Q [ / ] [ / ]/ where Q is a lift of Q to #Zq that is monic with same monomials in support. This is possibly an issue for g 5, see Tuitmans paper. See also [29] for heuristics, possible solutions in higher≥ genus. Let V Spec S, U Spec R. The ring of overconvergent functions on U is

R† Zq x, 1 r, y † Q h / i /

Goal: compute a lift of Frobenius on R† in an explicit and fast way. Let Fq x, y denote the ﬁeld of fractions of R Z Fq and Qq x, y the ﬁeld ( ) ⊗ q ( ) of fractions of R Z Qq. ⊗ q Assumption 0 : The polynomial r x is separable over Fq, recall that ( ) R dx R dy Ω1 † ⊕ † R† dQ

1 and if we write d: R† Ω we have → R† 1 H U coker d Qq. ri g( ) ( ) ⊗ [10] We will compute H X H1 U ??????????????? ( ) ⊆ ( ) Proposition 2.33 R is a free module of rank dx over S Zq x, 1 r . A basis is † † h / i†

dx 1 1, y,..., y − . [ ]

18 p Theorem 2.34 There is a lift of Frobeius φ on R† that sends x to x . idea compute φ y by Hensel lifting, using equation ( ) Qσ xp , φ y ( ( )) ∂Q note that this is possible since we’ve removed zeroes of ∂y from the curve by deleting r x . ( ) 2 dx 1 After precomputing φ y , φ y , . . . , φ y − and φ 1 r it is easy to com- 1 ( ) ( ) ( ) ( / ) pute φ on R†, Ω . R†

Proposition 2.35 Let G Mdx dx Zq x, 1 r denote the matrix s.t. ∈ × ( [ / ])

dx 1 j Õ− i d y Gi+1 j+1 y dx ( ) , i0 for j 0,..., dx 1. then G M r where M Matdx dx Zq x . − / ∈ × ( [ ]) 0 Assumption 1: Let W GLdx Zq x, 1 r , W∞ GLdx Zq x, 1 x, 1 r be matrices such that if we denote∈ ( [ / ]) ∈ ( [ / / ])

dx 1 − 0 Õ 0 i b j Wi+1,j+1 y i0

dx 1 Õ− i b∞j Wi∞+1,j+1 y i0

0 dx 1 for all 0 j dx 1. Then b − is an integral basis for Q x, y over Q x . ≤ ≤ − { j }j0 ( ) [ ] dx 1 b∞ − is an integral basis for Q x, y over Q 1 x . { j }j0 ( ) [ / ] Remark 2.36 Magma can compute these integral bases. Once we compute the action of Frobenius on 1-forms we need to reduce, Tuitman uses Lauder’s ﬁbration algorithm. 1. Reduce pole order of points not lying over . ∞ 2. Reduce pole order of points lying over . ∞ Let r denote dr dx for points not over . 0 / ∞ dx Proposition 2.37 For all l N and every w Qq x ⊕ there exist vectors u, v dx ∈ ∈ [ ] ∈ Qq x ⊕ such that [ ] deg v < deg r ( ) ( ) and Ídx 1 0 Ídx 1 0 ! Ídx 1 0 ! − wi b dx − vi b − ui b dx i 0 i d i 0 i + i 0 i l l l 1 r r r r − r

Proof. Since r is separable, r is invertible in Qq x r Check that there exists a 0 [ ]/ unique solution v to the dx dx linear system. × M w lI v mod r r0 − ≡ r0 ( ) over Qq x r . Then take [ ]/( ) w M lr I v dv u − ( − 0 ) . r − dx

19 For points over inﬁnity, similar proposition. ?????????

Theorem 2.38 Deﬁne the Qq-vector spaces

( dx 1 ! ) − Õ 0 dx dx E0 ui x b : u Qq x ⊕ ( ) i r ∈ [ ] i0

( dx 1 ! ) − Õ dx dx E ui x, 1 x b∞ : u Qq x, 1 x ⊕ ∞ ( / ) i r ∈ [ / ] i0

(dx 1 ) − Õ 0 dx B0 vi x b : v Qq x ⊕ ( ) i ∈ [ ] i0

(dx 1 ) − Õ dx B vi x, 1 x b∞ : v Qq x, 1 x ⊕ ∞ ( / ) i ∈ [ / ] i0 then E0 E and d B0 B are ﬁnite dimensional Qq-vector spaces and ∩ ∞ ( ∩ ∞) 1 H U E0 E d B0 B . ri g( )'( ∩ ∞)/ ( ∩ ∞) Theorem 2.39 There is an exact sequence

res0 res Qq 0 H1 X H1 U )( ⊕ ∞)⊗ → ( ) → ri g( ) −−−−−−−−−−−−−→ Lecture 8 3/10/2019 I lovingly stole this from Angus’ notes, ty Angus.

1. Determine a basis for cohomology. We want to ﬁnd ω0, . . . , ωk 1 1 − ∈ E0 E Ω U such that ( ∩ ∞) ∩ ( ) (a) ω0, . . . , ωk 1 is a basis of { − } 1 Hrig E0 E d B0 B ( ∩ ∞)/ ( ∩ ∞) 1 1 (b) The class of every element of E0 E Ω U in H U has ( ∩ ∞) ∩ ( ) rig( ) p-adically integral coeﬃcients with respect to ω0, . . . , ωk 1 . { − } (c) ω0, . . . , ωk 1 is a basis for the kernel of res0 res and hence for the{ subspace− H} 1 X of H1 U . ⊕ ∞ rig( ) rig( )

2. Compute lift of Frob φ, and compute the action of Frob on ω0, . . . , ωk 1 . { − } 3. Reduce pole orders so that we have Õ φ∗ωi d fi + Mji ωj j

where d fi d fi,0 + d fi, +d fi,end |{z} |{z}∞ ﬁnite pole reduction inﬁnite pole reduction Remark 2.40

1. Let X be a genus 3 smooth plane quartic, say X Xs 13 , the split Cartan curve of level 13. Then dim H1 X 6, but dim H1( U) 45 6 + 39, rig( ) rig( )

20 where 39 3 deg r x . ( ) 2. For applications to Coleman integrals between “good points”, proceed as before φ Q Q ¹ ( ) ¹ ωi φ∗ωi φ P P ( ) and correct endpoints. 3. For Coleman integration from a very bad point (a point above or a point with x-coordinate such that r x 0) B, split up the integral∞ ( )

¹ Q ¹ B0 ¹ Q ωi ωi + ωi B B B0

for B0 a point near the boundary of the residue disk of B.

Finite pole order reduction:. For i 0,..., 2g 1, ﬁnd fi,0 † Qp such that − ∈ R ⊗ dx φ ω d f + G ∗ i i,0 i r where Gi Qp only has poles above . ∈ R ⊗ ∞

Infinite pole order reduction:. For i 0,..., 2g 1, find fi, Qp such that − ∞ ∈ R ⊗ φ∗ωi d fi,0 + d fi, + Hi ∞ where Hi only has poles at point P above , and ordP Hi ord0 W deg r + 0 1 ∞ ( ) ≥ ( ( )− 2 ep where W W W and ep is the index of ramification of x-map at P. ) ( )− ∞

Final reduction:. For i 0,..., 2g 1, ﬁnd fi Qp such that − ,end ∈ R ⊗ Õ φ∗ωi d fi,0 + d fi, + d fi,end + Mji ωj. ∞ | {z } j d fi

2.4 Iterated Coleman Integrals Let X Q be a smooth, projective curve, p a prime of good reduction. /

Goal:. Describe an iterated Coleman integral on XQp and applications to rational points. Roughly speaking, an iterated Coleman integral is an iterated path integral

¹ Q ¹ 1 ¹ t1 ¹ tn 1 − ηn . . . η1 ... fn tn ... f1 t1 dtn ... dt1. P 0 0 0 ( ) ( )

Idea:. Want to apply Kedlaya/Tuitman as before, by computing action of Frob and reducing to simpler integrals. Earlier we had

¹ Q d f f Q f P P ( ) − ( ) and now reduce n-fold integral to n 1 -fold integral. ( − )

21 Notation:. P, Q X Qp , η1, . . . , ηn are 1-forms of the 2nd kind, without poles at P, Q. ∈ ( )

¹ Q ¹ R1 ¹ Rn 2 ¹ Rn 1 Q − − ηP η1 . . . ηn η1 R1 η2 R2 ... ηn 1 Rn 1 ηn P ( ) P ( ) P − ( − ) P for dummy variables Ri.

2.5 Algorithm for tiny iterated integral

Input:. Points P, Q X Qp in same residue disk. ∈ ( ) ∫ Q Output:. P η1 . . . ηn 1. Compute a local coordinate at P, x t , y t ( ( ) ( )) 2. For each k, write ηk x, y as ηk t dt. ( ) ( ) 3. Let In+1 1. Compute for k n, n 1,..., 2 − ¹ Rk 1 ¹ t Rk 1 − ( − ) Ik ηk Ik+1 ηk t Ik+1dt P P ( )

where t Rk 1 is parameterising points in the residue disk of P. ( − ) ∫ Q ∫ t Q 4. η1 . . . ηn ( ) η1 t I2 t dt. P P ( ) ( ) To compute more general iterated Coleman integrals, we’ll use the following properties.

Proposition 2.41 Let ωi , . . . ωi be forms of the second kind, regular at P, Q 1 n ∈ X Qp . ( ) ¹ P 1. ωi1 . . . ωin 0 P ¹ Q n ¹ Q Õ Ö 2. ωσ i1 . . . ωσ in ωi j ( ) ( ) all permutations σ P j1 P

¹ Q ¹ P n 3. ωi1 . . . ωin 1 ωin . . . ωi1 P (− ) Q n ∫ Q 1 ∫ Q Corollary 2.42 P ωi . . . ωi n! P ωi

References. For classical theory of iterated integrals • K-T. Chen, Algebras of iterated path integrals and fundamental groups, Trans. of AMS 156 (1971) [30] For p-adic theory • Coleman, Dilogarithms, regulators and p-adic L-functions, Invent. Math 1982 [34]. • Coleman, de Shalit, p-adic regulators on curves and special values of p-adic L-functions, Invent. Math. 1988 [33] • Besser, Coleman integration using the Tannakian formalism, Math. Ann. 2002 [15]

22 Remark 2.43 Still have linearity in the integrand, change of variables under rigid analytic maps. Be careful about additivity in endpoints.

Lemma 2.44 Let P, P , Q X Qp . Then 0 ∈ ( )

¹ Q n ¹ Q ¹ P0 Õ ωi1 . . . ωin ωi1 . . . ωi j ωi j+1 . . . ωin P j0 P0 P For all algorithms, we’ll restrict to the case n 2 (double Coleman integrals).

Example 2.45 Let φ P , φ Q be images of P, Q X Qp under Frobenius φ, then ( ) ( ) ∈ ( )

Q φ P φ Q Q φ P Q φ Q Q ¹ ¹ ( ) ¹ ( ) ¹ ¹ ( ) ¹ ¹ ( ) ¹ ωi ωk ωi ωk+ ωi ωk+ ωi ωk+ ωk ωi+ ωk ωi P P φ P φ Q P φ P φ P φ Q ( ) ( ) ( ) (2.1)( ) ( )

2.6 Strategy for computing double Coleman integrals of words 1 ( ) in HdR X ∫ Q Input:. P ωi ωj 1. Compute φ P , φ Q ( ) ( ) 2. Compute action of Frob and do some linear algebra to simplify

φ Q Q ¹ ( ) ¹ ωi ωj φ∗ ωi φ∗ ωj φ P P ( ) ( ) ( ) 3. Correct endpoints using equation(2.1).

φ Q Q ¹ ( ) ¹ ωi ωk φ∗ ωi ωk φ P P ( ) ( ) ¹ Q φ∗ωi φ∗ωk P ( )( ) ¹ Q © + Õ ª © + Õ ª d fi Mji ωj® d fk Mjk ωj® P j j « ¬ « ¬ ¹ Q © + Õ + Õ + Õ Õ ª d fi d fk d fi Mjk ωj Mji ωj d fK Mji ωj Mjk ωj® P j j j j « ¬ Lecture 9 8/10/2019 Last time: we were computing Coleman integrals

φ Q Q ¹ ( ) ¹ ωi ωj φ∗ ωi ωj φ P P ( ) ( ) ¹ Q ¹ Q Õ ¹ Q Õ ¹ Q Õ Õ d fi d fk+ Mji ωj d fk+ d fi Mjk ωj+ Mji ωj Mjk ωj P P j P j P j j

23 expand each of the ﬁrst three terms into expressions involving single Coleman integrals

¹ Q ¹ Q ¹ R ¹ Q ¹ Q d fi d fk d fi R d fk d fi R fk R fk P fk d fi fk P fi Q fi P P P ( ) P P ( )( ( )− ( )) P − ( )( ( )− ( ))

¹ Q Õ ¹ Q Õ ¹ R ¹ Q Õ ¹ Q Õ ¹ Q Õ Mji ωj d fk Mji ωj R d fk Mji ωj R fk R fk P fk Mji ωj fk P Mji ωj ( ) ( )( ( )− ( )) − ( ) P P P P P j P ¹ Q Õ ¹ Q ¹ R Õ d fi Mjk ωj d fi R Mjk ωj ( ) P j P P ¹ R Õ ¹ Q Õ fi R Mjk ωj fi R Mjk ωj R ( ) | − ( ) ( ) P j P ¹ Q Õ ¹ Q Õ fi Q Mjk ωj fi Mjk ωj ( ) − P j P then collect terms If cik is the sum of the expanded above, then the vector of double coleman integrals is a solution of a linear system involving all of the above.

