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Sudip Pandit

Publications and source records attributed to Sudip Pandit.

8 recordsLinked to original sources
arXiv2026

Delta theory of Anderson Modules II: Hodge-Pink structure

In this article, using the theory of $\delta$-geometry, we construct a canonical $z$-isocrystal $(\mathbf{H}_\delta(E), \mathfrak{f}^*)$ admitting a Hodge-Pink structure for any abelian Anderson module $E$. The Hodge-Pink structure on $\mathbf{H}_\delta(E)$ induces a natural filtration $(\mathbf{H}_\delta(E) \supset \mathbf{X}_{\mathrm{prim}}(E) \supset \{0\})$. The elements of $\mathbf{X}_{\mathrm{prim}}(E)$ are represented by primitive delta characters associated to $E$. We establish a natural morphism from $\mathbf{H}_\delta(E)$ to the associated de Rham cohomology module $\mathbf{H}^*_{\mathrm{dR}}(E)$, which is strictly compatible with the aforementioned filtration and the classical Hodge filtration $(\mathbf{H}^{*}_{\mathrm{dR}}(E)\supset {\mathrm{Lie}(E)^{*}}\supset \{0\})$ on $\mathbf{H}^*_{\mathrm{dR}}(E)$. Moreover, we show that the map induces an isomorphism between $\mathbf{X}_{\mathrm{prim}}(E)$ and $\mathrm{Lie}(E)^*$. Hence our isomorphism provides an interesting interpretation of the invariant differentials of $E$ as primitive delta characters of $E$. Furthermore, when $E$ is a Drinfeld module, we show that the constructed $z$-isocrystal $\mathbf{H}_\delta(E)$ is weakly admissible. Consequently, the positive equal characteristic analogue of the Fontaine functor associates a crystalline $z$-adic Galois representation to the $\delta$-geometric object $\mathbf{H}_\delta(E)$. In the case, when $E$ is the Carlitz module, we show that the Galois representation associated to $\mathbf{H}_\delta(E)$ is indeed the usual one coming from the Tate module.

math.NT↗
arXiv2026

Delta characters and crystalline cohomology of abelian schemes

We provide an explicit description of the smallest filtered sub-isocrystal generated by the Hodge filtered piece of the crystalline cohomology for an abelian scheme over a $p$-adic ring. Our method is based on the theory of arithmetic jet spaces and delta characters associated to the abelian scheme, introduced by Buium and later studied by Borger and Saha using a functor of points approach. In particular, we prove that the delta isocrystal constructed by Borger and Saha is indeed isomorphic to the fundamental smallest sub-isocrystal of the crystalline cohomology in the category of filtered $F$-isocrystals. As an application, we establish a comparison isomorphism between the delta isocrystal and the crystalline cohomology of abelian schemes, which is governed by the group of order $1$ delta characters of the abelian scheme.

math.AG↗
arXiv2025

A Buium--Coleman bound for the Mordell--Lang conjecture

For $X$ a hyperbolic curve of genus $g$ with good reduction at $p\geq 2g$, we give an explicit bound on the Mordell--Lang locus $X(\mathbb{C})\cap \Gamma $, when $\Gamma \subset J(\mathbb{C})$ is the divisible hull of a subgroup of $J(\mathbb{Q} _p ^{\mathrm{nr}})$ of rank less than $g$. Without any assumptions on the rank (but with all the other assumptions) we show that $X(\mathbb{C})\cap \Gamma $ is unramified at $p$, and bound the size of its image in $X(\overline{\mathbb{F} }_p )$. As a corollary, we obtain a new proof that Mordell implies Mordell--Lang for curves.

