arXiv subjects
Publications and source records attributed to Syed Waqar Ali Shah.
Kolyvagin introduced the method of Euler systems to study the structure of Selmer groups of elliptic curves. In this semi-expository article, we prove the horizontal norm relations for the CM points on modular curves underlying Kolyvagin's Euler system, with a view toward higher-dimensional generalizations.
We derive an explicit formula for the action of a geometric Hecke correspondence on special cycles on a Shimura variety in terms of such cycles at a fixed neat level and compare it with another closely related expression sometimes used in literature. We provide evidence that the two formulas do not agree in general.
We show that for Siegel modular varieties of arbitrary genus, the natural distribution relations satisfied by certain integral Eisenstein cohomology classes defined by Kings admit an adelic refinement. This generalizes the classical relations for Siegel units on modular curves.
We present a novel axiomatic framework for establishing horizontal norm relations in Euler systems that are built from pushforwards of classes in the motivic cohomology of Shimura varieties. This framework is uniformly applicable to the Euler systems of both algebraic cycles and Eisenstein classes. It also applies to non-spherical pairs of groups that fail to satisfy a local multiplicity one hypothesis, and thus lie beyond the reach of existing methods. A key application of this work is the construction of an Euler system for the spinor Galois representations arising in the cohomology of Siegel modular varieties of genus three, which is undertaken in two companion articles.
We establish abstract horizontal norm relations involving the unramified Hecke-Frobenius polynomials that correspond under the Satake isomorhpism to the degree eight spinor $L$-factors of $ \mathrm{GSp}_{6} $. These relations apply to classes in the degree seven motivic cohomology of the Siegel modular sixfold obtained via Gysin pushforwards of Beilinson's Eisenstein symbol pulled back on one copy in a triple product of modular curves. The proof is based on a novel approach that circumvents the failure of the so-called multiplicity one hypothesis in our setting, which precludes the applicability of an existing technique. In a sequel, we combine our result with the previously established vertical norm relations for these classes to obtain new Euler systems for the eight dimensional Galois representations associated with certain non-endoscopic cohomological cuspidal automorphic representations of $ \mathrm{GSp}_{6} $.
We prove a conjecture of Miller and Reiner on the existence of Smith normal form for the $DU$-operators for a certain class of $r$-differential posets.
Let $n \geq 1$ be an odd integer. We construct an anticyclotomic Euler system for certain cuspidal automorphic representations of unitary groups with signature $(1, 2n-1)$.