Papers › Iwasawa Theory for p-torsion Class Group Schemes in Characteristic p
Iwasawa Theory for p-torsion Class Group Schemes in Characteristic p
Jeremy Booher, Bryden Cais
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We investigate a novel geometric Iwasawa theory for 𝐙ₚ-extensions of function fields over a perfect field k of characteristic p>0 by replacing the usual study of p-torsion in class groups with the study of p-torsion class group schemes. That is, if ⋯→X₂ →X₁ →X₀ is the tower of curves over k associated to a 𝐙ₚ-extension of function fields totally ramified over a finite non-empty set of places, we investigate the growth of the p-torsion group scheme in the Jacobian of Xₙ as n→∞. By Dieudonn\'e theory, this amounts to studying the first de Rham cohomology groups of Xₙ equipped with natural actions of Frobenius and of the Cartier operator V. We formulate and test a number of conjectures which predict striking regularity in the k[V]-module structure of the space Mₙ:=H⁰(Xₙ, Ω¹_(Xₙ/k)) of global regular differential forms as n→∞. For example, for each tower in a basic class of 𝐙ₚ-towers we conjecture that the dimension of the kernel of Vʳ on Mₙ is given by aᵣ p²ⁿ + λᵣ n + cᵣ(n) for all n sufficiently large, where aᵣ, λᵣ are rational constants and cᵣ : 𝐙/mᵣ 𝐙 →𝐐 is a periodic function, depending on r and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on 𝐙ₚ-towers of curves, and we prove our conjectures in the case p=2 and r=1.
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