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Raju Krishnamoorthy

Publications and source records attributed to Raju Krishnamoorthy.

15 recordsLinked to original sources
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

Constructing abelian varieties from rank 2 Galois representations

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let $\mathbb L$ be a rank $2$, geometrically irreducible $\bar{\mathbb Q}_\ell$-local system on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $E\subset \bar{\mathbb Q}_\ell$, and has bad, infinite reduction at some closed point $x$ of $X\setminus U$. We show that $\mathbb L$ occurs as a summand of the cohomology of a family of abelian varieties over $U$. The argument follows the structure of the proof of a recent theorem of Snowden-Tsimerman, who show that when $E=\mathbb Q$, then $\mathbb L$ is isomorphic to the cohomology of an elliptic curve $E_U\rightarrow U$.

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arXiv2022

Periodic de Rham bundles over curves

In this article, we introduce the notion of periodic de Rham bundles over smooth complex curves. We prove that motivic de Rham bundles over smooth complex curves are periodic. We conjecture that irreducible periodic de Rham bundles over smooth complex curves are motivic. We show that the conjecture holds for rank one objects and rigid objects.

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arXiv2022

Periodicity of Hitchin's uniformizing Higgs bundles

We link the periodicity of Hitchin's uniformizing Higgs bundle with the arithmetic geometry of its underlying curve. Some new relations are discovered. We also speculate on the whole class of periodic Higgs bundles.

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arXiv2022

The Manin-Mumford conjecture in genus 2 and rational curves on K3 surfaces

Let $A$ be a simple abelian surface over an algebraically closed field $k$. Let $S\subset A(k)$ be the set of torsion points $x$ of $A$ such that there exists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in the image of $f$, and $f$ sends a Weierstrass point of $C$ to the origin of $A$. The purpose of this note is to show that if $k$ has characteristic zero, then $S$ is finite -- this is in contrast to the situation where $k$ is the algebraic closure of a finite field, where $S=A(k)$, as shown by Bogomolov and Tschinkel. We deduce that if $k=\bar{\mathbb{Q}}$, the Kummer surface associated to $A$ has infinitely many $k$-points not contained in a rational curve arising from a genus $2$ curve in $A$, again in contrast to the situation over the algebraic closure of a finite field.

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arXiv2022

Rank 2 local systems and abelian varieties II

Let $X/\mathbb{F}_{q}$ be a smooth, geometrically connected, quasiprojective variety. Let $\mathcal{E}$ be a semisimple overconvergent $F$-isocrystal on $X$. Suppose that irreducible summands $\mathcal{E}_i$ of $\mathcal E$ have rank 2, determinant $\bar{\mathbb{Q}}_p(-1)$, and infinite monodromy at $\infty$. Suppose further that for each closed point $x$ of $X$, the characteristic polynomial of $\mathcal{E}$ at $x$ is in $\mathbb{Q}[t]\subset \mathbb Q_p[t]$. Then there exists a non-trivial open set $U\subset X$ such that $\mathcal{E}|_U$ comes from a family of abelian varieties on $U$. As an application, let $L_1$ be an irreducible lisse $\bar{\mathbb{Q}}_l$ sheaf on $X$ that has rank 2, determinant $\bar{\mathbb{Q}}_l(-1)$, and infinite monodromy at $\infty$. Then all crystalline companions to $L_1$ exist (as predicted by Deligne's crystalline companions conjecture) if and only if there exists a non-trivial open set $U\subset X$ and an abelian scheme $π_U\colon A_U\rightarrow U$ such that $L_1|_U$ is a summand of $R^1(π_U)_*\bar{\mathbb{Q}}_l$.