Application (preview). Let Z be the minimal regular model of an elliptic curve, and let 0. Let E/ X E − dx ω0 , ω1 xω1 2y + a1x + a3 in Weierstrass coordinates, let b be 0 (really a tangential base-point at 0). Or an integral 2-torsion point. Let p be an odd prime of good reduction, suppose has analytic rank 1, and Tamagawa product 1. E Consider ¹ z ¹ z log z ω0, D2 z ω0ω1, ( ) b ( ) b can think of log as extending the log in the formal group. Theorem 2.46 Kim ’10, B.-Kedlaya-Kim ’11. Suppose P is a point of inﬁnite order in Z then Z Z is in the zero set of E( ) X( ) ⊆ E( ) 2 2 f z log P D2 z log z D2 P . ( ) ( ( )) ( ) − ( ( )) ( )

Chabauty-Coleman wrap up. What if Coleman’s bound

#X Q #X Fp + 2g 2 ( ) ≤ ( ) − is larger than #X Q known. If we carry out Chabauty-Coleman, what can we do if we seem to ﬁnd( “extra”) p-adic points that don’t look like they live if X Q ? Try using the Mordell-Weil sieve (developed by Scharashkin in his thesis( ) 99, adapted by Flynn (’04), Poonen-Schaefer-Stoll 07, Bruin-Stoll. See (((Unre- solved xref, reference "bib-bruin-stoll-mw"; check spelling or use "provisional" attribute))) (((Unresolved xref, reference "bib-siksek-mw"; check spelling or use "provisional" attribute))) . Set-up X Q /

24 a curve of genus g 2, M > 0 an integer ≥ i : X , J → suppose cM J Q MJ Q is a set of residue classes for which we want to ⊆ ( )/ ( ) show that no rational point P X Q maps to cM under π i. Simplest case: pick a good∈ prime( )v ◦

X Q / J Q MJ Q ( ) ( )/ ( )

X Fv / J Fv MJ Fv ( ) ( )/ ( ) if αv cM βv X Fv then done. Typically this is not enough: More generally( ) consider ∩ ( ( set)) S of∅ good primes and the commutative diagram

X Q / J Q MJ Q ( ) ( )/ ( )

Î Î X Fv / J Fv MJ Fv v ( ) v ( )/ ( ) Î then it suﬃces to show that αs cM βs v X Fv . The goal is then to ﬁnd a good( ) set ∩ of(S s.t. ( )) ∅ ( ) Ö A S, cM c cM : αS c βs X Fv ( ) ∈ ( ) ∈ ( v ( )) is empty. Heuristically the size of A S, cM is as follows: For a good prime ( )

XM v βv X Fv , ( ( )) #XM,v γ v, M ( ) #J Fv M J Fv ( )/ ( ( )) Note : v is only useful if γ v, M < 1. Expected size of A S, M( is ) ( ) Ö #cM γ v, M c ( ) want this to be small. Difficulties in using the sieve, the set, A S, cM can be large. The com- ( ) putation of images of X Fv in J Fv can become infeasible (the computation ( ) ( ) requires v discrete logs in J Fv ). Can be mitigated by using primes for which ( ) #J Fv is sufficiently smooth. ( Two) variations on Chabauty-Coleman: Pass to a collection of covering curves, (difficulty: constructing covers) Elliptic Chabauty method: (difficulty computing E K for K Q a larger degree number field. These methods( ) can/ potentially compute X Q when rk J Q g. ( ) ( ) ≥ Theorem 2.47 Chevalley-Weil. Let f : Y X be an unramified morphism of curves Q then there is a computable finite extension→ K Q such that / / 1 f − X Q Y K . ( ( )) ⊆ ( )

25 Theorem 2.48 Wetherell ’97. There is a ﬁnite set of unramiﬁed covering curves Yi X over Q (all isomorphic over Q), such that → Øn X Q fi Yi Q . ( ) ⊆ ( ( )) i1 Remark 2.49 When X is an elliptic curve this is descent in the proof of the Mordell-Weil theorem.

2.7 Introduction to p-adic heights Lecture 11 17/10/2019 The theory of p-adic heights has developed over 20 years, with work due to Bernardi, Neron, Schneider, Perrin-Riou, Mazur, Tate, Zarhin, Iovita, Werner, Coleman, Gross, Nekovar. We will begin with p-adic heights on elliptic curves. Let E Q be an elliptic curve, then a p-adic height h is a function /

h : E Q Qp ( ) → that plays a similar role to the canonical height

hE Q R. ˆ ( ) → Let p 5 be a prime of good ordinary reduction for E. Let 0 , P E Q , then write≥ ∈ ( ) a P b P P x P , y P ( ) , ( ) ( ( ) ( )) d P 2 d P 2 ( ) ( ) and assume a P , d P b P , d P 1 and d 1. d P is the denominator of P. ( ( ) ( )) ( ( ) ( )) ≥ ( ) Suppose P satisﬁes two conditions

1. P reduces to 0 in E Fp . ( ) 2. P reduces to a non-singular point of E Fl for all bad primes l. ( ) 1. Implies E P x P y P is divisible by P. ( ) − ( )/ ( ) Deﬁnition 2.50 Let E Q and p 5 be good ordinary, the cyclotomic p-adic height h on such points/ P in E Q≥ is ( ) 1 σ P h P log ( ) . ( ) p p d P ( )

σ P is the p-adic sigma function associated to E Zp. ♦ ( ) / What is σ P ? Mazur-Stein-Tate( ) “Computation of p-adic heights and log convergence”, Doc. Math 2006. Mazur-Tate “The p-adic sigma function”, Duke 1991. Mazur- Tate gave 11 diﬀerent characterizations of the p-adic sigma function. We’ll describe one characterization, let

2 x t t− + Zp t ( ) · · · ∈ (( )) be x in the formal group of E Zp, then / 3 y t t− + Zp t ( ) · · · ∈ (( )) (Silverman AEC, Ch IV).

26 Theorem 2.51 Mazur-Tate. There is exactly on odd function

σ t t + tZp t ( ) · · · ∈ [[ ]] and constant c Zp that together satisfy the p-adic diﬀerential equation ∈ d 1 dσ x t + c − ( ) ω σ ω

ω is the invariant diﬀerential associated to a chosen Weierstrass model for E

dx ω 2y + a1x + a3 and 2 a + 4a2 E2 E, ω c 1 − ( ) 12 we’ll say more about E2 E, ω in a bit! ( ) Lemma 2.52 The height function h extends uniquely to the full Mordell-Weil group E Q and satisfies h nQ n2h Q for all n Z, and Q E Q for P, Q E Q setting( ) ( ) ( ) ∈ ∈ ( ) ∈ ( ) P, Q h P + h Q h P + Q ( ) ( ) ( ) − ( ) we get a pairing on E Q . ( ) To compute h Q for arbitrary Q E Q . Let n1 #E Fp , n2 lcmv cv , ( ) ∈ ( ) ( ) ({ }) let n lcm n1, n2 then P nQ satisfies 1) and 2) from earlier. So compute h P h nQ( and) then ( ) ( ) 1 1 h Q h nQ h P . ( ) n2 ( ) n2 ( ) Remark 2.53 The p-adic regulator Reg of E Q is the determinant of the matrix p / of pairings on E Q E Q tors. ( )/ ( ) Conjecture 2.54 Schneider ’82. The (cyclotomic) height pairing is non-degenerate, equivalently Regp is non-zero. Contrast with canonical height h P 0 P is torsion ˆ( ) ⇐⇒ Remark 2.55 This p-adic regulator fits into a p-adic BSD conjecture. Conjecture 2.56 Special case of Mazur-Tate-Teitelbaum, ’86, Inventionnes. Suppose E has good ordinary reduction at p, let p E, T be the p-adic L-function attached to E Q. L ( ) / 1. ordT0 p E, T rk E Q L ( ) ( ) 2. Î p cv III E Q Reg | ( / )| p p∗ E, 0 L ( ) #E Q 2 ( )tors the leading coefficient.

1 2 2 p 1 α where α is the unit root of x ap x + p 0. ( − − ) − Remark 2.57 See also Stein-Wuthrich, “Algorithms for the arithmetic of elliptic curves using Iwasawa theory”, Math. Comp. 2013.

27 Back to E2 E, ω , Katz ’73, gave an interpretation of E2 E, ω as a direc- tion of the unit-root( ) eigenspace of frobenius acting on Monsky-Washnitzer( ) cohomology. Suppose E : y2 f x , then the basis for ( ) 1 1 dx x dx H E H E0 − , dR( ) MW ( ) y y

1 0 Ω1 moreover HdR E H E, U with U the unit root subspace. Compute the matrix of p(-power) frobenius( ) ⊕F, want to ﬁnd a basis for U. We know that

dx F pH1 E y ∈ dR( )

n n x dx so that mod p , U is the span of F y so for each n write

x dx dx x dx Fn a + b y n y n y then E E, ω 12 an mod pn . 2 bn What( does) this− have( to do with) rational/integral points on E?

E : y2 f x ( ) Recall d x + c − ω in formal group 1 dσ ω x + c d ⇒ ( ) − σ ω ¹ x dx c dx dσ d + . ⇒ y y − σω Lecture 1? 22/10/2019 p-adic heights after Coleman-Gross, “p-adic heights on curves” 1989. Let X Q be a smooth projective curve of genus g 1 and p a prime of good / ≥ reduction for X and ordinary reduction for J. We’ll be thinking of X Qp K. 1 0 /1 Important: ﬁx a subspace of H Y K complementary to H X, Ω call dR( / ) ( X K) 1 0 1 / this W. H X K H X, Ω W. dR( / ) ( X) ⊕ We’d like to understand the p-adic height of two degree 0 divisors D1, D2. 0 Assume D1, D2 Div X have disjoint support. ∈ ( ) Deﬁnition 2.58 The cyclotomic p-adic height is a symmetric bilinear pairing

0 0 h : Div X Div X Qp ( ) × ( ) →

D1, D2 h D1, D2 ( ) 7→ ( ) disjoint support. where

1. Õ Õ h D1, D2 hv D1, D2 hp D1, D2 + hv D1, D2 ( ) ( ) ( ) ( ) v v,p ¹ Õ ωD + mv log v 1 p( ) D2 v,p

28 with the left term a coleman integral of the third kind and mv some intersection multiplicities. 2. h D, β 0 ( ÷( )) for β Q X , so h is well deﬁned on J J. ∈ ( )× × ♦ Remark 2.59 The local heights can be extended non-uniquely to give the self- pairing Õ h D, D hv D, D ( ) ( ) this needs a choice of tangent vector at each point in the support of D. Remark 2.60 The local height at p is a Coleman integral of a normalized diﬀerential w.r.t. W of the third kind. e.g. if X : y2 f x ( ) is a hyperelliptic curve

D1 P Q , P, Q X Q , y P , y Q , 0 ( ) − ( ) ∈ ( ) ( ) ( ) can we construct ω with Res ω D1? We want residue 1 at P and -1 at Q. So ω has simple poles at P, Q. e.g.( )

dx y + y P y + y Q ω ( ) ( ) 2y x x P − x x Q − ( ) − ( ) has Res ω D1. But there are lots more! ( ) 1 Recall: HdR X has a canonical non-degenerate form given by cup-product pairing ( ) H1 X K H1 X K K dR( / ) × dR( / ) → µ1 , µ2 µ1 µ2 ([ ] [ ]) 7→ [ ] ∪ [ ] where Õ ¹ µ1 µ2 ResQ µ2 µ1 [ ] ∪ [ ] Q note that µ1, µ2 are diﬀerentials of the second kind (residue 0). So the residue does not depend on the choice of local integral for µ1 since µ2 has no residue at any point. Let T K be the space of diﬀerentials of the third kind on X, at most simple poles, integer( ) residues. We have a residue divisor hom

Res: T K Div0 X ( ) → ( ) Õ ω Res ω Resp ω 7→ ( ) p we have the short exact sequence

Res 0 H0 X, Ω1 T K Div0 X 0. → ( X) → ( ) −−→ ( ) →

We’re interested in Tl K these are log diﬀerentials ( ) d f , f K X ×. f ∈ ( )

29 Since 0 Ω1 Tl K H X, X K 0 ( ) ∩ ( / ) { } and Res d f f div f we get from the above sequence ( / ) ( ) 0 1 0 H X, Ω T K Tl K J K 0 → ( X) → ( )/ ( ) → ( ) → Proposition 2.61 There is a canonical homomorphism

1 Ψ: T K Tl K H X . ( )/ ( ) → dR( ) Note: Ψ is the identity on differentials of the first kind. Ψ sends third kind differentials to second kind or exact differentials. Ψ sends log differentials to 0. 0 Definition 2.62 Let D Div X then ωD is the unique form of the third kind with ∈ ( ) Res ωD D ( ) and Ψ ωD W ( ) ∈ recall we fixed H1 X H0 X, Ω1 W. dR( ) ( X) ⊕ ♦

Deﬁnition 2.63 The local height at p of D1, D2 is ¹ hp D1, D2 ωD1 . ( ) D2 ♦ Remark 2.64 When X has good reduction and J has ordinary reduction then there is a canonical choice for W, the unit root subspace for the action of frobenius. 1 Proposition 2.65 If ω0, . . . , ω2g 1 is a basis for H X with ω0, . . . , ωg 1 { − } dR( ) { − } ⊆ H0 X, Ω1 . Then ( ) n n φ∗ ωg ,..., φ∗ ω2g 1 {( ) ( ) − } is a basis for W mod pn where φ is a lift of frobenius. Algorithm 2.66 Coleman integral of diﬀerential of the third kind, with poles in non-weierstrass disks. Input: ω with Res ω P Q . P, Q X Q ( ) ( )−( ) ∈ ( ) non-weierstrass. R, S X Qp , R, S < disk P , disk Q . Output: ∈ ( ) ( ) ( ) ¹ R ω S

Ψ 1 1. Compute ω HdR X . Let φ be a lift of Frobenius. Let α φ∗ω pω. Use Ψ ω and( ) frobenius ∈ ( equivariance.) We have − ( ) Ψ α Ψ φ∗ω pω ( ) ( − ) Ψ φ∗ω Ψ pω ( ) − ( ) φ∗Ψ ω pΨ ω ( ) − ( ) 2. Let β be a 1-form with Res β R S . Compute Ψ β . ( ) ( ) − ( ) ( ) Ψ Ψ 1 3. Compute α β , easy since both are elements in HdR that we just computed. ( ) ∪ ( )

30 4. Compute ¹ S ω φ S ( ) and φ R ¹ ( ) ω R tiny 5. Compute Õ ¹ ResA α β . ( ) A 6.

