arXiv subjects

Yeuk Hay Joshua Lam

Publications and source records attributed to Yeuk Hay Joshua Lam.

At least 19 recordsLinked to original sources

arXiv2026

Murphy's law in non-abelian Hodge theory

We construct explicit examples of non-integral variations of $\mathbb{Q}$-Hodge structures. Our approach leverages Fenchel--Nielsen-type parameterizations, due to Kabaya and Maskit, of the Teichmüller component of relative character varieties. Additionally, we discuss various Diophantine results concerning the $\mathcal{O}_{K,S}$-integral points of such character varieties, and give a new proof of Beauville's classical theorem on families of elliptic curves. We conclude by collecting \emph{pathological behaviors} of the Hodge locus of non-integral $\mathbb{Q}$VHS, in particular the failure of the Cattani--Deligne--Kaplan theorem and the André--Oort conjecture; our results indicate that for $\mathbb{Q}$VHS which are not $\mathbb{Z}$VHS, what can go wrong must go wrong.

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arXiv2026

p-Curvature and Non-Abelian Cohomology

Let $X\to S$ be a smooth projective morphism. Katz proved the Grothendieck-Katz $p$-curvature conjecture for the Gauss-Manin connection on the $i$-th cohomology of $X/S$: if its $p$-curvature vanishes mod $p$ for infinitely many $p$, then the action of $π_1(S,s)$ on $H^i(X_s, \mathbb{Z})$ factors through a finite group. We prove a non-abelian analogue of this statement: if the $p$-curvature of the isomonodromy foliation on the moduli of flat bundles of rank $r$ on $X/S$ vanishes mod $p$ for infinitely many $p$, then the action of $π_1(S,s)$ on the rank $r$ integral characters of $π_1(X_s)$ factors through a finite group. We deduce many new cases of the Bost/Ekedahl--Shepherd-Barron--Taylor conjecture. The proofs rely on a non-abelian version of Katz's formula, and a non-abelian version of the Hodge index theorem.

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arXiv2025

A characteristic p analogue of the André--Pink--Zannier conjecture

We investigate the analogue of the André--Pink--Zannier conjecture in characteristic $p$. Precisely, we prove it for ordinary function field-valued points with big monodromy, in Shimura varieties of Hodge type. We also prove an algebraic characteristic $p$ analogue of Hecke-equidistribution (as formulated by Mazur) for Shimura varieties of Hodge type. We prove our main results by a global and local analysis of prime-to-$p$ Hecke correspondences, and by showing that Weyl special points are abundant in positive characterstic.

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arXiv2025

Local systems which do not come from abelian varieties

For each smooth curve over a finite field, after puncturing it at finitely many points, we construct local systems on it of geometric origin which do not come from a family of abelian varieties. We do so by proving a criterion which must be satisfied by local systems which do come from abelian varieties, inspired by an analogous Hodge theoretic criterion in characteristic zero.

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arXiv2025

On Siegel's problem and Dwork's conjecture for $G$-functions

We answer in the negative Siegel's problem for $G$-functions, as formulated by Fischler and Rivoal. Roughly, we prove that there are $G$-functions that cannot be written as polynomial expressions in algebraic pullbacks of hypergeometric functions; our examples satisfy differential equations of order two, which is the smallest possible. In fact, we construct infinitely many non-equivalent rank-two local systems of geometric origin which are not algebraic pullbacks of hypergeometric local systems, thereby providing further counterexamples to Dwork's conjecture and answering a question by Krammer. The main ingredients of the proof are a Lie algebra version of Goursat's lemma, the monodromy computations of hypergeometric local systems due to Beukers and Heckman, as well as results on invariant trace fields of Fuchsian groups.

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arXiv2025

Algebraicity and integrality of solutions to differential equations

We formulate a conjecture classifying algebraic solutions to (possibly non-linear) algebraic differential equations, in terms of the primes appearing in the denominators of the coefficients of their Taylor expansion at a non-singular point. For linear differential equations, this conjecture is a strengthening of the Grothendieck-Katz $p$-curvature conjecture. We prove the conjecture for many differential equations and initial conditions of algebro-geometric interest. For linear differential equations, we prove it for Picard-Fuchs equations at initial conditions corresponding to cycle classes, among other cases. For non-linear differential equations, we prove it for isomonodromy differential equations, such as the Painlevé VI equation and Schlesinger system, at initial conditions corresponding to Picard-Fuchs equations. We draw a number of algebro-geometric consequences from the proofs.