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arXiv2022

Rank 2 Local Systems, Barsotti-Tate Groups, and Shimura Curves

We develop a descent criterion for $K$-linear abelian categories. Using recent advances in the Langlands correspondence due to Abe, we build a correspondence between certain rank 2 local systems and certain Barsotti-Tate groups on complete curves over a finite field. We conjecture that such Barsotti-Tate groups "come from" a family of fake elliptic curves. As an application of these ideas, we provide a criterion for being a Shimura curve over $\mathbb{F}_q$. Along the way, we formulate a conjecture on the field-of-coefficients of certain compatible systems.

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arXiv2021

Rank 2 local systems and abelian varieties

Let $X/\mathbb{F}_{q}$ be a smooth geometrically connected variety. Inspired by work of Corlette-Simpson over $\mathbb{C}$, we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on $X$ come from families of abelian varieties. When $X$ is a projective variety, we prove a Lefschetz-style theorem for abelian schemes of $\text{GL}_2$-type on $X$, modeled after a theorem of Simpson. If one assumes a strong form of Deligne's ($p$-adic) \emph{companions conjecture} from Weil II, this implies that our conjecture for projective varieties also reduces to the case of projective curves. We also answer affirmitavely a question of Grothendieck on extending abelian schemes via their $p$-divisible groups.

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arXiv2020

Finiteness of logarithmic crystalline representations II

Let $K$ be an unramified $p$-adic local field and let $W$ be the ring of integers of $K$. Let $(X,S)/W$ be a smooth proper scheme together with a simple normal crossings divisor and fix positive integers $r$ and $f$. We show that the set of absolutely irreducible representations $π_1(X_{\bar K})\rightarrow \mathrm{GL}_r(\mathbb{Z}_{p^f})$ that come from log crystalline $\mathbb Z_{p^f}$-local systems over $(X_K,S_K)$ of rank $r$ is finite. The proof uses $p$-adic nonabelian Hodge theory and a finiteness result due Abe/Lafforgue.

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arXiv2020

Finiteness of logarithmic crystalline representations

Let $K$ be an unramified $p$-adic local field and let $W$ be the ring of integers of $K$. Let $(X,S)/W$ be a smooth proper scheme together with a normal crossings divisor. We show that there are only finitely many log crystalline $\mathbb Z_{p^f}$-local systems over $X_K\setminus S_K$ of given rank and with geometrically absolutely irreducible residual representation, up to twisting by a character. The proof uses $p$-adic nonabelian Hodge theory and a finiteness result due Abe/Lafforgue.

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arXiv2020

Deformation theory of periodic Higgs-de Rham flows

In this note we study the deformation theory of periodic (logarithmic) Higgs-de Rham flows. Under suitable numerical assumptions, this is equivalent to the deformation theory of torsion (logarithmic) Fontaine-Faltings modules. As an application, we formulate an \emph{ordinarity} condition, which provides a sufficient condition for a $p^n$-torsion crystalline representation to deform to a $p^{n+1}$-torsion crystalline representation.

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arXiv2020

A Lefschetz theorem for crystalline representations

As a corollary of nonabelian Hodge theory, Simpson proved a strong Lefschetz theorem for complex polarized variations of Hodge structure. We show an arithmetic analog. Our primary technique is $p$-adic nonabelian Hodge theory. Conditional on certain foundational results in \emph{logarithmic} $p$-adic Hodge theory, we also show a logarithmic analog.

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arXiv2017

Correspondences without a Core

We study the formal properties of correspondences of curves without a core, focusing on the case of étale correspondences. The motivating examples come from Hecke correspondences of Shimura curves. Given a correspondence without a core, we construct an infinite graph $\mathcal{G}_{gen}$ together with a large group of "algebraic" automorphisms $A$. The graph $\mathcal{G}_{gen}$ measures the "generic dynamics" of the correspondence. We construct specialization maps $\mathcal{G}_{gen}\rightarrow\mathcal{G}_{phys}$ to the "physical dynamics" of the correspondence. We also prove results on the number of bounded étale orbits, in particular generalizing a recent theorem of Hallouin and Perret. We use a variety of techniques: Galois theory, the theory of groups acting on infinite graphs, and finite group schemes.

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