¹ R ¹ ¹ S ¹ φ R ! 1 Õ ( ) ω Ψ α Ψ β + ResA α β ω ω S 1 p ( ) ∪ ( ) ( ) − φ S − R − A ( ) Remark 2.67 Idea behind this algorithm is that α is almost of the second kind, in that the sum of the residues of α in each annulus is 0.

Algorithm 2.68 Local height at p. Output: hp D1, D2 . ( ) 1. Let ω be a diﬀerential in T K with Res ω D1. ( ) ( ) Í2g 1 1 2. Compute Ψ ω − ai ωi H X . Then ( ) i0 ∈ dR( ) g 1 Õ− Ψ ω ai ωi W ( ) − ∈ i0 Íg 1 let ωD ω − ai ωi. 1 − i0 3. Compute using the previous algorithm ¹ hp D1, D2 ωD1 . ( ) D2 More details in Balakrishnan-Besser “computing local p-adic heights on hyperelliptic curves”. IMRN 2012. What if D1 and D2 have common support? e.g. hp D, D . The local height at P would be ( ) ¹ hp D, D ωD ( ) D idea of Gross “local heights on curves” Arithmetic Geometry ’86. At each point x in the common support of your divisors, choose a basis t, tx for the tangent space. Let z z be a uniformizer at x ∞

∂t z 1 any function f K X × ∈ ( ) then has a well-deﬁned “value” at x f f x x [ ] zm ( )

31 where m ordx f . This depends only on t and not z. To do this for local p-adic heights use( Besser’s) p-adic Arakelov theory, JNT 2005. Balakrishnan-Besser Coleman-Gross height pairings and p-adic sigma function, Crelle, 2015. Proposition 2.69 Let X Q be a hyperelliptic curve with odd degree model monic. / D P ( ) − (∞) ¹ h D, D “ ω00D ( ) D g 1 P ¹ − ¹ ωi ωi i0 ∞ ωi self dual basis for cup. Lecture 1? 29/10/2019 How do we use local heights on Jacobians of curves to study integral points. Theorem 2.70 Quadratic Chabauty for integral points on hyperelliptic curves B.-Besser-Muller. Let f x Z x be monic separable polynomial of degree 2g + 1 3, that does not reduce( ) ∈ to[ a] square modulo q for any prime q. (in the paper monic≥ is not used, this condition then restricts the reduction type) Let U Spec Z x, y y2 f x and let X be the normalization of the proj closure of the generic( ﬁber[ of]/(U. Let− J(be))) the Jacobian on X and assume rk J Q g, choose a prime p of good ordinary reduction. Suppose that the p-adic Coleman( ) integrals

¹ z ¹ z xi dx fi z ωi ( ) 2y ∞ ∞ then there exists explicitly computable constants αij Qp s.t. the locally analytic function ∈ Õ ρ z θ z αij fi z fj z , ( ) ( ) − ( ) ( ) 0 i j g 1 ≤ ≤ ≤ − where θ z hp z , z is an extension of the Coleman-Gross local height at(p)which(( takes) − values (∞) ( in) − (∞))

Z 1 p U( [ / ]) in an effectively computable finite set S Qp. ⊆ Refs, Balakrishnan, Jennifer S., Amnon Besser, and J. Steffen Müller. “Quadratic Chabauty: P-Adic Heights and Integral Points on Hyperelliptic Curves.” Jour- nal Für Die Reine Und Angewandte Mathematik (Crelles Journal) 2016, no. 720 (January 1, 2016). https://doi.org/10.1515/crelle-2014-0048. Balakrish- nan, Jennifer S., Amnon Besser, and J. Steffen Müller. “Computing Integral Points on Hyperelliptic Curves Using Quadratic Chabauty.” Mathematics of Computation 86, no. 305 (October 12, 2016): 1403–34. https://doi.org/ 10.1090/mcom/3130. Müller, Jan Steffen. “Computing Canonical Heights Using Arithmetic Intersection Theory.” Mathematics of Computation 83, no. 285 (2014): 311–336. https://doi.org/10.1090/S0025-5718-2013-02719-6. Proof. Sketch: Recall the Coleman-Gross p-adic height for X Q is a symmetric bilinear pairing / h : J Q J Q Qp ( ) × ( ) →

32 the global height decomposes as a sum of local heights Õ h D1, D2 hv D1, D2 ( ) v ( ) in particular we have Õ h D1, D2 hp D1, D2 + hv D1, D2 ( ) ( ) ( ) v,p

¹ Õ ωD + hv D1, D2 1 ( ) D2 v,p where ωD1 is a normalized diﬀerential of the 3rd kind (depends on a splitting 1 of the Hodge ﬁltration on H X Qp ) and for v , p dR( / )

hv D1, D2 mv log v, mv Q ( ) p ∈ computed using arithmetic intersection theory. See Muller and: Bommel, Raymond van, David Holmes, and J. Steﬀen Müller. “Explicit Arithmetic Intersection Theory and Computation of Néron-Tate Heights.” Mathematics of Computation, 2019. https://arxiv.org/abs/1809.06791v1. + Í Then look at h hp v,p hv note 1. h is a quadratic form, so can be written in terms of a basis of space of quadratic forms for J Q and this can be done using Coleman integrals. ( ) Õ h z , z αij fi z fj z , ( − ∞ − ∞) ( ) ( )

2. hp is a locally analytic function and in the extension to self-pairing:

g 1 Õ− ¹ z xi dx hp z , z 2 ωi ω¯ i , ωi , ω¯ i cup product duals. ( − ∞ − ∞) − 2y i 0 ∞

3. The sum Õ hv z , z ( − ∞ − ∞) v,p takes on ﬁnitely many values in S when restricted to p-integral points. The set S can be computed by knowing the reduction of X at bad primes.

4. Then rewrite Õ h hp hv S − ∈ { } v,p

5. this ρ can be computed as a convergent power series in each residue disk. So now pretend we are working in classical Chabauty Coleman. Expand ρ in each disk, set equal to each value in S solve for all t U Zp ∈ ( ) s.t. ρ z Z. Take all such points call that X Zp 2. ( ) ∈ ( ) 6. It’s possible that X Zp 2 is strictly larger than the known points in U Z . In this case run 1-4( on) a collection of good ordinary p and run Mordell-( ) Weil sieve. More details on each step

33 0 1. Let D1,..., Dg Div X representing basis elements of J Q Q then ∈ ( ) ( ) ⊗ compute global height pairings. h Di , Dj using B.-Besser-Muller. A basis for spaces of bilinear forms on(J Q is) ( ) 1 fk fl + fl fk 2( ) so compute 1 fk Di fl Dj + fl Di fk Dj 2( ( ) ( ) ( ) ( )) do linear algebra to compute αk,l Õ 1 h Di , Dj αk l fk Di fl Dj + fl Di fk Dj ( ) , (2( ( ) ( ) ( ) ( ))) 2. Want to compute ω¯ i { } for 0 1 g 1 such that π¯ i ωj δij ≤ ≤ − [ ] ∪ [ ] (a) Compute splitting of 1 0 1 H X Qp H X, Ω W, dR( / ) ( X) ⊕ where W is the unit root eigenspace of frob, recall that modulo pn a basis for W is n n φ∗ ωg ,..., φ∗ ω2g 1 {( ) ( ) − } 0 1 (b) Let ω˜ j be a projection of ωj onto W along H X, Ω . i.e. ( ) g 1 Õ− ω˜ j ωj ai ωi. − i0 (c) use cup product matrix to compute

2g 1 Õ− ω¯ 0 b0i ω˜ i ig . . 2g 1 Õ− ω¯ g 1 bg 1,i ω˜ i − − ig then let g 1 Õ− ¹ θ z 2 ωi ω¯ i ( ) − i0 to compute this as a power series in each residue disk for each residue disk compute a Zp point P, the value θ P local coordinate ( ) zP at P. g 1 Õ− ¹ z θ z 2 ωi ω¯ i ( ) − i 0 ∞ g 1 g 1 g 1 ! Õ− ¹ P Õ− ¹ zP Õ− ¹ zP ¹ P 2 ωi ω¯ i + ωi ω¯ i + ωi ω¯ i − P P i 0 ∞ i 0 i 0 ∞ g 1 g 1 ! Õ− ¹ zP Õ− ¹ zP ¹ P θ P 2 ωi ω¯ i + ωi ω¯ i ( ) − P P i 0 i 0 ∞

34 3. Prop. There is a proper regular model of X Qq over Zq (q , p) such that if is p-integral then X ⊗ X

hq x , x (( ) − (∞) ( ) − (∞))

depends solely on the component of the special fibre q that the section X in Zz corresponding to x intersects. e.g. in g 2 special fibres have beenX( classified) by Namikawa-Ueno and if is semistable then the types are X

In1 In2 m or In1 n2 n3 [ − − ] [ − ] pos integers ni , m. Need to compute regular models (implementation in Magma by Donnelly) and grobner bases of ideals of divisors to compute intersection multiplicities. Remark 2.71 Roughly speaking intersections are computing denominators which is why its not obvious how to go beyond integral points using this construction. e.g. for elliptic curves have Mazur-Stein-Tate p-adic height.

1 1 h P log σ P log D P ( ) p p( ( )) − p p( ( ))

Coleman-Gross Õ h P hp P + hv P ( − ∞) ( − ∞) ( − ∞) v,p extended appropriately. Example 2.72 Let

X : y2 x3 + x + 1 x4 + 2x3 3x2 + 4x + 4 ( )( − ) J C496 new modular curve studied by Baker Gonzalez-Jimenez Gonzalez, Poonen

J Q Z3 Z 2 ( ) ⊕ / let P 1, 2 , Q 0, 2 , R 2, 12 , S 3, 62 want to show that up to hyperelliptic(− involution) ( these) are(− the only) integral( points.) Gens for J Q Q. ( ) ⊗

P1 P , P2 S w Q , P3 w S R . { [( ) − (∞)] [( ) − ( ( ))] [( ( )) − ( )]} Lecture 1? 31/10/2019 Goal today: Give more context for quadratic chabauty, discuss Kim’s nonabelian Chabauty program. References

1. “p-adic approaches to rational and integral points on curves” - Poonen 2. From chabauty’s method to kim’s non-abelian chabauty method - Cor- win Let X be a smooth projective curve over K a number ﬁeld and let Z be a 0-dimensional subscheme. Let U X Z, d #Z K . The topological Euler − ( )

35 characteristic of U is χ U χ X d 2g 2 d. If χ U < 0 we say that U is hyperbolic (we want( ) to consider( ) − hyperbolic− − curves( because) they have nonabelian π1.) Example 2.73 g 0, d 3 e.g. P1 r 0, 1, . ≥ { ∞} Example 2.74 g 1, d 1 e.g. punctured elliptic curve E r 0 . ≥ { } Example 2.75 g 2, d 0 e.g. smooth projective curves of genus g 2 or punctured versions, integral≥ points. ≥ Fix a prime p of good reduction for X. Recall the classical Chabauty-Coleman diagram

X K / X Kp ( ) ( )

J K / J Kp / Lie JK ( ) ( ) p in the classical chabauty-coleman method, the image of J K in the g-dimensional ( ) space Lie JK spans a Kp subspace of dimension at most r rk J K . So if r < g p ( ) there exists a non-zero Kp valued functional on Lie JK vanishing on J K . This p ( ) pulls back to d non-zero locally analytic functions on X Kp that vanish on X K . ( ) ( Problem:) By using the geometry of J impose too much structure, can’t extend this to r g. Kim’s idea: Replace J with references to X and various homology groups,≥ then generalize the diagram by replacing homology with various deeper quotients of (nonabelian) π1. Gives construction of Selmer varieties. References for Kims work: Kim, Minhyong. “The Motivic Fundamental Group of P1 r 0, 1, and the Theorem of Siegel.” Inventiones Mathematicae 161, no. 3 (September{ ∞} 2005): 629–56. https://doi.org/10.1007/s00222-004- 0433-9. Kim, Minhyong. “The Unipotent Albanese Map and Selmer Varieties for Curves.” Publications of the Research Institute for Mathematical Sciences 45, no. 1 (2009): 89–133. https://doi.org/10.2977/prims/1234361156. Kim, Minhyong. “Massey Products for Elliptic Curves of Rank 1.” Journal of the American Mathematical Society 23, no. 3 (2010): 725–47. https://doi.org/ 10.1090/S0894-0347-10-00665-X.

X K / X Kp ( ) ( )

J K / J Kp / Lie JKp ( ) ( ) 7

J K 1 p / J Kp 1 p ( )[ / ] ( )[ / ]

1 1 dR 0 SelQ J H GK , V / H GK , V / H XK F p ( ) f ( ) f ( v ) 1 ( p )/

1 1 H GK , V H GK , V ( ) ( )

36 We want to start by removing J K from the diagram. Recall: Let M be an abelian group. The p-adic completion( )

Mb lim M pn M / ←−−n is a Zp-module. We can get a Qp-vector space by inverting p.

