Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 45 min ago
PreprintCC BY 4.0arXiv

Artin vanishing along the ℓ-adic tower of an abelian variety

  • Ashutosh Roy Choudhury
  • K. V. Shuddhodan
Open at sourcePDFDOICite

Abstract

Let A be an abelian variety of dimension g over an algebraically closed field and let ℓ be a prime invertible in the field. For every constructible 𝔽ℓ-sheaf F on A we prove that there is an integer e, depending on F, such that the pullback map [ℓe]*:Hi(A,[ℓn]*F)→Hi(A,[ℓn+e]*F) is zero for all n≥0 and all i>dimSuppF. In particular varinjlimnHi(A,[ℓn]*F)=0 for i>dimSuppF, which answers a question of Bhatt--Schnell--Scholze. Our methods also give sharp codimension estimates for the supports of the cohomology sheaves of the Fourier--Mellin transform of a perverse sheaf. With ℓ-adic coefficients, we remove the arithmeticity hypothesis from the estimates of Esnault--Kerz. The corresponding estimates for the completed transform with 𝔽ℓ-coefficients are new and hold even when Hard Lefschetz fails.

Related

  1. An abelian model for the Goodwillie tower of the circle
    Marco Nervo
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  2. Intersecting a curve in an abelian variety with multiples of another curve
    Fabrizio Barroero, Gabriel A. Dill, Lars Kuehne
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. On the residual finiteness of the non-abelian exterior square of some Artin groups, constructible soluble groups, generalised Baumslag-Solitar groups and SLn(Z)
    Lucas Barroso Rocha, Dessislava H. Kochloukova
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Vanishing of Sha(A/K)[P∞] and its consequences for the anticyclotomic Iwasawa theory of GL2-abelian varieties
    Luca Mastella, Ahmed Matar, Francesco Zerman
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Generic vanishing for Dolbeault cohomology of mixed Hodge modules on abelian varieties
    Ze Yun
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{choudhury2026artin,
  title = {{Artin vanishing along the $\ell$-adic tower of an abelian variety}},
  author = {Ashutosh Roy Choudhury and K. V. Shuddhodan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35404},
  eprint = {2609.35404},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35404},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Artin vanishing along the ℓ-adic tower of an abelian variety
AU  - Ashutosh Roy Choudhury
AU  - K. V. Shuddhodan
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.35404
UR  - https://arxiv.org/abs/2609.35404
AB  - Let A be an abelian variety of dimension g over an algebraically closed field and let ℓ be a prime invertible in the field. For every constructible 𝔽_ℓ-sheaf F on A we prove that there is an integer e, depending on F, such that the pullback map [ℓ^e]^*:H^i(A,[ℓ^n]^*F)→H^i(A,[ℓ^n+e]^*F) is zero for all n≥0 and all i>dimSuppF. In particular varinjlim_nH^i(A,[ℓ^n]^*F)=0 for i>dimSuppF, which answers a question of Bhatt--Schnell--Scholze. Our methods also give sharp codimension estimates for the supports of the cohomology sheaves of the Fourier--Mellin transform of a perverse sheaf. With ℓ-adic coefficients, we remove the arithmeticity hypothesis from the estimates of Esnault--Kerz. The corresponding estimates for the completed transform with 𝔽_ℓ-coefficients are new and hold even when Hard Lefschetz fails.
ER  -