Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1403.3962

The congruence relation in the non-PEL case

Oliver Bueltel
View original source ↗Read source document ↗Export metadata ↓

Abstract

This work settles the Eichler-Shimura congruence relation of Blasius and Rogawski for certain 5-dimensional Hodge-type Shimura varieties, that were not tractable by previously known methods. In a more general context we introduce a hypothesis called (NVC) on the behavior of Hecke correspondences, and show that it implies the congruence relation. A major ingredient in the proof of this result is a theorem of R.Noot on CM-lifts of ordinary points in characteristic p, along with an analysis of the mod p-reductions of various Hecke translates of that CM-lift. Finally we prove this (NVC)-hypothesis for our particular Shimura 5-folds, and in doing so we obtain an unconditional result for the congruence relation of these non-PEL examples.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Oliver Bueltel. 2014-03-16. The congruence relation in the non-PEL case. https://arxiv.org/abs/1403.3962

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Lawson--Deligne Classes and Applications

We construct the integral Lawson--Deligne map of weight $q=n-p-k-1$ on smooth complex projective $n$-folds using filtered currents. It lifts the Friedlander--Mazur cycle class, recovers the reduced generalized Abel--Jacobi invariant on homologically trivial classes, and is compatible with algebraic correspondences. A Picard--Fuchs separation argument applied to the conic and van Geemen normal functions on the mirror quintic determines explicit regulator subspaces modulo the full rational period group. For prescribed elliptic moduli and a suitable mirror-quintic fiber, the subspace generated by their $a$- and $b$-loop products has dimension twice the $\Q$-dimension of the period-monomial space. Moduli $i\sqrt{\ell_j}$ for distinct primes $\ell_j$ give $2^{k+1}$ independent images on varieties of dimension $p+k+2$; one repeated imaginary quadratic modulus gives dimension four for every $k\geq1$. Compatibility with known projective-bundle and blow-up decompositions yields independent exceptional subspaces on smooth rational varieties. We also compare the higher Chow composite with the Bloch--KLM regulator after lowering the Hodge filtration. The KLM representative reduces to a cut-current class, and equality with the Lawson composite is proved in degree zero and for constant-unit decomposable classes. The general positive-degree comparison is reduced to an explicit filtered-realization condition.

math.AG↗
arXiv2026

Border rank lower bounds for families of GL(V)-invariant tensors

We give non-trivial lower bounds for the border rank of families of $\mathbf{GL}(V)$-invariant tensors in $U\otimes \mathbf{S}_λV\otimes \mathbf{S}_μV$ where $U$ is $V$, $\mathrm{Sym}^2V$ or $\bigwedge^2V$. In particular, we provide a family of tensors with border rank reaching arbitrarily close to $2\ell$ in the unbalanced case, where $\ell$ is the largest ambient vector space dimension. We do this by resolving a conjecture introduced by Wu, and obtaining new results on $6j$-symbols as a byproduct. We then generalize our results to $\mathrm{Sym}^2V$ and $\bigwedge^2 V$ using novel techniques based on an application of a theorem of Kostant and Kempf collapsing.

math.AG↗
arXiv2026

Maximally nodal sextic surfaces and linear determinantal representations

We prove that every maximally nodal sextic surface (with 65 nodes) $X \subset \mathbb{P}_{\mathbb{C}}^3$ contains a symmetric half-even set of nodes of cardinality 35. It follows that the associated half-quadratic sheaf is the cokernel of a symmetric $6 \times 6$ matrix of linear forms, yielding a linear determinantal representation of $X$. In particular, after a suitable Serre twist, the half-quadratic sheaf is an Ulrich sheaf of rank 1. As an example, we exhibit an explicit $6 \times 6$ matrix of linear forms whose determinant defines the Barth sextic surface.

math.AG↗