math.NT↗
arXiv2023

Delta Characters and Crystalline Cohomology

The first part of the paper develops the theory of $m$-shifted $\pi$-typical Witt vectors which can be viewed as subobjects of the usual $\pi$-typical Witt vectors. We show that the shifted Witt vectors admit a delta structure that satisfy a canonical identity with the delta structure of the usual $\pi$-typical Witt vectors. Using this theory, we prove that the generalized kernels of arithmetic jet spaces are jet spaces of the kernel at the first level. This also allows us to interpret the arithmetic Picard-Fuchs operator geometrically. For a $\pi$-formal group scheme $G$, by a previous construction, one attaches a canonical filtered isocrystal $\mathbf{H}_\delta(G)$ associated to the arithmetic jet spaces of $G$. In the second half of our paper, we show that $\mathbf{H}_\delta(A)$ is of finite rank if $A$ is an abelian scheme. We also prove a strengthened version of a result of Buium on delta characters on abelian schemes. As an application, for an elliptic curve $A$ defined over $\mathbb{Z}_p$, we show that our canonical filtered isocrystal $\mathbf{H}_\delta(A) \otimes \mathbb{Q}_p$ is weakly admissible. In particular, if $A$ does not admit a lift of Frobenius, we show that $\mathbf{H}_\delta(A) \otimes \mathbb{Q}_p$ is isomorphic to the first crystalline cohomology $\mathbf{H}^1_{\mathrm{cris}}(A) \otimes \mathbb{Q}_p$ in the category of filtered isocrystals. On the other hand, if $A$ admits a lift of Frobenius, then $\mathbf{H}_\delta(A) \otimes \mathbb{Q}_p$ is isomorphic to the sub-isocrystal $H^0(A,\Omega_A) \otimes \mathbb{Q}_p$ of $\mathbf{H}^1_{\mathrm{cris}}(A) \otimes \mathbb{Q}_p$.

math.NT↗
arXiv2022

Delta Theory of Anderson Modules I: Differential Characters

In this article we develop the theory of differential or delta characters (the arithmetic analogue of Manin characters) of Anderson modules. Here we generalize the construction by Borger and Saha of the canonical finite rank $R$-module $\mathbf{H}(E)$ with a semilinear operator on it to any Anderson module $E$, where $R$ is the base ring which is a $\pi$-adically complete discrete valuation ring with a fixed lift of Frobenius $\phi$ on it. Then we show that $\mathbf{H}(E)$ admits a functorial map to the de Rham cohomology $\mathbf{H}_{\mathrm{dR}}^*(E)$ of $E$ which also preserves the Hodge filtration. We also prove that the module of delta characters $\mathbf{X}_{\infty}(E)$ is finite and free as an $R\{\phi^{*}\}$-module. This leads to a strengthened version of an analogous result by Buium on the generation of differential characters of abelian varieties. We also construct a family of differential modular functions that play the analogous role of $f_{\mathrm{jet}}$ constructed by Buium for elliptic curves. In a subsequent article, the finite rank $R$-module $\mathbf{H}(E)$ will lead to the construction of a canonical $z$-isocrystal $\mathbf{H}_{\delta}(E)$ with a Hodge-Pink filtration on it and we will show that $\mathbf{H}_{\delta}(E)$ is an admissible $z$-isocrystal.

math.NT↗
arXiv2021

On a certain divisor function in Number fields

The main aim of this paper is to study an analogue of the generalized divisor function in a number field $\mathbb{K}$, namely, $\sigma_{\mathbb{K},\alpha}(n)$. The Dirichlet series associated to this function is $\zeta_{\mathbb{K}}(s)\zeta_{\mathbb{K}}(s-\alpha)$. We give an expression for the Riesz sum associated to $\sigma_{\mathbb{K},\alpha}(n),$ and also extend the validity of this formula by using convergence theorems. As a special case, when $\mathbb{K}=\mathbb{Q}$, the Riesz sum formula for the generalized divisor function is obtained, which, in turn, for $\alpha=0$, gives the Vorono\"{\i} summation formula associated to the divisor counting function $d(n)$. We also obtain a big $O$-estimate for the Riesz sum associated to $\sigma_{\mathbb{K},\alpha}(n)$.

math.NT↗
arXiv2021

Numerical Semigroups with unique Ap\'{e}ry expansions

In this paper, we carry out a fairly comprehensive study of two special classes of numerical semigroups, one generated by the sequence of partial sums of an arithmetic progression and the other one generated by the partial sums of a geometric progression, in embedding dimension $4$. Both these classes have the common feature that they have unique expansions of the Ap\'{e}ry set elements.

math.AC↗