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 \(x^4+y^4=z^2\) has no rational solutions.
Mordell used it ot show \(X\mathbb{Q}\) is f.g. for \(g=1\)
\(g>1\), Jacobians
The section conjecture
\[1 \to \pi_1(X_{\overline{\mathbb{Q}}},\overline{x}) \to \pi_1(X,\overline{x}) \to G_\mathbb{Q}\to 1\]
(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 \(\mathbb{Q}\) (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:
\(X_{ns}^+(N)\) for \(N= 27,25,49,121\) and two other level \(49\) curves and \(X^+_{ns}(l)\) for \(l>19\).
Computig \(X^+_{ns}(27)(\mathbb{Q}\) would finish the classification of \(3\)-adic images of Galois.
What do we know about \(X_{ns}^+(27)\)?
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 \(X_H'\) over \(K=\mathbb{Q}(\zeta_3)\) together with with a degrer \(3\) morphism from \(X_{ns}^+(27)\to X_H'\) defined over \(K\).
\(Res_{K/\mathbb{Q}} (J(X_H')) \sim \mathbb{Q}\)-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) \(X_H'(K)\), since \(r\le [K:\mathbb{Q}] (g-1) + (r_2+1)(r_{ns}-1)\), which is \(2*(3-1) + (1+1) (3-1) = 8\).
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 \(\mathcal{O}_K\)-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 \(X/\mathbb{Q}\) be a hyperelliptic curve \(y^2=f(x)\) of genus \(g\), \(f\) monic and of odd degree, let \(r=\mathrm{rank}J(\mathbb{Q})\) and suppose \(r=g\) and \(f_i=\int_{\infty}^x\omega_i\) for \(\omega_i\in H^0(X,\Omega^1)\) are linear independent, then there is an explicitely computable finite set \(\Omega\subset \mathbb{Q}_p\) and explicitely computable constnts \(\alpha_{ij}\in \mathbb{Q}_p\) such that
\[ \theta(P) - \sum_{0\le i,j\le g-1} \alpha_{ij} f_i f_j (P) \]
takes values in \(\Omega\) in the integral points and
\[ \theta(P) = \sum_{i\le g-1} \int_{\infty}^P \omega_i \overline{\omega_i} \]
Proof(idea)
Use the Coleman-Gross height pariing \(h\) which is bilinear \(h:J\times J\to \mathbb{Q}_p\) and rewrite in terms of a natural basis for the space of bilinear forms on \(J\), this is given by \((\int \omega_i \int \omega_j)\) of Coleman integrals, This uses a finite index subgroup of \(J(\mathbb{Q})\).
Then \(h = h_p + \sum_{v} h_v\) decomposes as global height and sum of local heights, the local height \(h_p(P-\infty, P-\infty) = \sum \int_\infty^P \omega_i \overline{\omega_i}\).
\[h- h_p = \left\{\sum_{v\neq p} h_v\right\}\]
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 \(1-1\) correspondence with the set of \(\mathbb{Z}_p\) extensions \(L/K\).
The \(p\)-adic height depends on choice of idele class character.
Example \(K=\mathbb{Q}(\sqrt{\sigma})\)
with \(\sigma>0\), just have cyclic \(p\)-adic height. If
with \(\sigma<0\), have two independent \(p\)-adic heights cyclic and anti-cyclic.
Should Work Condition
Sisek 2013 gave Chabauty and RoS that should work when \(r-\mathrm{rank}J(K)\le d(g-1)\). Consider \(X/K\), \([K:\mathbb{Q}]=d\) adn \(r=\mathrm{rank}J(K)\), \(V=\mathrm{Res}_{K/\mathbb{Q}} X\), \(A = \mathrm{Res}_{K/\mathbb{Q}} J\). \(\dim_\mathbb{Q}A = d\) and \(\dim_\mathbb{Q}A = gd\).
Consider \(X(K)\cong V(\mathbb{Q})\subset V(\mathbb{Q}_p)\cap \overline{A(\mathbb{Q})} \subset A(\mathbb{Q}_p)\), with \(d, \le r, gd\) respectively. Since \(r+d\le gd\), we expect the intersetcion is finite.
This implies QC and RoS should work if \(r\le d(g-1) + (r_2 +1)\).
Theorem BBBm Let \(X = X^+_{\sigma}(91)\), given as \(y^2=x^6-3x^4+19x^2-1\). \(\mathrm{rank}J^+_0(91)(\mathbb{Q}) = 2\) and \(\mathrm{rank}J_0^+(91)(\mathbb{Q}(i)) = 4\).
Can apply QC (and MW sieve) to compute \(X^+_0(91)(\mathbb{Q}(i))\).
Part III
For \(\mathbb{Q}\)-points on modular curves, the idea is want to generalize \(h-h_p\) is som set of values of \(\sum_{v\neq p} h_v\).