Mb 1 p Mb Zp Qp. [ / ]' ⊗ n The group J Kp is compact so that the images of p J Kp in Lie J Kp 0, p-adically as(n ) . So the homomorphism ( ) ( ) → → ∞

J Kp Lie JK ( ) → p factors through JKp and hence also through JKp 1 p . Since log is a local ( ) ( )[ / ] diﬀeomorphism with ﬁnite kernel get a Qp-linear map

JKp 1 p Lie JK ( )[ / ] → p is an isomorphism. Recall the Kummer exact sequence

m 0 J m J · J 0 → [ ] → −→ → take GK Gal K K -cohomology to get a long exact sequence which leads to a short exact sequence( / )

κm 1 1 0 J K m J K H GK , J m H GK , J m 0 → ( )/ ( ) −−→ ( [ ]) → ( )[ ] → where κ is the Kummer map. There is a canonical GK-equivariant isomorphism J m Het J , Z m Het X , Z m . [ ]' 1 ( K / )' 1 ( K / ) The Kummer map gives an embedding

1 et J K m J K , H Gk , H X , Z m ( )/ ( ) → ( 1 ( K / )) to get an embedding of J K rather than just J K m J K take m pn and inverse limit. ( ) ( )/ ( ) Get a Zp-Tate module.

n et T lim J p H X , Zp [ ]' 1 ( K ) ←−−n let V T 1 p T Zp Qp. Let J K Zp J K Z Zp, J K Qp J K Z Qp. This gives[ / ] us embeddings⊗ ( ) ( ) ⊗ ( ) ( ) ⊗

1 J K Z , H GK , T ( ) p → ( ) 1 J K Q , H GK , V ( ) p → ( ) this is almost our replacement for J. 1 We still need to identify Qp-span of J K inside H GK , V using V and without reference to J. ( ) ( ) The vector space V has the structure of a GK-rep so we want a Galois 1 theoretic way of identifying the image of J K Qp in H GK , V . This is where the Bloch-Kato Selmer groups come in. Back( ) to the fundamental( ) short exact

37 sequence from the Kummer sequence let m pn and we have the local diagram as well.

κ 1 1 0 / J K m J K / H GK , J m / H GK , J m / 0 ( )/ ( ) ( [ ]) ( )[ ] α Î Î 1 Î 1 0 / J Kv m J Kv / H GK , J m / H GK , J m / 0 v ( )/ ( ) β ( v [ ]) ( v )[ ]

1 n the selmer group is a ﬁnite dimensional subspace of H GK , J p . ( [ ]) 1 Selpn J α− im β ( ) ( ) we have n J K p J K , Selpn J ( )/ ( ) → ( ) now taking inverse limits gives

1 JdK , SelZ J H GK , T ( ) → p ⊆ ( ) and inverting p gives

1 JdK 1 p , SelQ J H GK , V ( )[ / ] → p ⊆ ( ) let III be the Shafarevich-Tate group of J. we have

n n 0 J K p J K Selpn J III p 0 → ( )/ ( ) → ( ) → [ ] → take inverse limits and invert p

n 0 JdK 1 p SelQ J lim III p 1 p 0 → ( )[ / ] → p ( ) → ( [ ])[ / ] → ←−−n if III p∞ is ﬁnite, then [ ] lim III pn 0 [ ] ←−−n so JdK 1 p SelQ J ( )[ / ]' p ( ) 1 we want to determine the image of J K Q in H GK , V in terms of V using ( ) p ( ) V as a Galois rep. We’re almost there, we’ve replaced J K Qp with SelQp J . Remaining problem is that the local conditions giving us( ) ( )

1 SelQ J H GK , V p ( ) ⊆ ( ) Î 1 we use information from the geometry of J. (certain subgroups of H GKv , V ). This problem was solved by Bloch and Kato using Bloch-Kato Selmer groups.( ) use p-adic Hodge theory to re-interpret im β. Lecture 1? 5/11/2019 Recall the big chabauty-kim diagram. Last time we had this

n 1 n J K p J K / H GK , J p ( )/ ( ) ( [ ]) α Î n 1 n J Kv p J Kv / H GK , J p v ( )/ ( ) β ( [ ])

38 1 Selpn J α− im β ( ) ( ( )) lim Selpn J SelZ J, ( ) p ←−−n then we get 1 bJ K 1 p , SelQp J H GK , V . ( )[ / ] → ( ) ⊆ ( ) 1 Goal is to determine the image of J K Q in H GK , V in terms of the Galois ( ) p ( ) representation V. We replaced J K Qp by SelQp , defined by local conditions. ( ) Î 1 But the local conditions (subgroup of v H GKv , V are defined via geometry of J rather than via galois reps. ( ) Bloch and Kato defined im β using p-adic Hodge theory and defined SelQ J in terms of data from V. p ( ) 1. They wanted to extend Selmer and Shafarevich-Tate groups to motives beyond H1 of abelian varieties.

2. this construction will let us extend the idea of Selmer group from

et et ab T H X , Zp π X Zp 1 ( K )' 1 ( K) ⊗ et to certain (non-abelian) quotients of π1 . Crash course in Bloch-Kato Selmer groups. Let V be a ﬁnite dimensional Qq-vector space with a continuous action of GKv . Let, Kv Qp. Fontaine GK deﬁned Dcris V Bcris Q V v for a ring of p-adic periods Bcris we have ( ) ( ⊗ p )

dimK Dcris V dimQ V v ( ) ≤ p and if equality holds we say that V is crystalline. See Berger, or Caruso, or Andre. 1 An element ξ H GK , V corresponds to an isomorphism class of exten- ∈ ( v ) sions of Qp by V 0 V E Qp 0 → → → → 1 and ξ is said to be crystalline if the galois representation is. Let H GK , V f ( v ) 1 be the set of crystalline classes in H GKv , V . The “local Bloch-Kato Selmer group” at v. ( )

1 Deﬁnition 2.76 The Bloch-Kato Selmer group H GK , V is the set of ξ f ( v ) ∈ 1 1 H GK , V whose image in H GK , V is crystalline for each v p. ♦ ( ) ( v ) | Theorem 2.77 Bloch-Kato. The image of the Kummer map

1 κv : J Kv Q H GK , V ( ) → ( v ) 1 is H GK , V . f ( v ) Corollary 2.78 The Selmer group

SelQp J

1 1 coincides with H GK , V in H GK , V . f ( ) ( ) Need to also deﬁne log intrinsically, also needs a new target space in place of Lie J KK . ( p )

39 Deﬁnition 2.79 A ﬁltered module over a commutative ring #][m R] is an R- module M with a collection Gi M { } of submodules that is decreasing

+ Fi 1M Fi M, i. ⊆ ∀ If Ð Fi M M the filtration is exhaustive. For any filtered R-module M the associated graded module Ê gr M gri M , gri M Fi M Fi + 1M. ( ) ( ) ( ) / i ♦ Recall the Hodge filtration on de Rham cohomology, this is a decreasing i 1 filtration F on H JK with dR( p ) 2 1 0 1 0 1 0 F F H JK , Ω F H . ⊆ ( p ) ⊆ dR 1 dR The de Rham cohomology HdR JKp is the dual of H1 JKp and has a ( ) 1 ( ) decreasing hodge filtration dual to that of H JK , defined as dR( p ) i dR i+1 1 H H JK Ann F− H JK . 1 ( p ) ( dR( p ))]

0 1 0 1 1 1 1 dR H X, Ω ∨ H J, Ω ∨ gr H JK ∨ gr− H JK ( ) ' ( ) ( dR( p )) 1 ( p ) 1 dR dR 0 dR dR 0 dR gr− H JK H JK F H JK H XK F H XK 1 ( p ) 1 ( p )/ 1 ( p )' 1 ( p )/ 1 ( p ) this is our intrinsic definition of H0 X, Ω1 . ( )∨ More of our crash course on p-adic hodge theory. To define logBK. We assume Kp Qp, Fontaine’s theory defines a series of Qp-algebras ' with GQp -action and additional structure compatible with this, (i.e. frob, differential operators, hodge) we have

Bcr ys B ??? ⊆ dR 1 + 0 Facts, theres a descending filtration F on BdR, for which BdR F BdR is a DVR with fraction field BdR and residue field Cp. There is an action of frobenius on Bcr ys whose fixed subring is denoted

φ1 Bcr ys.

GKp GKp We have Bcr ys BdR Kp. For a continuous p-adic representation V of GQp . We deﬁne

D V B V GKp dR( ) ( dR ⊗ ) + + D V B V GKp dR( ) ( dR ⊗ ) the ﬁltration on B induces one on D V . dR dR( ) If Y is a smooth variety over Qp then p-adic height gives for

i V H Y , Qp et( Qp ) that D V ∼ Hi Y . dR( ) −→ dR( )

40 + Respecting the Hodge filtration on either side. Also D V is naturally iden- dR( ) tified with F0D V . dR( ) Fm D V Fm B V GKv dR( ) ( dR ⊗ ) i Frobenius on Bcrys induces a Frobenius map on Dcrys V and V H Y , Qp ( ) et( Qp ) and Y good reduction with special fibre YFp . Then there’s a natural isomorphism 1 Dcrys V ∼ H YF , Qp ( ) −→ crys( p ) respects frob on each side. DdR and Dcrys commute with taking ∨ so have corresponding results for homology.

et Example 2.80 Let V H X , Qp then 1 ( K ) dR H XK D V Dcrys V 1 ( p )' dR( )' ( )] compat with hodge Lecture 1? 7/11/2019

et T H X , Zp 1 ( K ) et V H X , Qp 1 ( K ) want to deﬁne the Bloch-Kato logarithm.

1 dR 0 log : H GK , V H XK F . BK f ( p ) → 1 ( p )/ Bloch-Kato 1990 give a short exact sequence

α φ1 + β 0 Qp B B B 0, → −→ cris ⊕ dR −→ dR → where α x x, x , β x, y x y. Tensor with V and take Galois cohomology to get( ) ( ) ( ) −

+ GKp φ 1 1 1 φ 1 0 V Dcris V D V D V H GK , V H GK , V B → → ( ) ⊕ dR( ) → dR( ) → ( p ) → ( p ⊗ cris ) → · · · Bloch-Kato show that

1 1 1 φ1 H GK , V ker H GK , V H GK , V B e ( p ) ( ( p ) → ( p ⊗ cris )) is equal to 1 H GK , V . f ( p ) This gives a surjection

+ 1 D V D V H GK , V dR( )/ dR( ) → f ( p ) this is the Bloch-Kato exponential map. (coincides with usual exp in the case of a p-adic formal lie group). The kernel of this map is

φ 1 GKp Dcris V V ( ) / this is trivial since V is the Tate module of an abelian variety (true because of the Weil conjecture, and what we know about eigenvalues of frobenius).

41 So in this case the Bloch-Kato exponential map has an inverse, our Bloch- Kato logarithm.

1 + dR 0 log : H GK , V ∼ D V D V H X F BK f ( p ) −→ dR( )/ dR( )' 1 ( )/ Kim’s generalization. Let G be a group (resp. topological group). For subgroups A, B G ⊆ let A, B denote the (the closure of) the commutator of the subgroups. [ ] Deﬁnition 2.81 Let 1 G[ ] G 2 1 G[ ] G[ ], G [ ] ... i+1 i G[ ] G[ ], G [ ] then 1 2 G[ ] D G[ ] D ··· is a descending chain of normal subgroups of G called the lower central series n+1 of G. Let Gn G G . ♦ / [ ] Example 2.82 1 G1 G G[ ] G G, G / /[ ] ab is the abelianization G of G. For n 2 the group Gn is an n-step nilpotent group, typically nonabelian. ≥

Now for various flavours of π1 X we have ( ) ab π1 X 1 π1 X H1 X ( ) ( ) ( ) so above we should think of H1 as the abelianization of π1. We can therefore try to replace it by π1 X n for n > 1. We’ll do this for the two homology groups associated to X( ) 1. p-adic étale homology 2. de Rham homology we can define the geometric étale fundamental group πet X such that 1 ( K) πet X ab Het X , bZ 1 ( K) ' 1 ( K ) et ab et et et 2g π X H X , bZ H X , Zp H X , Qp V Ga Qp 1 ( K) ' 1 ( K ) 1 ( K ) ⊆ 1 ( K ) ( ) 2g with action of GK in particular GK acts via Qp-linear automorphisms on Ga . et For n 2 there’s an analogous construction that transforms π X n into ≥ 1 ( K) a topological group Vn that is the group of Qp-points of a unipotent algebraic group n Qp, equipped with a GK-action. V / Definition 2.83 A pro-unipotent group over Qp is a group scheme of Qp that is a projective limit of unipotent algebraic groups over Qp. ♦

Then there is a notion of Qp-pro-unipotent completion

et π X Q 1 ( K) p

42 of πet X 1 ( K) there is a continuous homomorphism

et et π X π X Q Qp 1 ( K) → 1 ( K) p ( ) is universal among such maps. There are lower central series quotients

et et et et n+1 Un π X Qp,n π X Qp π X [ ]. 1 ( K) 1 ( K) / 1 ( K)Qp In the cases we consider, these are ﬁnite dimensional varieties. The coordinate ring et π X Q O( 1 ( K) p ) is a Qp-vector space. Recall that for a smooth curve X we had

dR et H XK D H X , Qp 1 ( p )' dR( 1 ( K )) we may deﬁne the de Rham fundamental group

dR π XK 1 ( p ) as et Spec D π X Q . ( dR(O( 1 ( K) p ))) dR π1 is equipped with a Hodge ﬁltration. Just like we took the lower central et series ﬁltration on π X Q . do the same thing for quotients by 1 ( K) p dR π XK n 1 ( p ) dR We’ll denote these as Un . We have

dR et U Dcris U n ( n ) so it is equipped with a frobenius action. Now we have a non-abelian version of the Chabauty-Coleman diagram

