Author: Eiko

Tags: p-adic, chabauty, diophantine geometry

Time: 2024-11-25 09:56:27 - 2024-11-25 09:56:27 (UTC)

These are notes the author took at the conference in Edinburgh, ICMS, 2024.

A brief History Of Nonabelian Chabauty

Talk given by Kiran S. Kedlaya, UCSD, at ICMS Edinburgh 2024 Nov 25.

Infinite Descent

Invented by Fermat, to prove x4+y4=z2 has no rational solutions.

  • Mordell used it ot show XQ is f.g. for g=1

  • g>1, Jacobians

The section conjecture

1π1(XQ,x)π1(X,x)GQ1

(ppt, unnoted)

Unexpected Algebraic Points In Non-Abelian Chabauty Loci

(ppt, unnoted)

Quadratic Chabauty For Modular Curves

Joint with Amnon Besser, Alexander Betts, Francesca Bianchi, Netan Dogra, Daniel Hast,

Motivation

Rouse0Sutherland-Zurieck-Brown) described the classification of possibpl images of l-adic Galois representation s attached to elliptic curves over Q (Mazur’s Program B). They do quite a lot technieuqes, adn identified some challenges for l=3,5,7,11. They end up classify rational points on almost all modular curves, aside from those dominating:

  • Xns+(N) for N=27,25,49,121 and two other level 49 curves and Xns+(l) for l>19.

  • Computig Xns+(27)(Q would finish the classification of 3-adic images of Galois.

What do we know about Xns+(27)?

  • genus 12, rank 12,

  • SAtisfy hypothesis of QC for modular curves (B-Dogra-Muller-Tuitman-Vonk), yet computationally challenging.

Prop RSZB

  • There exists a smooth plane quartic XH over K=Q(ζ3) together with with a degrer 3 morphism from Xns+(27)XH defined over K.

  • ResK/Q(J(XH))Q-simple abelian variety associated to newform orbit 729, 2.a.c. of rank 6.

The idea is by combinint QC with restriction of scalaers, should be possible to compute (a finite set containing) XH(K), since r[K:Q](g1)+(r2+1)(rns1), which is 2(31)+(1+1)(31)=8.

Goal

  • Talk about QC for r=g for integral points on affine hypier curves (B-Besser-Muller 2016).

  • Then restriction of scalers we get qC for OK-points on hyperelliptic curves K=points on biell curves with larger Jacobien rank. (B-Besser-Bianchi-Muller-2021)$.

  • By Nekovar heighte and Hecke correspondence we get QC for rational points on modulear curves (B-Dogra-Muller-Tuitman-Vonk 2019, 2023).

  • Combine them today we can have QC for K-rational points on modular curves with higher rank.

Part I

There is a concrete theorem we can write down (BBM). Let X/Q be a hyperelliptic curve y2=f(x) of genus g, f monic and of odd degree, let r=rankJ(Q) and suppose r=g and fi=xωi for ωiH0(X,Ω1) are linear independent, then there is an explicitely computable finite set ΩQp and explicitely computable constnts αijQp such that

θ(P)0i,jg1αijfifj(P)

takes values in Ω in the integral points and

θ(P)=ig1Pωiωi

Proof(idea)

Use the Coleman-Gross height pariing h which is bilinear h:J×JQp and rewrite in terms of a natural basis for the space of bilinear forms on J, this is given by (ωiωj) of Coleman integrals, This uses a finite index subgroup of J(Q).

Then h=hp+vhv decomposes as global height and sum of local heights, the local height hp(P,P)=Pωiωi.

hhp={vphv}

The RHS is a finite set of values on integral points, can calculate it. This amountes to solving p-adic power series equations.

Part II QC with restriction of scalaers

Remark D,H proved finiteness theorems for various nonabelian Chabauty sets for curves using unlikely intersections. In BBBM, they give explicit method, outline how to combine QC + restriction of scalars (RoS).

The idea is to use more (Coleman-Gross) p-adic heights.

Suppose X is a curve on K number field, the set of canonical p-adic height parings are in 11 correspondence with the set of Zp extensions L/K.

The p-adic height depends on choice of idele class character.

Example K=Q(σ)

  • with σ>0, just have cyclic p-adic height. If

  • with σ<0, have two independent p-adic heights cyclic and anti-cyclic.

