Γ(p)-Level Structure on p-Divisible Groups

• Main Author: Frimu, Andrei
• Format: Others
• Published: 2019
• Online Access: https://thesis.library.caltech.edu/11575/1/main_final.pdf; Frimu, Andrei (2019) Γ(p)-Level Structure on p-Divisible Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8JYH-KT84. https://resolver.caltech.edu/CaltechTHESIS:05302019-185557287 <https://resolver.caltech.edu/CaltechTHESIS:05302019-185557287>

<p>The main result of the thesis is the introduction of a notion of Γ(<i>p</i>)-level structure for <i>p</i>-divisible groups. This generalizes the Drinfeld-Katz-Mazur notion of full level structure for 1-dimensional <i>p</i>-divisible groups. The associated moduli problem has a natural forgetful map to the Γ₀(<i>p</i>)-level moduli problem. Exploiting this map and known results about Γ₀(<i>p</i>)-level, we show that our notion yields a flat moduli problem. We show that in the case of 1-dimensional <i>p</i>-divisible groups, it coincides with the existing Drinfeld-Katz-Mazur notion.</p> <p>In the second half of the thesis, we introduce a notion of epipelagic level structure. As part of the task of writing down a local model for the associated moduli problem, one needs to understand commutative finite flat group schemes <i>G</i> of order <i>p</i>² killed by <i>p</i>, equipped with an extension structures 0&#8594; H₁&#8594; G&#8594; H₂&#8594; 0, where H₁,H₂ are finite flat of order <i>p</i>. We investigate a particular class of extensions, namely extensions of <i>Z/pZ</i> by &#956;_p over Z_p-algebras. These can be classified using Kummer theory. We present a different approach, which leads to a more explicit classification.</p>