X K / X Kp ( ) ( )

& 1 et 1 et dR 0 H GK , π X Q n / H GK , U / U F f ( 1 ( K) p , ) f ( p n ) n /

An element of 1 et H GK , U f ( n ) is a scheme over Qp with continuous GK-action and GK-equivariant action of et et Un making it into a Un -torsor. Kim shows that 1 et 1 et H GK , U H GK , U f ( n ) ⊆ ( n ) has the structure of an algebraic variety the Selmer variety Sel Un . 1 et ( ) We refer to H GK , U as the local Selmer variety. f ( p n )

43 Theorem 2.84 Kim. For some n > 1 if

1 et dim Sel Un < dim H GK , U ( ) f ( p n ) then X K is contained in the zero locus of non-trivial locally analytic function on ( ) X Kp so X K is ﬁnite. ( ) ( ) Lecture 1? 12/11/2019 Lemma 2.85 Let X be a smooth projective curve over an algebraically closed ﬁeld k k and n 1 be invertible in k then ≥ 0 H X , µn µn et( k ) 1 0 H X , µn Pic X n . et( k ) ( k)[ ] Proof. Sketch: Recall the Kummer sequence

0 µn Gm Gm 0 → → → → we also have 0 H X, Gm k× et( ) 1 H X, Gm Pic X et( ) ( ) q H X, Gm 0 for q 2 et( ) ≥ take cohomology of the Kummer sequence to get

0 0 H X , µn k× k× 0 → et( k ) → → → 1 2 0 H X , µn Pic X H x, µn 0 → et( k ) → ( ) → et( ) → We also have 0 Pic0 X Pic X Z 0 → ( ) → ( ) → → and we can identify Pic0 X with the group of k-rational points of J of X. Note that multiplication by n (is surjective) and the kernel is Pic0 n . et[ ] More generally, for any galois stable quotient U of Un we have a similar dR diagram U Dcris U and corresponding maps ( ) et dR et j , j , locU p , j : X Q Sel U . U U , U ( ) → ( ) We have that X Q is contained in ( ) et, 1 X Qp U j − locU p Sel U X Qp ( ) U,p ( , ( ( ))) ⊆ ( ) when U Un we get X Qp n X Qp U ( ) ( ) n we have X Q X Qp n X Qp 1 X Qp . ( ) ⊆ · · · ⊆ ( ) ⊆ · · · ⊆ ( ) ⊆ ( ) Conjecture 2.86 Kim. For n 0 X Qp n is ﬁnite. ( ) Remark 2.87 This is implied by Beilinson-Bloch-Kato and other standard conjectures on motives.

44 Conjecture 2.88 Kim. For n 0 X Qp n X Q . ( ) ( ) The analogue of analytic properties of AJb (there is a non-zero functional we can construct that vanishes on J Q with Zariski dense image, and given by a convergent p-adic power series),( so) there are ﬁnitely many zeroes on each residue disk of X Qp ) is the following ( ) dR Theorem 2.89 Kim 09. The map jn has Zariski dense image and is given by convergent p-adic power series on every residue disk. The analogue of Chabauty-Coleman r < g hypothesis is non-density of locU,p.

Theorem 2.90 Kim 09. Suppose locU p is non-dominant, yhen X Qp U is finite. , ( ) Remark 2.91 All finiteness results come from bounding the dimension of Sel U . ( ) Coates and Kim did this: Theorem 2.92 Coates, Kim ’10. Let X Q be a smooth projective curve of genus / g 2 and suppose J is isogenous over Q to ≥ Ö J Ai ∼ of abelian varieties with Ai having CM by Ki of degree 2 dim Ai. Then for n 0 have X Qp n is finite. ( ) Examples: • Fermat curves xm + ym + zm 0.

• Twisted fermat curves

axm + b ym + czm 0.

• Van Wamelen’s list of genus 2 curves whose jacobians have CM and are simple. Coates, John, and Minhyong Kim. “Selmer Varieties for Curves with CM Jacobians.” Kyoto Journal of Mathematics 50, no. 4 (2010): 827-52. https:// doi.org/10.1215/0023608X-2010-015. Key idea is to use multivariable Iwasawa theory to bound the dimension of the Selmer variety. et et Recall π X Q is the Qp-pro-unipotent completion of π X . Let 1 ( Q) p 1 ( Q)

et et 2 et 2 W π X Q π X ( ) , π X ( ) 1 ( Q) p /[ 1 ( Q)Qp 1 ( Q)Qp ] be the quotient by the derived series

0 G( ) G

n n 1 n 1 G( ) G( − ), G( − ) [ ] W itself has a lower central series ﬁltration

1 2 n 1 n W W W W − W, W ⊇ ⊇ · · · [ ] ⊇ · · · n+1 and associated quotients Wn W W . /

45 Theorem 2.93 Coates-Kim. There is a bound B depends on X and T s.t.

n Õ 2 i i+1 2g 1 dim H GT , W W Bn − . ( / ) ≤ i1 This is done by controlling the zeroes of an algebraic p-adic L-function (really an annihilator of an ideal class group). A corollary of this gives a dimension bound on Selmer varieties.

1 dR 0 Corollary 2.94 For n 0 have dim H G, Wn < dim W F . f ( ) n / Remark 2.95 Structure of Selmer variety as an algebraic variety is a conse- et quence of Qp-pro-unipotent completion of π X . 1 ( Q) Qp-pro-unipotent completion: Given a ﬁnitely presented discrete group E, take the group algebra

Q E [ ] complete it with respect to the augmentation ideal K

Q E lim Q E Kn [[ ]] [ ]/ ←−−n the coproduct ∆: Q E Q E Q E [ ] → [ ] ⊗ [ ] deﬁned by g g g takes K to the ideal 7→ ⊗ K Q E + Q E K ⊗ [ ] [ ] ⊗ then there’s an induced coproduct on

∆: Q E Q E Q E . [[ ]] → [[ ]]⊗ˆ [[ ]] The unipotent completion U E ( ) can be realized as the grouplike elements in Q E . [[ ]] U E g Q E : ∆ g g g ( ) { ∈ [[ ]] ( ) ⊗ } This defines the Q-points of a pro-algebraic group over Q. When E is topologically finitely presented profinite group the Qp-pro- unipotent completion is defined analogously. Lecture 1? 15/11/2019 Missed a lecture Lecture 1? 19/11/2019 Definition 2.96 Filtered φ modules. A filtered φ-module is a finite dimensional Qp-vector space W, equipped with an exhaustive and separated decreasing filtration Fili and an automorphism φ. ♦

Remark 2.97 We have seen instances of this already, VdR is a ﬁltered φ-module.

Definition 2.98 Mixed extensions. We define MFil,φ Qp , VdR, Qp 1 to be the category of mixed extensions of filtered φ-modules,( the objects are( )) triples M, M , ψ where ( • •) 1. M is a filtered φ-module.

46 2. M is a ﬁltration by sub-ﬁltered φ-modules •

M M0 M1 M2 M3 0. ⊇ ⊇ ⊇ 3. ψ are isomorphisms of ﬁltered φ-modules •

φ0 : M0 M1 ∼ Qp / −→ φ1 : M1 M2 ∼ V / −→ dR φ2 : M2 M3 ∼ Qp 1 / −→ ( ) and the morphisms of this category are morphisms of filtered φ-modules which also respect the filtrations M and commute with the isomorphisms ψi • and ψ0i. Let M Qp , V , Qp 1 Fil,φ( dR ( )) denote the set of isomorphism classes of objects in this category. ♦ Lemma 2.99 For any filtered φ-module W for which

W φ 1 0 we have an isomorphism 1 0 Ext Qp , W W Fil . ( )' /

Idea. Given an extension of Qp by W

0 W E Qp 0 → → → → φ choose a splitting s : Qp E which is φ-invariant. Choose a splitting Fil → φ s : Qp E which respects the ﬁltration. s chosen earlier. s is unique → by sFil is determined up to an element of

Fil0 W.

So sφ sFil W mod Fil0 W − ∈ ( ) is independent of choices. Recall M Dcris Met . dR ( ) Then the structure of a mixed extension of ﬁltered φ-modules on MdR allows us to deﬁne extensions

E1 E1 M M Qp 1 ( ) dR/ ( ) E2 E2 M ker M Qp . ( ) ( dR → ) We have 0 0 0 Qp 1 E2 Fil V Fil 0. (2.2) → ( ) → / → dR/ → The image of the extension class

1 0 M H GQ , E2 E2 Fil [ ] ∈ f ( p )' / inside 0 1 V Fil H GQ , V dR/ ' f ( p dR)

47 is E1 . [ We] deﬁne δ to be

0 s 0 δ : V Fil V E2 E2 Fil dR/ −→ dR → → / where s is our ﬁxed splitting of the Hodge ﬁltration on VdR and

V E2 dR → is the unique frobenius equivariant splitting of

0 Qp 1 E2 V 0 → ( ) → → dR → 0 by construction M and δ E1 have the same image in V Fil so by(2.2) [ ] [ ( )] dR/ their difference defines an element of Qp 1 . Since the filtered φ-module ( )

1 Qp 1 H GQ , Qp 1 ( ) f ( p ( )) by Lemma 2.99 we can think of

1 M δ E1 H GQ , Qp 1 . [ ] − [ ( )] ∈ f ( p ( )) Finally, taking the local component

χp : Q× Qp p → of idèle class character ﬁxed at the beginning of the discussion of Nekovář heights. This gives a map

1 χp : H GQ , Qp 1 Qp f ( p ( )) → and hp M χp M δ E1 . ( ) ([ ] − [ ( )]) For applications to rational points we need to make this more explicit. The splitting s of the Hodge filtration we fixed earlier defines idempotents

s1, s2 : V V dR → dR projecting onto s V Fil0 and Fil0 ( dR/ ) components, respectively. Suppose that we are given a vector space splitting of

s0 : Qp V Qp 1 ∼ M. ⊕ dR ⊕ ( ) −→ The split mixed extension

Qp V Qp 1 ⊕ dR ⊕ ( ) has the structure of a ﬁltered φ-module as a direct sum. So we choose two further splittings φ s : Qp V Qp 1 ∼ M ⊕ dR ⊕ ( ) −→ Fil s : Qp V Qp 1 ∼ M ⊕ dR ⊕ ( ) −→

48 sφ is frobenius-equivariant and sFil respects the ﬁltrations, sφ is unique and sFil is not. Now choose bases for Qp , VdR, Qp 1 such that with respect to these bases we have ( ) 1 0 0 1 φ ©α 1 0ª s0− s φ ◦ T ® γφ β 1 « φ ¬ 1 0 0 1 Fil © 0 1 0ª s0− s ® ◦ γ βT 1 « Fil Fil ¬ then with respect to these choices

hp M χp M δ E1 ( ) ([ ] − [ ( )]) T T χp γ γ β s1 α β s2 α . ( φ − Fil − φ ( φ) − Fil ( φ)) Goal: Compute these components explicitly and construct M as

“AZ b, x 00 ( ) where Z is a choice of endomorphism. Then

E1 x E1 M E1 AZ b, x AJ x . ( ) ( ) ( ( )) b( ) Remark 2.100 By work of Olsson

dR Dcris AZ b, x A b, x . ( ( )) Z ( )

E2 Z x E2 AZ b, x E AJ x + c. , ( ) ( ( )) ( b( )) this is done by a twisting construction and some input from a quadratic Chabauty pair. Definition 2.101 Quadratic Chabauty pairs. Let X Q be a smooth projective curve of genus g. Suppose rk J Q g and that / ( ) J Q J Qp ( ) ⊆ ( ) has finite index. A quadratic Chabauty pair θ, S is a function ( ) θ : X Qp Qp ( ) → and a finite set S such that

1. On each residue disk of X Qp the map ( ) 0 1 AJ , θ : X Qp H X, Ω ∗ Qp ( b ) ( ) → ( ) × has Zariski dense image and is given by a convergent p-adic power series. 2. There exists an endomorphism E of 0 1 H XQ , Ω ∗, ( p ) 0 1 a functional c H XQ , Ω and a bilinear form B ∈ ( p )∗ 0 1 0 1 B : H XQ , Ω ∗ H XQ , Ω ∗ Qp ( p ) ⊗ ( p ) → such that for all x X Q , we have ∈ ( ) θ x B AJ x , E AJ x + c S. (2.3) ( ) − ( b( ) ( b( )) ) ∈ ♦

49 This gives us a finite set of points containing X Q , since Item 1 implies that only finitely many points satisfy(2.3) and Item 2( implies) that all rational points satisfy(2.3). Remark 2.102 We will use the Nekovář height to do this, the function θ will be the local height at p and B will be the global p-adic height. If we further add the hypothesis that X has everywhere potentially good reduction then S 0 . { } Now we describe how knowing θ and S gives us a method for determining a finite subset of X Qp containing X Q . For α S define( ) ( ) ∈

X Qp x X Qp : θ z B AJ x , E AJ x + c α . ( )α { ∈ ( ) ( ) − ( b( ) ( b( )) ) } By deﬁnition Ä X Q X Qp ( ) ⊆ ( )α α S ∈ so it suﬃces to describe X Qp . ( )α Let 0 1 0 1 : H XQ , Ω ∗ H XQ , Ω ∗. E ( p ) ⊗ ( p ) Lemma 2.103 Suppose we have P1,..., Pm X Q s.t. ∈ ( )