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arXiv2024

Boundedness of trace fields of rank two local systems

Let $p$ be a fixed prime number, and $q$ a power of $p$. For any curve over $\mathbb{F}_q$ and any local system on it, we have a number field generated by the traces of Frobenii at closed points, known as the trace field. We show that as we range over all pointed curves of type $(g,n)$ in characteristic $p$ and rank two local systems satisfying a condition at infinity, the set of trace fields which are unramified at $p$ and of bounded degree is finite. This proves observations of Kontsevich obtained via numerical computations, which are in turn closely related to the analogue of Maeda's conjecture over function fields. The key ingredients of the proofs are Chin's theorem on independence of $\ell$ of monodromy groups, and the boundedness of abelian schemes of $\mathrm{GL}_2$-type over curves in positive characteristics, obtained using partial Hasse invariants; the latter is an analogue of Faltings' Arakelov theorem for abelian varieties in our setting.

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arXiv2024

On the locus of curves mapping to a fixed target

Suppose $Y$ is a smooth variety equipped with a top form. We prove a simple theorem giving a sharp lower bound on the geometric genus of a family of subvarieties of $Y$, in terms of the dimension of this family. Two elementary applications are presented. On the one hand, we show that for a very general curve $C$ and a very general hypersurface $Y\subset \mathbb P^{n+1}$ of degree $\ge 2n+1$, any map $C \to Y$ is constant. On the other hand, we give a lower bound on the genus of a family of curves with an isotrivial factor in the associated family of Jacobians; we also characterize the families of curves attaining this bound as the families of degree $2$ branched covers of a fixed curve.

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arXiv2024

Constructing abelian varieties from rank 3 Galois representations with real trace field

Let $U/K$ be a smooth affine curve over a number field and let $L$ be an irreducible rank 3 $\overline{\mathbb Q}_{\ell}$-local system on $U$ with trivial determinant and infinite geometric monodromy around a cusp. Suppose further that $L$ extends to an integral model such that the Frobenius traces are contained in a fixed totally real number field. Then, after potentially shrinking $U$, there exists an abelian scheme $f\colon B_U\rightarrow U$ such that $L$ is a summand of $R^2f_*\overline{\mathbb Q}_{\ell}(1)$. The key ingredients are: (1) the totally real assumption implies $L$ admits a square root $M$; (2) the trace field of $M$ is sufficiently bounded, allowing us to use recent work of Krishnamoorthy-Yang-Zuo to construct an abelian scheme over $U_{\bar K}$ geometrically realizing $L$; and (3) Deligne's weight-monodromy theorem and the Rapoport-Zink spectral sequence, which allow us to pin down the arithmetizations using the total degeneration.

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arXiv2023

Frobenius trace fields of cohomologically rigid local systems

Let $X/\mathbb{C}$ be a smooth variety with simple normal crossings compactification $\bar{X}$, and let $L$ be an irreducible $\overline{\mathbb{Q}}_{\ell}$-local system on $X$ with torsion determinant. Suppose $L$ is cohomologically rigid. The pair $(X, L)$ may be spread out to a finitely generated base, and therefore reduced modulo $p$ for almost all $p$; the Frobenius traces of this mod $p$ reduction lie in a number field $F_p$, by a theorem of Deligne. We investigate to what extent the fields $F_p$ are bounded, meaning that they are contained in a fixed number field, independent of $p$. We prove a host of results around this question. For instance: assuming $L$ has totally degenerate unipotent monodromy around some component of $Z$, then we prove that $L$ admits a spreading out such that the $F_p$'s are bounded; without any local monodromy assumptions, we show that the $F_p$'s are bounded as soon as they are bounded at one point of $X$. We also speculate on the relation between the boundedness of the $F_p$'s, and the local system $L$ being strongly of geometric origin, a notion due to Langer-Simpson.