Should Work Condition

Sisek 2013 gave Chabauty and RoS that should work when rrankJ(K)d(g1). Consider X/K, [K:Q]=d adn r=rankJ(K), V=ResK/QX, A=ResK/QJ. dimQA=d and dimQA=gd.

Consider X(K)V(Q)V(Qp)A(Q)A(Qp), with d,r,gd respectively. Since r+dgd, we expect the intersetcion is finite.

This implies QC and RoS should work if rd(g1)+(r2+1).

Theorem BBBm Let X=Xσ+(91), given as y2=x63x4+19x21. rankJ0+(91)(Q)=2 and rankJ0+(91)(Q(i))=4.

Can apply QC (and MW sieve) to compute X0+(91)(Q(i)).

Part III

For Q-points on modular curves, the idea is want to generalize hhp is som set of values of vphv.

The problem is that we cannot control the local heights away from p via Coleman-Gross use Nekovars heights.

Theorem (BDMTV) Tehre exists an algorithmn that takes as input eht following objects:

  • A modular curve X/Q of genus g and rank rankJ(Q)=g and rankNS(JQ)>1.

  • A covering of X by affines that are birational to a plane curve cut out by an equation that is monic in 1 variable, p-integral, satisfy some interesting technical conditions (TA1).

  • A prime p of good reduction such that Tp generates End(J).

  • A supply of non-rational points in X(Q)

  • for all l, the local heights hl.

We will get a finite set containing X(Q).

Part IV QC for modular curves + RoS

Combine II + III, seem hopeful if rd(g1)+(rNS+1). Try to carry out III under RoS with Nekovar heights.

Theorem (BBHJM) Let XH over Q(ζ3) be the genus 3 quotient of Xns+(27) of RSZB.

We have #XH(Q(ζ3))=13, #Xns+(27)(Q(ζ3))=10, Xns+(27)(Q)=8.

A finiteness Conjecture For Abelian Varieties Over Number Fields

Given by Akio Tamagawa (RIMS Kyoto) at ICMS Edinburgh 2024 Nov 26.

Let K be a number field, X is a geometrically connected variety over K, there is a fundamental exact sequence of fundamental groups

rendering math failed o.o

where GK=Gal(K/K).

X=PK1{0,1,}

  • ker(ρX)={1} (Belyi)
  • Define a character (K,l):=Kker(ρXl)

    (K,l):=maxpro-l-torsion unramified outside l.

    Gal((K,l)/K)π1(OK[1l]).

Reidemeister Torsion In Arithmetic

Let X=Spec(OF), cH3(A,Z/nZ). ρHomG(π,A)/A

ρcH3(π,Z/nZ)H3(X,Z/n)1nZ/Z,CS[ρ]=inv(ρc).

Does this action funciton give rises to some L-functions?

We will consider smooth projective curves over finite fields. The spaces of representations will be representations of etale fundamental groups. Let t=qs.

L(X,ρ,s)=xXdet(Itdeg(x)ρ(Frobx))|V)1=i=02det(ItF|Hi(Xk,ρ))(1)i

no H^0, no H^2

L(X,ρ,s)=det(ItF|H1(Xq,ρ))1

We want to understand the L-value L(X,ρ,12)=det(IF|Hnl/l2.

The Trivial locus of Ql local systems

Anna Cadoret, joint work with A. Tamagawa

Let k be a number field and X a smooth geoemtrically connected variety over k. l a prime and Vl a Ql-lcoal system,

rendering math failed o.o

Degeneracy locus / Take locus

|X|Vl:={x|X|:Gl,xGl}

Main problem is to understand the arithmetic geometric properties of |X|Vl.

Langlands Duality For Skein Modules

Arithmetic TQFT

Number theory Topology.

Algorithms For Certifying Nontriviality Of Ceresa Cycles

Joint work with Jordan Ellenberg, Adam Logan.

Let X be a nice genus g curve over a field with a point PX(K). This point is used to define an embedding

iP:XJ=Jac(X)=Div0(X)/PDiv(X)=Z0hom(X)Z0rat(X):X[XP]

which is a g dimensional group variety.

History Motivating question how many points does J(K) have? How do we understand points on this variety that have this functorial construction? Mapping them to other groups. Depends on K.

  • K=C there is

    J(C)Abel-JacobiCg/Z2g=H0(X,Ω1)H1(X,Z)

    [XP](ωγω)

  • K a number field. We have Mordell-Weil theorem: J(K) is finitely generated.