AJ Pi E AJ Pi + c { b( ) ⊗ ( b( )) } forms a basis for . Suppose E ψ1, . . . , ψm { } forms a basis for . E∗ Assume that Pi X Qp αi where αi S. ∈ ( ) ∈ For x X Qp deﬁne the matrix T x T θ,S x via ∈ ( ) ( ) ( )( )

θ z α Ψ1 x Ψm x © ( ) − ( )··· ( ) ª θ P1 α1 Ψ1 P1 Ψm P1 ® T x ( ). − (. )··· .( ) ® . ( ) . . . . . ® . . . . ® θ Pm αm Ψ1 Pm Ψm Pm « ( ) − ( )··· ( )¬ Where Ψi x ψi AJ Pi E AJ Pi + c ( ) ( b( ) ⊗ ( b( )) ) if x X Qp α then ∈ ( ) det T x 0. ( ( )) Remark 2.104 Computing det T x 0 and ﬁnding its zeroes is what gives us our ﬁnite set of points containing( )X Q . ( ) We need enough rational points so that the ψi forms a basis for and we need a base point so which means we need g2 + 1 points with this approach,E It can be possible to make do with only g + 1 if height is taken to be equivalent with respect to some other structure. Lecture 1? 21/11/2019

Theorem 2.105 B.-Dogra. Let X Q be a smooth projective curve of genus g 2. Let r rk J Q and / ≥ ( ) ρ ρ JQ rk NS JQ ( ) ( )

50 suppose r < g + ρ 1. Then − X Qp 2 ( ) is ﬁnite. Remark 2.106 This is also true for X K, with K an imaginary quadratic ﬁeld. / To prove this, look at Kim’s diagram

X Q / X Qp ( ) ( )

& 1 et dR 0 Sel Un / H GQ , U / U F ( ) f ( p n ) n /

In this case take n 2. One interpretation of U2 is

Û2 1 coker Qp 1 V U2 V 1. → ( ( ) → ) → → → The main idea is not to work with the full U2, which is hard, but instead to use a suitable Galois stable quotient U for which 1 1 dim H GT , U < dim H GQ , U . f ( ) f ( p ) | {z } Sel U ( ) Then the result follows since

X Qp 2 X Qp U . ( ) ⊆ ( ) We’ll take U to be a quotient of U2 surjecting onto V, in particular, m U, U Qp 1 , for m 1. [ ] ( ) ≥ m The reason for considering such quotients (extensions of V by Qp 1 ) is to more easily prove the necessary dimension hypotheses for Selmer varieties.( ) Let K Q or quadratic imaginary, p split prime in K 1 H GK , Qp 1 × Qp Qp f ( p ( )) Op ⊗ (Bloch-Kato) 1 H GT , Qp 1 × Qp 0 f ( ( )) OK ⊗ then 1 1 dim H GT ,... < dim H GK ,... . f ( ) f ( p ) m Lemma 2.107 Let U be a quotient of U2 that is an extension of V by Qp 1 , let p be a prime of good reduction for X. ( ) 1. dim Sel U rk J Q ( ) ≤ ( ) 2. 1 dim H GQ , U g + m f ( p ) Then apply lemma when m ρ 1 to get the ﬁniteness result. − We want to construct this quotient U of U2, that’s nonabelian but small enough to make computations practical. Let τ : X X X X × → × x1, x2 x2, x1 ( ) 7→ ( ) be the canonical involution.

51 Deﬁnition 2.108 Correspondences. A correspondence Z Pic X X is ∈ ( × ) symmetric if there are Z1, Z2 Pic X such that ∈ ( )

τ Z Z + π∗ Z1 + π∗ Z2 ∗ 1 2 where π1, π2 are the canonical projections

X X X. × → ♦ Deﬁnition 2.109 Nice correspondences. A nice correspondence Z is one that is non-trivial symmetric and for which the cycle class

1 1 1 ξZ H X H X 1 End H X ∈ ( ) ⊗ ( )( )' ( ) has trace zero. ♦ Lemma 2.110 Suppose J is absolutely simple, and let Z Pic X X a symmetric correspondence. Then the class associated to Z lies in the subspace∈ ( × )

Û2 H1 X 1 H1 X H1 X 1 . ( )( ) ⊆ ( ) ⊗ ( )( ) Moreover Z is nice iﬀ the image of this class in H2 X 1 under the cup product is zero. ( )( ) By the lemma, if Z is a nice correspondence we get a homomorphism

2 Û 1 2 cZ : Qp 1 ker H XQ , Qp ∪ H X , Qp (− ) → ( et( p ) −→ ( Q )) and using that U2 was a central extension

2 ∗ Û 1 coker Qp 1 ∪ V U2 V 1 → ( ( ) −→ ) → → → take the quotient UZ U2 ker c∗ and we have a short exact sequence / ( Z)

1 Qp 1 UZ V 1. → ( ) → → → Twisting and p-adic heights. On 11/12 we deﬁned the Q-pro-unipotent completion. Now say a little more about the Qp-pro-unipotent completion.

et et Zp π X , b Q lim Zp π X , b N [[ 1 ( Q ) p ]] [ 1 ( Q )]/ ←−− where the limit is over all ﬁnite quotients of p-power order. Let I denote the augmentation ideal of

et Qp Zp π X , b Q . ⊗ [[ 1 ( Q ) p ]] Deﬁne the algebra

et et n+1 A b Qp Zp π X , b Q I . n ( ) ⊗ [[ 1 ( Q ) p ]]/

Fix a correspondence Z Pic X X and let UZ denote the corresponding et ∈ ( × ) quotient of U2 .

52 et Deﬁnition 2.111 The mixed extension AZ b is the pushout of A b by ( ) 2 ( )

∗ 2 cl∗ : coker Qp 1 ∪ V ⊗ Qp 1 . Z ( ( ) −→ ) → ( ) ♦ Remark 2.112 AZ b M f GT , Qp , V, Qp 1 ( ) ∈ ( ( )) with respect the I-adic ﬁltration. Recall: Sel U is deﬁned using nonabelian cohomology, satisfying certain conditions. Our( goal) is to get an equation for

X Qp U ( ) and use Nekovář’s construction to deﬁne a map

Sel U Qp. ( ) → A natural analogue of Nekovář’s construction is to start with the input of a 1 cohomology class ξ in H GT , Qp and deﬁne at all bad primes v an algebraic function ( ) 1 H GQ , U Qp ( v ) → which when restricted to 1 H GQ , Qp 1 ( v ( )) is the cup product with ξ. Given a splitting of the Hodge ﬁltration can do this. But to get equality for Selmer varieties better to have a construction with linearity properties analogous to the global height pairing. So here we describe how to embed Sel U into ( ) 1 1 H GT ,... H GT ,... st( ) ⊆ ( ) via twisting. Then apply Nekovář’s construction giving local functions from

Sel U Qp. ( ) → For more on twisting see Serre, Galois Cohomology [79]. Let G be a proﬁnite group U a G-group. Let V be a continuous representation of G with an equivariant action of U. Then for any G-equivariant U-torsor P and also U acts on V on the left. We form the twist of V by P as follows, let denote the equivalence. ∼ 1 p, v p u, u− v ( ) ∼ ( · ) on P V for all u U. Then the twist of V by P × ∈ P V( ) P U V × /∼ has a continuous G-representation structure. Remark 2.113 For all p P the map ∈ v pv 7→ P is a bĳection of V onto V( ) for this reason we say that this is the twist of V by P.

53 Now deﬁne maps 1 τ : H GT , UZ M f GT , Qp , V, Qp 1 f ( ) → ( ( )) P P U AZ b 7→ × Z ( ) 1 τp : H GT , UZ M f GQ , Qp , V, Qp 1 f ( ) → ( p ( )) P P U AZ b 7→ × Z ( ) Remark 2.114 Can show that τ is injective. Deﬁnition 2.115 Let x X Q and P πet X , b, x then ∈ ( ) ∈ 1 ( Q )

AZ b, x τ P P U AZ b . ( ) ([ ]) × Z ( ) ♦

Remark 2.116 For x X Qp , similarly. ∈ ( )

AZ b, x τp P ( ) ([ ]) is the mixed extension of GQp -modules obtained by twisting AZ b . Can also et ( ) deﬁne A1 b, x , IAZ b, x by twisting A b and IAZ b respectively. ( ) ( ) 1 ( ) ( ) Now we can deﬁne our quadratic Chabauty pair. We have

θ : X Qp Qp ( ) → x hp AZ b, x 7→ ( ( )) and using local heights at all bad v T0 we deﬁne ∈ ( ) Õ Ö S hv AZ b, xv : xv X Qv ( ( )) ( ) ∈ ( ) v T0 v T0 ∈ ∈ Theorem 2.117 Kim-Tamagawa ’08. The set S is ﬁnite. Lemma 2.118 Let X be as before, then θ, S is a quadratic Chabauty pair. The endomorphism is that induced by the correspondence( ) Z. The constant is

IAZ b [ ( )] and the bilinear pairing B is the global Nekovář height. Corollary 2.119 If further X has everywhere potentially good reduction, then

θ, S hp AZ b, , 0 ( ) ( ( ( ·) { }) is a quadratic Chabauty pair. Lecture 1? 26/11/2019 Recall: if X Q is a smooth projective curve of genus g 2 then we assume / ≥ r rk J Q g ( ) ρ rk NS JQ 2 ( ) ≥ and J Q J Qp ( ) ⊆ ( ) is ﬁnite index with p good for X. X has everywhere potential good reduction. Then hp AZ b, , 0 ( ( ( ·)) { }) is a quadratic Chabauty pair.

54 Remark 2.120 Hypotheses may seem rather restrictive. Nevertheless there are some interesting examples

1. X Xs 13 the “split Cartan” modular curve of level 13. ( ) + 2. X X0 p X0 p wp the Atkin-Lehner quotient. For p prime. ( ) ( )/ To compute X Qp U we need to compute hp AZ b, x so we need to choose a nice correspondence( ) Z and write the locally analytic( ( )) function

θ : X Qp Qp ( ) → x hp AZ b, x 7→ ( ( )) as a power series on every residue disk of X Qp . By our formula ( )

T T hp AZ b, x χp γ γ β s1 α β s2 α , ( ( )) ( φ − Fil − φ · ( φ) − Fil ( φ)) we see that we want to compute an explicit description of AZ b, x as a filtered φ-module. This we do in two parts: The Hodge filtration( (today)) and Frobenius structure (via solving a p-adic differential equation using Tuitman’s dR algorithm, see later), of certain universal objects AZ . Remark 2.121 Unlike the Frobenius structure which is p-adic, the Hodge filtration has a global meaning and can be computed over Q. Let X Q be a smooth projective curve of genus g 2. And Y X an affine / ≥ ⊆ open, let b Y Qp be integral at p. Suppose ∈ ( ) # X r Y Q d ( )( ) and let L Q be a finite extension over which all points of D XrY are defined. / 0 1 Choose a set ω0, . . . , ω2g+d 2 H YQ, Ω s.t. − ∈ ( ) 1. The differentials ω0, . . . , ω2g 1 are of the second kind on X and form 1 − a symplectic basis for HdR XQ s.t. the cup product is the standard symplectic form with respect( to) this basis. 2. The differentials ω2g , . . . , ω2g+d 2 − are of the third kind on X (i.e. all poles are of order 1).

Let V Y H1 Y and dR( ) dR( )∗ T0,..., T2g+d 2 − be the dual basis. Let AdR AdR b n n ( ) be the universal n-step unipotent object associated to πdR X, b -representation 1 ( ) AdR b this vector bundle carries a Hodge filtration. For our applications to n ( ) computing p-adic heights, take n 2 and let AZ AZ b be a certain quotient dR ( ) of A2 . We will compute the Hodge filtration on AZ via characterization of the Hodge filtration universal property by Hadian. Definition 2.122 Filtered connections. A filtered connection V, is a connection on X together with an exhaustive, descending filtration( ∇)

+ Fili V Fili 1 V · · · ⊇ ⊇ ⊇ · · ·

55 satisfying Griﬃths transversality

i i 1 1 Fil V Fil − V Ω . ∇( ) ⊆ ⊗ ♦ dR Theorem 2.123 Hadian ’11. For all n > 0 the Hodge filtration on An is the unique filtration s.t. dR 1. Fil• makes A , into a filtered connection. ( n ∇) 2. The natural maps induce a sequence of filtered connections

n dR V ⊗ X An An 1 0. dR ⊗ O → → − →

3. the identity element of AdR b lies in Fil0 AdR b . n ( ) n ( ) Remark 2.124 The important part of this theorem is the uniqueness of the Hodge filtration satisfying the above properties, in what we do the sub-bundle 0 Fil of AZ is determined in an explicit trivialization on Y, by writing down a general form for a basis, solving for the coefficients using the fact that it extends uniquely to X and satisfies the 3 conditions. We choose a trivialization

s0 b, : Q V Q 1 Y ∼ AZ b Y ( ·) ( ⊕ dR ⊕ ( )) ⊗ O −→ ( )| s.t. the connection on AZ via this trivialization is given by ∇ 1 s− s0 d + Λ 0 ∇ where 0 0 0 © ª Λ ω 0 0® − η ωTZ 0 « ¬ for some η of the third kind on X in space spanned by

ω2g , . . . , ω2g+d 2 { − }

ω ω0, . . . , ω2g 1 . { − } Z is the matrix of the Tate class with respect to

H1 X H1 X dR( ) ⊗ dR( ) (associated to the correspondence Z). Fix a basis

1, T0,..., T2g 1, S { − } to write down s0. The trivialization allows us to describe the connection on Y; to keep track of the fact that it extends to X, introduce the gauge transformation at all

x D. ∈ Recall. all points x are defined over L Q finite. In a formal neighborhood of / such a point x with local coordinate tx we can find a trivialization of AZ

