Rank 2 local systems and abelian varieties

Krishnamoorthy, Raju; Pál, Ambrus

doi:10.1007/s00029-021-00669-8

Cited by 5 publications

(13 citation statements)

References 47 publications

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“…In this paper, we only deal with non-proper varieties and we use infinitely many (space-filling, affine) curves together with a result of Drinfeld, which is only known for affine curves. In particular, the main results of [KP21] do not imply the main result of this paper.…”

Section: Introductionmentioning

confidence: 66%

“…For a thorough summary about the definitions and also what is known, we refer the reader to [Ked18]. Alternatively, the reader may see [KP21,§ 4].…”

Section: Preliminariesmentioning

confidence: 99%

“…Our motivation for formulating this conjecture was a celebrated theorem of Corlette and Simpson over [CS08, Theorem 11.2], the proof of which uses non-abelian Hodge theory. In contrast to our earlier work [KP21], this paper does not use Serre–Tate deformation theory nor does it use the algebraization/globalization techniques of [Har70].…”

Section: Introductionmentioning

confidence: 99%

“…We comment on the relation of this paper to [KP21]. In [KP21] we prove a Lefschetz-style theorem for families of -type abelian schemes over finite fields.…”

Section: Introductionmentioning

confidence: 99%

“…In [KP21] we prove a Lefschetz-style theorem for families of -type abelian schemes over finite fields. This has the following implication for [KP21, Conjecture 1.2]: if is a smooth projective variety, then there exists an ample curve such that if and comes from an abelian scheme of -type, then there is an open subset such that comes from an abelian scheme of -type. (It follows from Zarhin's work on the Tate isogeny conjecture that is indeed isogenous to .)…”

Section: Introductionmentioning

confidence: 99%

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Rank 2 local systems and abelian varieties II

Krishnamoorthy¹,

Pál²

2022

Compositio Math.

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Let $X/\mathbb {F}_{q}$ be a smooth, geometrically connected, quasi-projective scheme. Let $\mathcal {E}$ be a semi-simple 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 dense open subset $U\subset X$ 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 exist a dense open subset $U\subset X$ and an abelian scheme $\pi _U\colon A_U\rightarrow U$ such that $L_1|_U$ is a summand of $R^{1}(\pi _U)_*\bar {\mathbb {Q}}_l$ .

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Section: Introductionmentioning

confidence: 66%

“…For a thorough summary about the definitions and also what is known, we refer the reader to [Ked18]. Alternatively, the reader may see [KP21,§ 4].…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We comment on the relation of this paper to [KP21]. In [KP21] we prove a Lefschetz-style theorem for families of -type abelian schemes over finite fields.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rank 2 local systems and abelian varieties II

Krishnamoorthy¹,

Pál²

2022

Compositio Math.

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Constructing abelian varieties from rank 2 Galois representations

Krishnamoorthy,

Yang,

Zuo

2024

Compositio Math.

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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 lisse $\overline {{\mathbb {Q}}}_\ell$ -sheaf on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $E\subset \overline {\mathbb {Q}}_{\ell }$ , 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 and 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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Rank 2 local systems, Barsotti–Tate groups, and Shimura curves

Krishnamoorthy

2022

Alg. Number Th.

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Rank 2 local systems and abelian varieties

Abstract: Let $$X/\mathbb {F}_{q}$$ X / F q be a smooth, geometrically connected variety. For X projective, we prove a Lefschetz-style theorem for abelian schemes of $$\text {GL}_2$$ GL 2 -type on X, modeled after a theorem of Simpson. Inspired by wo… Show more

Cited by 5 publications

References 47 publications

Rank 2 local systems and abelian varieties II

Rank 2 local systems and abelian varieties II

Constructing abelian varieties from rank 2 Galois representations

Rank 2 local systems, Barsotti–Tate groups, and Shimura curves

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