An approach to the Tate conjecture for surfaces over a finite field

Bruno Kahn

doi:10.48550/arxiv.2505.07337

May 12, 2025Open Access

An approach to the Tate conjecture for surfaces over a finite field

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Key Points

To reformulate the Tate conjecture for surfaces over a finite field and explore proof attempts.
Reformulated the Tate conjecture in terms of affine open subsets.
Proposed three attempts to prove the reformulation.
Related the last two attempts to techniques for Gersten's conjecture.
Each proof attempt fell short of confirming the reformulation.
The attempts provided insights into the connections with Gersten's conjecture.

Implication

The reformulated Tate conjecture shows limitations in proofs regarding surfaces over finite fields, suggesting further exploration is needed.

Abstract

We give a reformuation of the Tate conjecture for a surface over a finite field in terms of suitable affine open subsets. We then present three attempts to prove this reformulation, each of them falling short. Interestingly, the last two are related to techniques used in proofs of Gersten's conjecture.

Discussion

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Cite This Study

Bruno Kahn (2025) studied this question.

synapsesocial.com/papers/69473b64db9c958d0dfca881 https://doi.org/10.48550/arxiv.2505.07337

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An approach to the Tate conjecture for surfaces over a finite field

Key Points

Implication

Abstract

Discussion

Cite This Study

An approach to the Tate conjecture for surfaces over a finite field

Key Points

Implication

Abstract

Discussion

Cite This Study

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