∼ sx : Q VdR Q 1 L tx , d AZ L tx , (( ⊕ ⊕ ( )) ⊗ [[ ]] ) −→( | [[ ]] ∇)

56 since AZ is unipotent and any unipotent connection on a formal disk is trivial. We have a gauge transformation

1 Cx sx− s0 this is unipotent and satisﬁes

1 C−x dCx Λ.

Expanding out this equation we get that Cx is of the following form:

1 0 0 ©Ω ª Cx x 1 0® . g ΩTZ 1 « x x ¬ Where dΩx ω − T dgx Ω Z dΩx η. x − Lemma 2.125 BDMTV 4.10. The diﬀerential η in Λ is the unique diﬀerential satisfying 1. η is in the space spanned by

ω2g , . . . , ω2g+d 2 . { − } 2. The connection extends to a holomorphic connection on the whole of X. ∇ Proof. The deﬁning equations of the gauge transformation Cx imply that

T Res Ω Z dΩx η 0 ( x − ) for all x D. Since the kernel of ∈ 1 d H YQ L dR( ) → 1 given by the residues at all d points x D is precisely HdR X , so the ﬁrst condition implies that η is unique. ∈ ( )

Now we can describe the Hodge ﬁltration on AZ with respect to our chosen trivialization, s0 i.e. an explicit isomorphism

Fil s : Q V Q 1 Y ∼ AZ (( ⊕ dR ⊕ ( )) ⊗ O ) −→ that respects the Hodge ﬁltration on both sides, where the LHS is given by

1 Fil− Q V Q 1 Y ( ⊕ dR ⊕ ( )) ⊗ O 0 0 Fil Q Fil V Y ( ⊕ dR) ⊗ O Fil1 0.

So we just need to describe Fil0, deﬁne

γ Y Fil ∈ O T g bFil bg ,..., b2g 1 Q ( − ) ∈ by the requirement that γ b 0 Fil( )

57 and for all x D: ∈ T T T T gx + γ b N Ωx Ω ZNN Ωx L tx Fil − Fil − x ∈ [[ ]] where T N 0g , 1g M2g g Q . ( ) × ( ) 0 Then a basis for Fil AZ w.r.t. s0 is given by

1 + γFilS, Tg + bg S,..., T2g 1 + b2g 1S . { − − } That is we may choose sFil s.t. the restriction of

1 Fil s0− s

0 Q Fil V Y given by the 2g + 2 g + 1 matrix ( ⊕ dR) ⊗ O ( ) × ( ) 1 0 © ª 0 N ® . γ bT « Fil Fil¬ Algorithm for Hodge ﬁltration

1. Compute local coordinates at each x D ∈ 2. For each x D compute Laurent series expansions ω at x to large precision, ∈

3. Compute Ωx deﬁned by dΩx ωx − 4. solve for η as the unique

ω2g , . . . , ω2g+d 2 − s.t. dΩ has residue zero at all x D. ∈ 5. Solve the system of equations for gx.

Remark 2.126 For hyperelliptic curves βFil and γFil are both zero. Lecture 1? 3/12/2019 Recall: Goal is to compute hp AZ b, x : Consider certain rank(2g +( 2 vector)) bundle

Z A our working quotient of 2. A

Z A has the structure of a ﬁltered connection which we computed last time.

58 The importance of Z having ﬁltered connection is that the base change A of Z to Qp has a Frobenius structure and then we have an isomorphism of ﬁlteredA φ-modules

x∗ Z Dcris Z b, x for all x X Qp . A (A ( )) ∈ ( ) Today: describe frobenius structure on isocrystal

ri g A b¯ Z ( ) and compute this using universal properties. What is an isocrystal? (for now affine story, for more details on the rigid picture, see Appendix to B.-Dogra-Müller-Tuitman-Vonk) Let A be an affinoid algebra and A† its weak completion. Let A¯ A† π. The idea behind isocrystals and its relation to iterated Coleman integrals/ is that the iterated Coleman integral can be thought of as a solution to a certain p-adic differential equation (with certain additional structure). The iterated Coleman integral ¹ ωn ωn 1 ω1 − ··· is the yn coordinate of a solution to the following system of differential equations: 0 0 0 ··· ©ω1 0 0ª ··· ® 0 ω2 0® ··· ® dy Ωy, Ω . . . .® . . . . . .® ® 0 0 0® ··· 0 0 ωn 0 « ··· ¬ With y0 1. This is a unipotent differential equation

y0 © . ª y . ® . . ® y « n¬ Deﬁnition 2.127 A unipotent isocrystal on A¯ is a A†-module M together with an (integrable) connection.

: M Ω1 K ∇ → ⊗A† (⊗ ) that is an iterated extension of trivial connections where a trivial connection is

1 A†, d . ( )

A morphism of unipotent isocrystals is a map of A†-modules, that is horizontal, i.e. commutes with connection. We denote the category of unipotent isocrystals as Un A¯ . By Berthelot it only depends on A¯ as the notation suggests (in ( ) particular it doesn’t depend on A†. ♦ Example 2.128 Let M Un A¯ be rank 2 then it sits in ∈ ( ) 0 1 M 1 0 → → → → which is non-canonically split. Its isomorphic to the object having underlying

59 module 2 A† ( ) and connection 0 0 d ∇ − ω 0 we have Ext1 1, 1 H1 A¯ . ( )' MW ( ) For more see Besser’s Heidelberg lectures on Coleman integration. There is an analogous category of unipotent isocrystals via rigid triples ri g (see Appendix of BDMTV). Let Fp be the category of (rigid) unipotent isocrystals on the special ﬁbre ofCX. (X ) There is an action of frobenius on path torsors

ri g π F , b¯ , x 1 (X p ) ri g rig of π1 , and n-step unipotent quotients of π1 .

φn : An b¯ , x¯ An b¯ , x¯ ( ) → ( ) rig ¯ there is a frobenius structure on n b the universal n-step object. The frobenius structure is an isomorphismA ( )

rig rig Φn : φ∗ b¯ ∼ b An ( ) −→An ( ) of overconvergent unipotent isocrystals. There is a corresponding de Rham realization, the pull back of frobenius to F gives frobenius on X p dR π XQ , b, x 1 ( p ) and a frobenius operator on the n-step unipotent quotients

dR dR φn b, x : A b, x A b, x . ( ) n ( ) → n ( ) Chiarellotto, B., & Le Stum, B. (1999). F-isocristaux unipotents. Compositio Mathematica, 116(1), 81-110. doi:10.1023/A:1000602824628 Theorem 2.129 Chiarellotto-Le Stum 99. There is an equivalence of categories

dR rig XQ ∼ F C ( p ) −→C (X p ) an given by the analytiﬁcation functor . For any x X Qp with reduction x¯ we have a canonical isomorphism of ﬁbre(·) functors ∈ ( )

an ix : x¯∗ x∗ ◦ (·) ' such that if x, y belong to the same residue disks the canonical isomorphism τx,y 1 ix i is given by parallel transport along the connection. ◦ −y Now let b0 and x0 be Teichmuller representatives of b, x respectively. This is how we relate our frobenius operator φn b, x to the isocrystal rig ¯ ( ) An b we have ( ) 1 φn b, x τb x φn b0, x0 τ− ( ) , ◦ ( ) ◦ b,x where τb,x is from Chiarellotto-Le Stum.

60 dR dR We can describe τb,x on An b, x is a quotient of An Y b, x so suﬃces to describe parallel transport here.( ) ( )( ) Recall we’ve ﬁxed s0

n Ê i dR s0 b, x : V Y ⊗ ∼ A b, x ( ) dR( ) −→ n ( ) i0 for any x1, x2 X Qp that lie in the same residue disk. Deﬁne ∈ ( ) Õ ¹ x2 I x1, x2 1 + w ω0, . . . , ω2g+d 2 ( ) ( − x x1 in n Ê i V Y ⊗ dR( ) i0 where the sum is over all words in T0,..., T2g+d 2 of length at most n make { − } the substitution with ωi for Ti. Here the integrals are given by formally integrating power series (since dR x1, x2 are in the same residue disk). Then τb x when considered on Y , An ( ) via s0.

n n Ê i Ê i τb x : V Y ⊗ ∼ V Y ⊗ , dR( ) −→ dR( ) i0 i0

v I x0, x I b, b0 7→ ( ) ∨ ( ) then by Besser’s theory of Coleman integration on unipotent connections, we have that for any b, b0, x, x0 Y Qp ∈ ( ) the same formula describes the unique frobenius equivariant isomorphism

dR dR A b, b0 ∼ A x, x0 n ( ) −→ n ( ) if the above is interpreted via Coleman integration.

Finally by taking quotient of the various A2∗ by our nice correspondence Z we get frobenius operators

dR dR φZ b, x : A b, x A b, x ( ) Z ( ) → Z ( ) rig ¯ and a quotient AZ b of the universal 2-step object. Chiarellotto-Le( Stum) give us an isomorphism

rig rig ΦZ : φ∗A b¯ A b¯ Z ( ) → Z ( ) we have the following equalities.

φZ b0, x0 x∗ ΦZ ( ) 0 1 φZ b, x τb x φZ b0, x0 τ− . ( ) , ◦ ( ) ◦ b,x The connections on dR an A b Y Z ( ) | and dR an φ∗A b Y Z ( ) |

61 are described with respect to s0 and are equal to d + Λ (as before) and d + Λφ, where 0 0 0 © ª Λφ φ∗ω 0 0® − φ η φ ω Z 0 « ∗ ∗ ∗ ¬ so to make the frobenius structure explicit we must compute G s.t.

ΛφG + dG GΛ Φ 1 where G −Z (inverse of frobenius structure). Proposition 2.130 Can take G as follows

1 0 0 © ª G f F 0® h gT p « ¬ where + φ∗ω Fω d f   f b0 0  ( ) dgT d f TZF  + T + dh ωFZ f d f Z f gω φ∗η pη  − − h b0 0  ( ) Here is the algorithm for frobenius structure on AZ 1. Use Tuitman’s algorithm to compute matrix of frobenius F and overconvergent functions f s.t. φ∗ω Fω + d f .

2. Compute the matrix + A I x, x0 I b0, b − ( ) ( ) where we deﬁne for any pair x1, x2 X Qp parallel transport matrices ∈ ( ) 1 0 0 ∫ x2 © ω 1 0ª I± x1, x2 x1 ® ( ) ∫ x2 ∫ x2 ® η + ωTωZω 0 1 « x1 x1 ¬ solve p-adic diﬀerential equation.

References References

[1] Apostol, Tom M. Introduction to analytic number theory. Springer Science & Business Media, 2013. [2] Arul, V., Best, A. J., Costa, E., Magner, R., & Triantaﬁllou, N. Computing zeta functions of cyclic covers in large characteristic.Proceedings of the Thirteenth Algorithmic Number Theory Symposium, The Open Book Series, 2(1), 37-53, 2019. [3] Balakrishnan, Jennifer S., Robert W. Bradshaw, and Kiran S. Kedlaya. Ex- plicit Coleman Integration for Hyperelliptic Curves. In ANTS-IX 2010, LNCS 6197, pp. 16-31, 2010.

62 [4] Balakrishnan, Jennifer S. Coleman integration for even-degree models of hyperelliptic curves. LMS Journal of Computation and Mathematics 18.1 (2015): 258-265. [5] Balakrishnan, Jennifer S. Iterated Coleman integration for hyperelliptic curves. The Open Book Series 1.1 (2013): 41-61.

[6] Balakrishnan JS, Dogra N. Quadratic Chabauty and rational points I: p-adic heights. arXiv preprint arXiv:1601.00388. 2016 Jan 4. [7] Balakrishnan JS, Dogra N, Müller JS, Tuitman J, Vonk J. Explicit Chabauty- Kim for the Split Cartan Modular Curve of Level 13. arXiv preprint arXiv:1711.05846. 2017 Nov 15. [8] Balakrishnan JS, Tuitman J. Explicit Coleman integration for curves. arXiv preprint arXiv:1710.01673. 2017 Oct 4. [9] Balakrishnan, J., Dan-Cohen, I., Kim, M., and Wewers, S. (2012). A nonabelian conjecture of Birch and Swinnerton-Dyer type for hyperbolic curves. arXiv preprint arXiv:1209.0640. [10] Baldassarri, Francesco, and Bruno Chiarellotto. Algebraic Versis Rigid Cohomology with Logarithmic Coeﬃcients. In Barsotti Symposium in Al- gebraic Geometry, edited by Valentino Cristante and William Messing, 15:11–50. Perspectives in Mathematics. Academic Press, 1994. https:// doi.org/10.1016/B978-0-12-197270-7.50007-3. [11] Banagl, Markus. Topological invariants of stratiﬁed spaces. Springer Science & Business Media, 2007. [12] Berkovich, Vladimir G. Integration of One-forms on P-adic Analytic Spaces. (AM-162). No. 162. Princeton University Press, 2007.