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 \(X/\mathbb{Q}\) of genus \(g\) and rank \(\mathrm{rank}J(\mathbb{Q})=g\) and \(\mathrm{rank}NS(J_\mathbb{Q})>1\).
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 \(T_p\) generates \(\mathrm{End}(J)\).
A supply of non-rational points in \(X(\mathbb{Q})\)
for all \(l\), the local heights \(h_l\).
We will get a finite set containing \(X(\mathbb{Q})\).
Part IV QC for modular curves + RoS
Combine II + III, seem hopeful if \(r\le d(g-1) + (r_{NS}+1)\). Try to carry out III under RoS with Nekovar heights.
Theorem (BBHJM) Let \(X_H'\) over \(\mathbb{Q}(\zeta_3)\) be the genus \(3\) quotient of \(X_{ns}^+(27)\) of RSZB.
We have \(\# X_H'(\mathbb{Q}(\zeta_3)) = 13\), \(\# X_{ns}^+(27)(\mathbb{Q}(\zeta_3)) = 10\), \(X_{ns}^+(27)(\mathbb{Q}) = 8\).
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
where \(G_K = \mathrm{Gal}(\overline{K}/K)\).
\(X=\mathbb{P}^1_K-\{0,1,\infty\}\)
- \(\ker(\rho_X) = \{1\}\) (Belyi)
Define a character \(\text{山}(K,l):= \overline{K}^{\ker(\rho_X^l)}\)
\(\text{天}(K,l):= \max\text{pro-}l\)-torsion unramified outside \(l\).
\(\mathrm{Gal}(\text{天}(K,l)/K) \to \pi_1(\mathcal{O}_K[\frac{1}{l}])\).
Reidemeister Torsion In Arithmetic
Let \(X=\mathrm{Spec}(\mathcal{O}_F)\), \(c\in H^3(A,\mathbb{Z}/n\mathbb{Z})\). \(\rho\in \mathrm{Hom}_{G}(\pi, A)/A\)
\[ \rho^* c\in H^3(\pi,\mathbb{Z}/n\mathbb{Z}) \to H^3(X,\mathbb{Z}/n)\to \frac{1}{n}\mathbb{Z}/\mathbb{Z}, \quad CS[\rho]=inv(\rho^*c).\]
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 \(t=q^{-s}\).
\[\begin{align*} L(X,\rho,s) &= \prod_{x\in X} \det(I-t^{deg(x)}\rho(\mathrm{Frob}_x)) | V)^{-1} \\ &= \prod_{i=0}^2 \det(I - t F | H^i(X_{\overline{k}},\rho))^{(-1)^{i}} \end{align*}\]
no H^0, no H^2
\[L(X,\rho,s) = \det(I-tF | H^1(X_{\overline{q}},\rho))^{-1}\]
We want to understand the \(L\)-value \(L(X,\rho,\frac{1}{2}) = \det(I-\overline{F}|H^n\in l^*/l^{*2}\).
The Trivial locus of \(\mathbb{Q}_l\) 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 \(\mathcal{V}_l\) a \(\mathbb{Q}_l\)-lcoal system,
Degeneracy locus / Take locus
\[|X|_{\mathcal{V}_l}:= \{ x\in |X| : G_{l,x}^\circ \not\subset G_l^\circ \}\]
Main problem is to understand the arithmetic geometric properties of \(|X|_{\mathcal{V}_l}\).
Langlands Duality For Skein Modules
Arithmetic TQFT
Number theory \(\to\) 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 \(P\in X(K)\). This point is used to define an embedding
\[i_P: X\xhookrightarrow{} J=\text{Jac}(X)=\mathrm{Div}^0(X)/\text{PDiv}(X) = \frac{Z_0^{hom}(X)}{Z_0^{rat}(X)}: \quad X\mapsto [X-P]\]
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 \(K\).
\(K=\mathbb{C}\) there is
\[J(\mathbb{C})\xrightarrow{\text{Abel-Jacobi}} \mathbb{C}^g/\mathbb{Z}^{2g} = \frac{H^0(X,\Omega^1)^\vee}{H_1(X,\mathbb{Z})}\]
\[[X-P]\mapsto \left(\omega\mapsto \int_\gamma \omega\right)\]
\(K\) a number field. We have Mordell-Weil theorem: \(J(K)\) is finitely generated.
\[J(K)\xrightarrow{\delta} \text{Sel}(J) \subset H^1(G_K, H^1_{et}(X_{\overline{K}}, \mathbb{Z}_l)).\]
Today we are going to study \(1\)-dimensional subvarieties of \(J=\text{Jac}(X)\).
\[Z_1(J) = \mathbb{Z}\langle \text{One dimensional subvarieties of } J \text{ defined over } K\rangle\]
Example
\(X\xhookrightarrow{i_P} J, x\mapsto [X-P]\)
(if \(X\) is not a rational point, a higher degree closed point, you can subtract more \(P\) to make it degree \(0\)).
