Finiteness of Frobenius Traces of a Sheaf on a Flat Arithmetic Scheme

Shimizu, Koji

doi:10.1093/imrn/rny145

Search citation statements

Order By: Relevance

Paper Sections

Select...

Introduction1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2024

Publication Types

Select...

Article1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

(1 citation statement)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, to prove the relative Fontaine–Mazur conjecture for rank local systems that have infinite monodromy around some point, it suffices to bound the field generated by Frobenius traces. This task seems to be quite difficult in general; for some progress on this question, see [Shi20].…”

Section: Introductionmentioning

confidence: 99%

Constructing abelian varieties from rank 2 Galois representations

Krishnamoorthy,

Yang,

Zuo

2024

Compositio Math.

View full text Add to dashboard Cite

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 $E_U\rightarrow U$ .

show abstract

Section: Introductionmentioning

confidence: 99%