Papers and preprints

PDF versions of preprints linked here are more up to date than the current arXiv posting.

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Syntomification and crystalline local systems [pdf] [arXiv] I show that the category of crystalline \(\mathrm{Z}_p\)-local systems is equivalent to a category of reflexive \(F\)-gauges, which resolves a conjecture of Bhatt. Prior to this, the question was partially answered by Guo-Li, who gave a way to lift analytic prismatic \(F\)-crystals to \(F\)-gauges. One can define a canonical \(t\)-structure on perfect complexes on the syntomification for smooth \(\mathrm{X}\), and characterize the output of Guo-Li's construction as the reflexive objects in the heart of this \(t\)-structure. Rationally this implies \(\mathsf{Coh}(\mathrm{X}^{\mathrm{Syn}})[1/p]\) is equivalent to crystalline \(\mathrm{Q}_p\)-local systems, and we use this integral result to produce a derived variant of this result. Namely, we show that \(\mathsf{Perf}(\mathrm{X}^{\mathrm{Syn}})[1/p]\) is equivalent to a category of admissible filtered \(F\)-isocrystals -- morally one can think of this as a derived category of crystalline local systems, although this is generally not the case unless \(\mathrm{X}=\mathcal{O}_K\). The integral underived result allows us to show essential surjectivity easily, while the harder component of full faithfulness comes from a variant of the Beilinson fiber square with coefficients for smooth proper formal schemes over \(\mathcal{O}_K\).
Appendix 2 to Takumi Watanabe's A remark on an integral structure of the imperfect coefficient ring of \((\varphi,\Gamma)\)-modules. [arXiv] This appendix proves that for \(K=\mathbb{Q}_p(p^{1/e})\) with \(e>1\) and \(\gcd(e,p(p-1))=1\), no Frobenius-stable subring \(R\subset \mathbb{A}_K\) has \(R/p\simeq \mathbb{E}_K^+\subset \mathbb{E}_K=\mathbb{A}_K/p\).
Extensions of mod \(p\) representations of local division algebras with Drew Keisling. [JNTB] [arXiv] Determines the extension groups of smooth irreducible representations of a division algebra \(D\) over a non-Archimedean local field.
Filtrations on block subalgebras of restricted universal enveloping algebras with Andrei Ionov. [Journal of Algebra and its Applications] [arXiv] Studies the associated graded algebras for the PBW filtration and related filtrations on blocks of restricted universal enveloping algebras.
Computing L-polynomials of Picard curves in polylogarithmic time with Sualeh Asif and Francesc Fité. [Mathematics of Computation] [arXiv] Develops and implements a practical algorithm to compute the zeta function of a curve in genus >2.
Coefficients of Gaussian Polynomials Modulo N [EJC] [arXiv] Resolves and extends a conjecture of Prof. Richard Stanley on periods of coefficients in \(q\)-binomial coefficients modulo \(N\).

Talk notes

These are some rough notes I wrote for seminar talks.

  • The Klein quartic [pdf]
  • Derived prismatic cohomology [pdf]
  • \(F\)-gauges and crystalline Galois representations [pdf]
  • The Fargues-Fontaine curve [pdf]
  • The arc topology [pdf]
  • The Emerton-Kisin correspondence [pdf]
  • Proetale cohomology [pdf]
  • Rigid flat connections and \(p\)-curvature [pdf]
  • The Beilinson fiber square [pdf]
  • Nearby cycles [pdf]
  • Shtukas [pdf]
  • Honda-Tate theory [pdf]
  • Twistor \(\mathbf{P}^1\) [pdf]
  • Almost purity [pdf]

I also taught a course on Bass-Serre theory. The notes are here.

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