Five-dimensional compatible systems and the Tate conjecture for elliptic surfaces

[2406.03617] Five-dimensional compatible systems and the Tate conjecture for elliptic surfaces Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2406.03617 (math) [Submitted on 5 Jun 2024 ( v1 ), last revised 31 Aug 2026 (this version, v3)] Title: Five-dimensional compatible systems and the Tate conjecture for elliptic surfaces Authors: Lian Duan , Xiyuan Wang , Ariel Weiss View a PDF of the paper titled Five-dimensional compatible systems and the Tate conjecture for elliptic surfaces, by Lian Duan and 2 other authors View PDF HTML (experimental) Abstract: Let $(\rho_\lambda\colon G_{\mathbb Q}\to \operatorname{GL}_5(\overline{E}_\lambda))_\lambda$ be a strictly compatible system of Galois representations such that no Hodge--Tate weight has multiplicity $5$. Under mild assumptions, we show that if $\rho_{\lambda_0}$ is irreducible for some $\lambda_0$, then $\rho_\lambda$ is irreducible for all but finitely many priimes $\lambda$. More generally, if $(\rho_\lambda)_\lambda$ is essentially self-dual, we show that either $\rho_\lambda$ is irreducible for all but finitely many $\lambda$, or the compatible system $(\rho_\lambda)_\lambda$ decomposes as a direct sum of lower-dimensional compatible systems. We apply our results to study the Tate conjecture for elliptic surfaces. For example, if $X_0\colon y^2 + (t+3)xy + y= x^3$, we prove the codimension one $\ell$-adic Tate conjecture for all but finitely many $\ell$, for all but finitely many general, degree $3$, genus $2$ branched multiplicative covers of $X_0$. To prove this result, we classify the elliptic surfaces into four one-dimensional families and two isolated classes. For each of the four families, we prove, using perverse sheaf theory and a result of Cadoret--Tamagawa, that if the relevant $5$-dimensional Galois representation is irreducible for one surface in a family, then it is irreducible for all but finitely many surfaces in that family. We then verify this irreducibility for one representative of each family by making our irreducibility result explicit: for the compatible system arising from the transcendental part of $H^2_{\mathrm{et}}(X_{\overline{\mathbb Q}}, \mathbb{Q}_\ell(1))$ for a representative $X$, we formulate an algorithm that takes as input the characteristic polynomials of Frobenius, and terminates if and only if the compatible system is irreducible. Comments: 57 pages. Final version, to appear in Advances in Mathematics Subjects: Number Theory (math.NT) ; Algebraic Geometry (math.AG) MSC classes: 11F80, 11F70, 14C25, 14D05, 14J27 Cite as: arXiv:2406.03617 [math.NT] (or arXiv:2406.03617v3 [math.NT] for this version) https://doi.org/10.48550/arXiv.2406.03617 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Ariel Weiss [ view email ] [v1] Wed, 5 Jun 2024 20:26:30 UTC (84 KB) [v2] Fri, 10 Apr 2026 02:36:46 UTC (84 KB) [v3] Mon, 31 Aug 2026 12:42:21 UTC (84 KB) Full-text links: Access Paper: View a PDF of the paper titled Five-dimensional compatible systems and the Tate conjecture for elliptic surfaces, by Lian Duan and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2024-06 Change to browse by: math math.AG References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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