[13] Berthelot, Pierre.Cohomologie Cristalline des Schémas de Caractéristique p > 0 LNM 407, Springer, 1974. [14] Berthelot, Pierre. Finitude et pureté cohomologique en cohomologie rigide https://perso.univ-rennes1.fr/pierre.berthelot/publis/Finitude.pdf

[15] Besser, Amnon. Coleman integration using the Tannakian formalism. Mathe- matische Annalen 322, no. 1 (2002): 19-48. [16] Besser, Amnon. Heidelberg lectures on Coleman integration. In The Arith- metic of Fundamental Groups, pp. 3-52. Springer, Berlin, Heidelberg, 2012.

p [17] Besser, Amnon, and Rob De Jeu. Li( )-Service? An Algorithm for Comput- ing p-Adic Polylogarithms. Mathematics of Computation 77, no. 262 (2008): 1105–34. [18] Besser, Amnon. Syntomic regulators and p-adic integration I: Rigid syntomic regulators. Israel Journal of Mathematics 120, no. 2 (2000): 291-334. [19] Besser, Amnon. Syntomic regulators and p-adic integration II: K 2 of curves. Israel Journal of Mathematics 120, no. 2 (2000): 335-359. [20] Best, A. Explicit Coleman integration in larger characteristic. Proceedings of the Thirteenth Algorithmic Number Theory Symposium, The Open Book Series, 2(1), 85-102, 2019.

63 [21] S. Bosch, U. Güntzer, R. Remmert, Non-Archimedean Analysis, Grundlehren der mathematischen Wissenschaften 261, Springer, 1984. MR 86b:32031 [22] Bloch, Spencer. Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. Vol. 11. American Mathematical Soc., 2000.

[23] Bochner, S. Formal Lie Groups. Annals of Mathematics, vol. 47, no. 2, 1946, pp. 192–201. JSTOR, JSTOR, www.jstor.org/stable/1969242. [24] Borel, Armand. Sur La Cohomologie Des Espaces Fibres Principaux Et Des Espaces Homogenes De Groupes De Lie Compacts. Annals of Mathematics, Second Series, 57, no. 1 (1953): 115-207. doi:10.2307/1969728. [25] Bostan, Alin, Gaudry, Pierrick, and Schost, Éric. Linear recurrences with polynomial coeﬃcients and application to integer factorization and Cartier-Manin operator. SIAM Journal on Computing 36, no. 6 (2007): 1777-1806. [26] Boston, Nigel. Explicit deformation of Galois representations. Inventiones mathematicae 103.1 (1991): 181-196. [27] Boston, Nigel, and Barry Mazur. Explicit universal deformations of Galois representations. Algebraic Number Theory—in honor of K. Iwasawa. Math- ematical Society of Japan, 1989.

[28] Booker, Andrew R., Jeroen Sĳsling, Andrew V. Sutherland, John Voight, and Dan Yasaki. A Database of Genus 2 Curves over the Rational Numbers. LMS Journal of Computation and Mathematics 19, no. A (2016): 235–54. https://doi.org/10.1112/S146115701600019X. [29] Castryck, Wouter, and Jan Tuitman. Point Counting on Curves Using a Gonality Preserving Lift. ArXiv:1605.02162 [Math], May 7, 2016. http:// arxiv.org/abs/1605.02162. [30] Chen, Kuo-Tsai. Algebras of iterated path integrals and fundamental groups. Transactions of the American Mathematical Society 156 (1971): 359-379. [31] Coleman, Robert F. Torsion points on curves and p-adic abelian integrals. Annals of Mathematics 121.1 (1985): 111-168. [32] Coleman, Robert F. Eﬀective Chabauty Duke Math. J 52.3 (1985): 765-770. [33] Coleman, Robert, and Ehud De Shalit. p-adic regulators on curves and special values of p-adic L-functions. Inventiones mathematicae 93, no. 2 (1988): 239- 266. [34] Coleman, Robert F. Dilogarithms, regulators and p-adic L-functions. Inven- tiones mathematicae 69, no. 2 (1982): 171-208. [35] Coleman, Robert F., and B. Gross. p-adic Heights on Curves Math. Sciences Research Inst., Berkeley, Calif. (1987).

[36] Conrad, Brian David. Arithmetic algebraic geometry. Vol. 9. American Math- ematical Soc., 2001. [37] Edixhoven, Bas. Point Counting after Kedlaya, EIDMA-Stieltjes Graduate Course, Leiden, September 22–26, 2003, n.d., 23.

[38] Fresnel, Jean, and Marius Van der Put Rigid analytic geometry and its applications Vol. 218. Springer Science & Business Media, 2012.

64 [39] Gabber O, Ramero L. Almost ring theory. Springer; 2003 Dec 15. [40] Gaudry, Pierrick, and Nicolas Gürel. An extension of Kedlaya’s point- counting algorithm to superelliptic curves. Advances in Cryptology - ASI- ACRYPT 2001, Springer, Berlin, Heidelberg, 2001.

[41] Gerritzen, L. and Van der Put, M., 2006. Schottky groups and Mumford curves (Vol. 817). Springer. [42] Gonçalves, Cécile. A point counting algorithm for cyclic covers of the projective line. Contemporary mathematics 637 (2015): 145. [43] Goncharov, Alexander. Mixed elliptic motives. London Mathematical Soci- ety Lecture Note Series (1998): 147-222. [44] Goncharov, Alexander B., and Andrey M. Levin. Zagier’s conjecture on L (E, 2). Inventiones mathematicae 132, no. 2 (1998): 393-432. [45] González, Josep, Jordi Guardia, and Victor Rotger. Abelian surfaces of GL2- type as Jacobians of curves. arXiv preprint math/0409352 (2004). [46] Harrison, Michael C. An extension of Kedlaya’s algorithm for hyperelliptic curves. Journal of Symbolic Computation 47.1 (2012): 89-101. [47] Hartshorne, Robin. Algebraic geometry. Vol. 52. Springer Science & Busi- ness Media, 2013.

[48] Harvey, David. Counting points on hyperelliptic curves in average polynomial time. Annals of Mathematics 179, no. 2 (2014): 783-803. [49] Harvey, David. Kedlaya’s Algorithm in Larger Characteristic. IMRN: Inter- national Mathematics Research Notices 2007 (2007).

[50] Harvey, David, and Andrew V.Sutherland. Computing Hasse–Witt matrices of hyperelliptic curves in average polynomial time. LMS Journal of Computation and Mathematics 17, no. A (2014): 257-273. [51] Harvey, David, and Andrew V. Sutherland. Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time II. Frobenius distributions: Lang–Trotter and Sato–Tate conjectures, Contemporary Mathematics 663 (2016): 127-148. [52] Hasegawa, Yuji, and Mahoro Shimura. Trigonal Modular Curves. Acta Arithmetica 88, no. 2 (1999): 129–40. https://doi.org/10.4064/aa-88-2- 129-140.

[53] Hida, Haruzo. Geometric modular forms and elliptic curves. World Scientiﬁc, 2012. [54] Hulsbergen, Wilfred WJ. Conjectures in arithmetic algebraic geometry. Braun- schweig: Vieweg, 1992.

[55] Jannsen, Uwe. Continuous Étale Cohomology. Mathematische Annalen 280.2 (1988): 207-246. [56] Jeﬀ Bezanson, Alan Edelman, Stefan Karpinski and Viral B. Shah, Julia: A Fresh Approach to Numerical Computing. (2017) SIAM Review, 59: 65–98. doi: 10.1137/141000671. https://julialang.org/research/ julia-fresh-approach-BEKS.pdf.

65 [57] Katz, Nicholas. Serre-Tate local moduli. In Surfaces algébriques, pp. 138- 202. Springer, Berlin, Heidelberg, 1981. [58] Katz, Eric, and David Zureick-Brown. The Chabauty–Coleman Bound at a Prime of Bad Reduction and Cliﬀord Bounds for Geometric Rank Functions. Compositio Mathematica 149, no. 11 (November 2013): 1818–38. https: //doi.org/10.1112/S0010437X13007410. [59] Kedlaya, Kiran S. Counting points on hyperelliptic curves using Monsky- Washnitzer cohomology, J. Ramanujan Math. Soc. 16 (2001), no. 4, 323-338; errata, ibid. 18 (2003), 417--418.

[60] Lang, Serge, Algebra, Graduate Texts in Mathematics 1.211 (2002): ALL- ALL. [61] Lang, Serge. Algebraic number theory. Vol. 110. Springer Science & Business Media, 2013. [62] Lee, John M. Smooth manifolds. Springer, New York, NY, 2003. 9780387954486. [63] Le Gall, François. Faster algorithms for rectangular matrix multiplication. In Foundations of Computer Science (FOCS), 2012 IEEE 53rd Annual Sym- posium on, pp. 514-523. IEEE, 2012.

[64] Qing Liu. Algebraic geometry and arithmetic curves. Vol. 6. Oxford University Press, 2002. [65] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235–265. [66] Matsumura, Hideyuki. Commutative ring theory. Vol. 8. Cambridge university press, 1989. [67] Matsumura, Hideyuki. Commutative algebra. Vol. 120. New York: WA Benjamin, 1970. [68] Mazur, Barry, William Stein, and John Tate. Computation of p-adic heights and log convergence. Doc. Math (2006): 577-614. [69] Mazur, Barry, John Tate, and Jeremy Teitelbaum. On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Inventiones mathematicae 84, no. 1 (1986): 1-48. [70] Milne, James S. Etale cohomology (PMS-33). Vol. 33. Princeton university press, 2016. [71] Minzlaﬀ, Moritz. Computing zeta functions of superelliptic curves in larger characteristic. Mathematics in Computer Science 3.2 (2010): 209-224. [72] Mumford, David. Abelian varieties. Vol. 5. Oxford University Press, USA, 1974. [73] Narkiewicz, Wladyslaw Elementary and Analytic Theory of Algebraic Num- bers. Springer, 2004. [74] Claus Fieker, William Hart, Tommy Hofmann and Fredrik Johansson, Nemo/Hecke: Computer Algebra and Number Theory Packages for the Julia Pro- gramming Language. In: Proceedings of ISSAC ’17, pages 157–164, New York, NY, USA, 2017. ACM.

66 [75] Polishchuk, Alexander. Abelian varieties, theta functions and the Fourier transform. Vol. 153. Cambridge University Press, 2003. [76] Rolshausen, Klaus, and Norbert Schappacher. On the second K-group of an elliptic curve. Journal für die reine und angewandte Mathematik 495 (1998): 61-77.

[77] SageMath, the Sage Mathematics Software System (Version 8.1.0), The Sage Developers, 2017, http://www.sagemath.org. [78] Serre, Jean-Pierre. A course in arithmetic. Vol. 7. Springer Science & Busi- ness Media, 2012.

[79] Serre, Jean-Pierre. Galois cohomology. Springer Science & Business Media, 2013. [80] Serre, Jean-Pierre. Linear representations of ﬁnite groups. Springer Science & Business Media, 1977.

[81] Serre, Jean-Pierre. Local algebra. Springer Science & Business Media, 2000. [82] Serre, Jean-Pierre, Martin Brown, and Michel Waldschmidt. Lectures on the Mordell-Weil theorem. Vol. 2. Braunschweig: Vieweg, 1990. [83] Shafarevich, Igor R., and Alexey O. Remizov. Linear algebra and geometry. Springer Science & Business Media, 2012.

[84] Silverman, Joseph H. The arithmetic of elliptic curves. Vol. 106. Springer Science & Business Media, 2009. [85] Silverman, Joseph H. Advanced topics in the arithmetic of elliptic curves. Vol. 151. Springer Science & Business Media, 2013.

[86] Stacks Project Authors, The Stacks Project, https://stacks.math.columbia. edu. [87] Stoll, Michael. Independence of Rational Points on Twists of a given Curve. Compositio Mathematica 142, no. 5 (September 2006): 1201–14. https: //doi.org/10.1112/S0010437X06002168.

[88] Stoll, Michael. Uniform Bounds for the Number of Rational Points on Hy- perelliptic Curves of Small Mordell–Weil Rank. Journal of the European Mathematical Society 21, no. 3 (December 12, 2018): 923–56. https: //doi.org/10.4171/JEMS/857.

[89] Sutherland, Andrew. Isogeny volcanoes. The Open Book Series 1, no. 1 (2013): 507-530. https://msp.org/obs/2013/1-1/obs-v1-n1-p25-s.pdf. [90] Tamme, Günter. Introduction to étale cohomology. Springer Science & Busi- ness Media, 2012. [91] Towse, Christopher. Weierstrass Points on Cyclic Covers of the Projective Line. Transactions of the American Mathematical Society 348, no. 8 (1996): 3355–3378. [92] Tuitman, Jan. Counting points on curves using a map to P1. Mathematics of Computation 85.298 (2016): 961-981.

[93] Tuitman, Jan. Counting points on curves using a map to P1, II. Finite Fields and Their Applications 45 (2017): 301-322.

67 [94] van der Geer, G., Moonen, B. Abelian Varieties, from https://www.math.ru. nl/~bmoonen/research.html#bookabvar. [95] Vélu, Jacques. Isogénies entre courbes elliptiques. CR Acad. Sci. Paris, Séries A 273 (1971): 305-347.

[96] Voight, John. Quaternion Algebras. http://quatalg.org [97] Washington, Lawrence C. Introduction to cyclotomic ﬁelds. Vol. 83. Springer Science & Business Media, 1997. [98] Waterhouse, William C. Proﬁnite groups are Galois groups. Proceedings of the American Mathematical Society 42.2 (1974): 639-640.

[99] Weng, L., and Nakamura, I. Arithmetic geometry and number theory, World Scientiﬁc, 2006. [100] Zagier, Don. The Bloch-Wigner-Ramakrishnan polylogarithm function. Math- ematische Annalen 286, no. 1 (1990): 613-624.

[101] Zagier, D. Modular Points, Modular Curves, Modular Surfaces and Modu- lar Forms. In Arbeitstagung Bonn 1984, edited by Friedrich Hirzebruch, Joachim Schwermer, and Silke Suter, 225–48. Lecture Notes in Mathemat- ics. Springer Berlin Heidelberg, 1985. [102] Pug Template Engine see pug-lang.com.