    J(K)δSel(J)H1(GK,Het1(XK,Zl)).

Today we are going to study 1-dimensional subvarieties of J=Jac(X).

Z1(J)=ZOne dimensional subvarieties of J defined over K

Example

  • XiPJ,x[XP]

    (if X is not a rational point, a higher degree closed point, you can subtract more P to make it degree 0).

  • XiPJ,x[PX]

  • Cerasa cycle (X,P):=iP(X)iP(X)

A natural filtration on Z1(J) (principal ones, algebraically trivial cycles, like deg zero divisors)

Z1rat(J)Z1alg(J)Z1hom(J)Z1(J)

ker(cyc:Z1(J)H2g2(JK,Zl(g1)))

Question: Hoo large is the quotient? (too hard). How deep in the filtration does C(X,P) live?

Some Answers

  • XXZ1hom(J) since [1] acts trivially on H2g2. Anser depends on X and p.

  • Hyperelliptic curves turns out choose p Weierstrass point, XX=0Z1(J) because hyperelliptic involution induces [1] on the Jacobian.

    rendering math failed o.o

  • Ceresa (1983) for X very general curve of genus g3, C(X,P) does not go deep in the filtration nC(X,P)Z1alg(J) for n0.

    The tool used are degeneration and C Hodge theory.

    (Betts-Liyongoiy) p-adic Hodge theory analogue.

  • B-Harris (1993) X Fermat quartic x4+y4=z4, C(X,P)Z1alg(J).

    Tool: Showing a certain period integral is not integer.

  • Bloch (1984) showed for X Fermat quartic, nC(X,P)Z1alg(J) for n0.

    Tool: Map homologically trivial cycles into someting like Selmer groups, Z1hom(J)H1(GK,H2g3(JK,))

  • Techniques of Block + B.Harris can be used to show degree Fermat curves and quotients.

  • Eskandan-Murty shoewe Fermat curve Xm for p>7 prime that n(C(Xp,P))Z1rat(J) for n0.

    Tool: Dimension reduction using symmetries. Z1hom(J)Z0hom(X).

    Arithmetic of J(Q) is infinite (Gross-Rohv lich).

  • Laga-Shnidman (2024) Picard curves g=3 with Z/3Z action, dimension reduction using Chow motives.

    Tool: Dimension reduction using special cycles.

All of these results apply to curves with automorphisms. Question: what can you say about the Ceresa cycle for a random curve?

Today’s Main Theorem

Input: A nice cerve over numebr field K.

Output: A certificate that C(X,P) has infinite order or does it terminate.

For example when run on 254704 height 1 smooth plane quartics with coefficients {1,0,1} on Magma, the algorithm failed to terminate in 102 cases. i.e. the remaining 254602 curves have provable infite order Ceresa cycles in Z1hom(J)Z1rat(J)=CH1(J).

Remarks

  1. All 102 exceptions have symmetries include known torsion for example y3=x41, y3=x4x.

  2. Our main tool is the l-adic Abel-Jacobi maps. Ongoing work with Besser-Balakrishnan, explicitly compute the p-adic AJ image using iterated Coleman integrals, certifying non-triviality in Gr(J)=Z1hom(J)Z1alg(J).

Tool: Bloch’s l-adic cycle class map

Z1hom(J)lH1(GK,H2g3(JK,Zl(g1))) C(X,b)(vl(x,b))l=v(x)

Ceresa class.

Final goal, show v(X) has infinite order.

Steps:

  1. Compute an upper bound on the number of cm(cycle)tors, using the idea of Kummer sequence and weights.

    Kummer sequence

    H0(GK,MΛM)H1(GK,M)[n] 0N:=det(FrobpI)|3H1(XK,)

  2. Choose an good prime and compute a lower bound Np on ord(v(x)).

    Algo: Compute N and Np for $p$ choosed bound. If there is p such that NpN, then output v(x) has infinite order.

    Theorem(Chebotarev) If GK acts on H1 is maximal and v(X) is inifite order, then Np is unbounded as ϕ vanishes.

    Curves acquire symmetries on reduction mod p,

    Np is the order of the following canonical point on J(XFp)

    [(2g2)PX(Fp)P|X(Fp)|KX]J(XFp)