arXiv subjects

Casimir Kothari

Publications and source records attributed to Casimir Kothari.

4 recordsLinked to original sources

arXiv2026

Dieudonné theory for $n$-smooth group schemes

For all $n \geq 1$, there is a notion of an $n$-smooth group scheme over any $\mathbb{F}_p$-algebra $R$, which may be thought of as a ``Frobenius analogue" of an $n$-truncated Barsotti--Tate group over $R$. We prove that the category of $n$-smooth commutative group schemes over $R$ is equivalent to a certain full subcategory of Dieudonné modules over $R$. As a consequence, we show that the moduli stack $\mathrm{Sm}_n$ of $n$-smooth commutative group schemes is smooth over $\mathbb{F}_p$ and that the natural truncation morphism $\mathrm{Sm}_{n+1} \to \mathrm{Sm}_n$ is smooth and surjective. These results affirmatively answer conjectures of Drinfeld.

math.AG↗

arXiv2024

Arbitrarily large jumps in the de Rham and Hodge cohomology of families in characteristic $p$

We construct smooth projective families of algebraic varieties in characteristic $p$ such that the dimensions of the de Rham and Hodge cohomology groups of the fibers can be made to jump by an arbitrarily large amount. To do this, we first construct an example using the classifying stack of a finite flat group scheme which degenerates $\mathbb{Z}/p^2\mathbb{Z}$ to $α_p \oplus α_p$. Along the way, we give a self-contained exposition of the construction of Godeaux--Serre varieties.

math.AG↗

arXiv2024

Duality of differential operators and algebraic de Rham cohomology

Given a smooth proper morphism $f\colon X\rightarrow S$, we introduce a certain derived category where morphisms are permitted to be $\mathcal{O}_S$-linear differential operators. We then prove a generalisation of Serre duality that applies to two-term complexes of this type. We apply this to give a new proof of Poincaré duality for relative algebraic de Rham cohomology.

math.AG↗

arXiv2021

The Explicit Sato-Tate Conjecture For Primes In Arithmetic Progressions

Let $τ(n)$ be Ramanujan's tau function, defined by the discriminant modular form \[ Δ(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}τ(n) q^n \,,q=e^{2πi z} \] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer's conjecture asserts that $τ(n)\neq 0$ for all $n\geq 1$; since $τ(n)$ is multiplicative, it suffices to study primes $p$ for which $τ(p)$ might possibly be zero. Assuming standard conjectures for the twisted symmetric power $L$-functions associated to $τ$ (including GRH), we prove that if $x\geq 10^{50}$, then \[ \#\{x < p\leq 2x: τ(p) = 0\} \leq 1.22 \times 10^{-5} \frac{x^{3/4}}{\sqrt{\log x}},\] a substantial improvement on the implied constant in previous work. To achieve this, under the same hypotheses, we prove an explicit version of the Sato-Tate conjecture for primes in arithmetic progressions.

math.NT↗