Current browse context:
math.AG
Change to browse by:
References & Citations
Mathematics > Algebraic Geometry
Title: Constructing abelian varieties from rank 3 Galois representations with real trace field
(Submitted on 26 Mar 2024)
Abstract: Let $U/K$ be a smooth affine curve over a number field and let $L$ be an irreducible rank 3 $\overline{\mathbb Q}_{\ell}$-local system on $U$ with trivial determinant and infinite geometric monodromy around a cusp. Suppose further that $L$ extends to an integral model such that the Frobenius traces are contained in a fixed totally real number field. Then, after potentially shrinking $U$, there exists an abelian scheme $f\colon B_U\rightarrow U$ such that $L$ is a summand of $R^2f_*\overline{\mathbb Q}_{\ell}(1)$.
The key ingredients are: (1) the totally real assumption implies $L$ admits a square root $M$; (2) the trace field of $M$ is sufficiently bounded, allowing us to use recent work of Krishnamoorthy-Yang-Zuo to construct an abelian scheme over $U_{\bar K}$ geometrically realizing $L$; and (3) Deligne's weight-monodromy theorem and the Rapoport-Zink spectral sequence, which allow us to pin down the arithmetizations using the total degeneration.
Submission history
From: Raju Krishnamoorthy [view email][v1] Tue, 26 Mar 2024 22:45:18 GMT (24kb,D)
Link back to: arXiv, form interface, contact.