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arXiv2023

Finite braid group orbits on $SL_2$-character varieties

Let X be a 2-sphere with n punctures. We classify all conjugacy classes of Zariski-dense representations $$ρ: π_1(X)\to SL_2(\mathbb{C})$$ with finite orbit under the mapping class group of X, such that the local monodromy at one or more punctures has infinite order. We show that all such representations are "of pullback type" or arise via middle convolution from finite complex reflection groups. In particular, we classify all rank 2 local systems of geometric origin on the projective line with n generic punctures, and with local monodromy of infinite order about at least one puncture.

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arXiv2023

Geometric local systems on the projective line minus four points

Let $J(m)$ be an $m\times m$ Jordan block with eigenvalue $1$. For $λ\in \mathbb{C}\setminus\{0,1\}$, we explicitly construct all rank $2$ local systems of geometric origin on $\mathbb{P}^1\setminus\{0,1,λ, \infty\}$, with local monodromy conjugate to $J(2)$ at $0,1,λ$ and conjugate to $-J(2)$ at $\infty$. The construction relies on Katz's middle convolution operation. We use our construction to prove two conjectures of Sun-Yang-Zuo (one of which was proven earlier by Lin-Sheng-Wang; the other was proven independently from us by Yang-Zuo).

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arXiv2022

Generalized Bockstein maps and Massey products

Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of the rational numbers and to give a new proof of the vanishing of triple Massey products in Galois cohomology.

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arXiv2022

Motivic local systems on curves and Maeda's conjecture

We show that only finitely many complex genus two curves and four punctured spheres admit rank two local systems of geometric origin, and moreover each carries finitely many. This gives further counterexamples to a conjecture of Esnault and Kerz: counterexamples over very general curves were recently obtained by Landesman and Litt. In the second part we prove an analogue of this result in positive characteristic, namely that over $\overline{\mathbb{F}}_p$, only finitely many genus two curves admit non-trivial rank two local systems pulled back from a fixed quaternionic Shimura variety, and the same for $\mathbb{P}^1$ minus four points; conjecturally, every rank two local system arises as such a pullback. This provides results towards Maeda's conjecture on Galois orbits of eigenforms over function fields. The proofs make use of ideas from the work of Landesman and Litt such as isomonodromy, as well as crucially the description of the Goren-Oort strata due to Tian and Xiao.

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arXiv2022

Lifts of supersingular abelian varieties with small Mumford-Tate groups

We investigate to what extent an abelian variety over a finite field can be lifted to one in characteristic zero with small Mumford-Tate group. We prove that supersingular abelian surfaces, respectively threefolds, can be lifted to ones isogenous to a square, respectively product, of elliptic curves. On the other hand, we show that supersingular abelian threefolds cannot be lifted to one isogenous to the cube of an elliptic curve over the Witt vectors.

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arXiv2021

Finiteness of reductions of Hecke orbits

We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimension $g$ only finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga-Satake construction, we also show that only finitely many supersingular $K3$-surfaces admit CM lifts. Our tools include $p$-adic Hodge theory and group theoretic techniques.

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arXiv2020

The Attractor Conjecture for Calabi-Yau variations of Hodge structures

We study attractor points for Calabi-Yau variations of Hodge structures. In particular, for certain moduli spaces which are Shimura varieties, we prove that the attractor points are CM points, thus proving Moore's Attractor Conjecture in these cases. We also study non-BPS examples of attractors, obtaining special points on locally symmetric spaces without hermitian structures, as well as locally symmetric spaces inside Shimura varieties; for the latter we point out a possible analogy with subspaces studied by Goresky-Tai. Finally we give an explicit geometric description of non-BPS attractors in the simplest case.

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arXiv2020

Attractors are not algebraic

The Attractor Conjecture for Calabi-Yau moduli spaces predicts the algebraicity of the moduli values of certain isolated points picked out by Hodge-theoretic conditions. We provide a family of counterexamples to the Attractor Conjecture in all suitably high, odd dimensions conditional on the Zilber-Pink conjecture.

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