\(X\xhookrightarrow{i^-_P} J, x\mapsto [P-X]\)
Cerasa cycle \((X,P):= i_P(X) - i^-_P(X)\)
A natural filtration on \(Z_1(J)\) (principal ones, algebraically trivial cycles, like deg zero divisors)
\[ Z^{rat}_1(J) \subset Z_1^{alg}(J) \subset Z^{hom}_1(J)\subset Z_1(J)\]
\[ \ker(cyc : Z_1(J)\to H^{2g-2}(J_{\overline{K}}, \mathbb{Z}_l(g-1)))\]
Question: Hoo large is the quotient? (too hard). How deep in the filtration does \(C(X,P)\) live?
Some Answers
\(X-X^-\in Z_1^{hom}(J)\) since \([-1]^*\) acts trivially on \(H^{2g-2}\). Anser depends on \(X\) and \(p\).
Hyperelliptic curves turns out choose \(p\) Weierstrass point, \(X-X^-=0\in Z_1(J)\) because hyperelliptic involution induces \([-1]^*\) on the Jacobian.
Ceresa (1983) for \(X\) very general curve of genus \(g\ge 3\), \(C(X,P)\) does not go deep in the filtration \(nC(X,P)\not\in Z_1^{alg}(J)\) for \(n\neq 0\).
The tool used are degeneration and \(\mathbb{C}\) Hodge theory.
(Betts-Liyongoiy) \(p\)-adic Hodge theory analogue.
B-Harris (1993) \(X\) Fermat quartic \(x^4+y^4=z^4\), \(C(X,P)\not\in Z_1^{alg}(J)\).
Tool: Showing a certain period integral is not integer.
Bloch (1984) showed for \(X\) Fermat quartic, \(nC(X,P)\not\in Z_1^{alg}(J)\) for \(n\neq 0\).
Tool: Map homologically trivial cycles into someting like Selmer groups, \(Z_1^{hom}(J)\to H^1(G_K, H^{2g-3}(J_{\overline{K}}, -))\)
Techniques of Block + B.Harris can be used to show degree Fermat curves and quotients.
Eskandan-Murty shoewe Fermat curve \(X_m\) for \(p>7\) prime that \(n(C(X_p,P))\not\in Z_1^{rat}(J)\) for \(n\neq 0\).
Tool: Dimension reduction using symmetries. \(Z^{hom}_1(J)\to Z_0^{hom}(X)\).
Arithmetic of \(J(\mathbb{Q})\) is infinite (Gross-Rohv lich).
Laga-Shnidman (2024) Picard curves \(g=3\) with \(\mathbb{Z}/3\mathbb{Z}\) 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 \(\frac{Z_1^{hom}(J)}{Z_1^{rat}(J)} = CH_1(J)\).
Remarks
All \(102\) exceptions have symmetries include known torsion for example \(y^3=x^4-1\), \(y^3=x^4-x\).
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 \(\mathrm{Gr}(J) = \frac{Z^{hom}_1(J)}{Z_1^{alg}(J)}\).
Tool: Bloch’s \(l\)-adic cycle class map
\[Z_1^{hom}(J) \to \prod_l H^1(G_K, H^{2g-3}(J_{\overline{K}}, \mathbb{Z}_l(g-1)))\] \[C(X,b)\mapsto (v_l(x,b))_l = v(x)\]
Ceresa class.
Final goal, show \(v(X)\) has infinite order.
Steps:
Compute an upper bound on the number of \(cm(cycle)_{tors}\), using the idea of Kummer sequence and weights.
Kummer sequence
\[H^0\left(G_K, \frac{M}{\Lambda M}\right) \to H^1(G_K, M)[n]\] \[0\neq N:=\det(\mathrm{Frob}_p - I) |_{\wedge^3 H^1(X_{\overline{K}}, )}\]
Choose an good prime and compute a lower bound \(N_p\) on \(\mathrm{ord}(v(x))\).
Algo: Compute N and \(N_p\) for $p$ choosed bound. If there is \(p\) such that \(N_p\not | N\), then output \(v(x)\) has infinite order.
Theorem(Chebotarev) If \(G_K\) acts on \(H^1\) is maximal and \(v(X)\) is inifite order, then \(N_p\) is unbounded as \(\phi\) vanishes.
Curves acquire symmetries on reduction mod \(p\),
\(N_p\) is the order of the following canonical point on \(J(X_{F_p})\)
\[\left[(2g-2)\sum_{P\in X(\mathbb{F}_p)} P - |X(\mathbb{F}_p)|K_X\right] \in J(X_{F_p})\]