arXiv subjects

Abhinandan

Publications and source records attributed to Abhinandan.

9 recordsLinked to original sources

arXiv2026

Log-crystalline representations and $(\varphi, \Gamma)$-modules

Let $F$ be an absolutely unramified mixed characteristic local field with rings of integers $O_F$. For a small affine algebra $R$ over $O_F$, we consider the logarithmic \'etale fundamental group $G_R$ of its generic fibre equipped with a "horizontal" log-structure, in the sense of Fujiwara--Kato. We study $p$-adic representations of $G_R$ and obtain a classification of these representations in terms of \'etale $(\varphi, \Gamma_R)$-modules. Moreover, we define and study the notion of log-crystalline representations of $G_R$, a generalisation of crystalline representations from the non-logarithmic/smooth case. Furthermore, in the logarithmic setting, we show that log-crystalline representations of $G_R$ are equivalent to Wach modules for $R$, extending our previous results from the (non-logarithmic) crystalline case.

math.NT↗

arXiv2026

Image Synthesis Using Spintronic Deep Convolutional Generative Adversarial Network

The computational requirements of generative adversarial networks (GANs) exceed the limit of conventional Von Neumann architectures, necessitating energy efficient alternatives such as neuromorphic spintronics. This work presents a hybrid CMOS-spintronic deep convolutional generative adversarial network (DCGAN) architecture for synthetic image generation. The proposed generative vision model approach follows the standard framework, leveraging generator and discriminators adversarial training with our designed spintronics hardware for deconvolution, convolution, and activation layers of the DCGAN architecture. To enable hardware aware spintronic implementation, the generator's deconvolution layers are restructured as zero padded convolution, allowing seamless integration with a 6-bit skyrmion based synapse in a crossbar, without compromising training performance. Nonlinear activation functions are implemented using a hybrid CMOS domain wall based Rectified linear unit (ReLU) and Leaky ReLU units. Our proposed tunable Leaky ReLU employs domain wall position coded, continuous resistance states and a piecewise uniaxial parabolic anisotropy profile with a parallel MTJ readout, exhibiting energy consumption of 0.192 pJ. Our spintronic DCGAN model demonstrates adaptability across both grayscale and colored datasets, achieving Fr'echet Inception Distances (FID) of 27.5 for the Fashion MNIST and 45.4 for Anime Face datasets, with testing energy (training energy) of 4.9 nJ (14.97~nJ/image) and 24.72 nJ (74.7 nJ/image).

physics.app-ph↗

arXiv2025

An integral comparison of crystalline and de Rham cohomology

Let $\mathcal{O}_K$ be a mixed characteristic complete DVR with perfect residue field $k$ and fraction field $K$. It is a celebrated result of Berthelot and Ogus that for a smooth proper formal scheme $X/\mathcal{O}_K$ there exists a comparison between the de Rham cohomology groups $\mathrm{H}^i_\mathrm{dR}(X/\mathcal{O}_K)$ and the crystalline cohomology groups $\mathrm{H}^i_\mathrm{crys}(X_k/W(k))$ of the special fibre, after tensoring with $K$. In this article, we use the stacky perspective on prismatic cohomology, due to Drinfeld and Bhatt--Lurie, to give a version of this comparison result with coefficients in a perfect complex of prismatic $F$-crystals on $X$. Our method is of an integral nature and suggests new tools to understand the relationship between torsion in de Rham and crystalline cohomology.

math.NT↗

arXiv2024

Prismatic $F$-crystals and Wach modules

We show that the category of analytic/completed prismatic $F$-crystals on the absolute prismatic site of a small (unramified at $p$) base ring is naturally equivalent to the category of relative Wach modules from the theory of $(\varphi, \Gamma)$-modules. The result is obtained by showing that the data of the Galois action on a Wach module is equivalent to the data of a prismatic stratification on the underlying $\varphi$-module. Along the way, we obtain new descent results for relative Wach modules.

math.NT↗

arXiv2024

Crystalline part of the Galois cohomology of crystalline representations

For $p \geqslant 3$ and an unramified extension $F/\mathbb{Q}_p$ with perfect residue field, we define a syntomic complex with coefficients in a Wach module over a certain period ring for $F$. We show that our complex computes the crystalline part of the Galois cohomology (in the sense of Bloch and Kato) of the associated crystalline representation of the absolute Galois group of $F$. Furthermore, we establish that Wach modules of Berger naturally descend over to a smaller period ring studied by Fontaine and Wach. This enables us to define another syntomic complex with coefficients, and we show that its cohomology also computes the crystalline part of the Galois cohomology of the associated representation.

math.NT↗

arXiv2023

Crystalline representations and Wach modules in the relative case II

We study relative Wach modules generalising our previous works on this subject. Our main result shows a categorical equivalence between relative Wach modules and lattices inside relative crystalline representations. Using this result, we deduce a purity statement for relative crystalline representations and provide a criteria for checking crystallinity of relative $p$-adic representations. Furthermore, we interpret relative Wach modules as modules with $q$-connections, and show that for a crystalline representation, its associated Wach module together with the Nygaard filtration is the canonical $q$-deformation (after inverting $p$) of the filtered $(\varphi,\partial)$-module associated to the representation.

math.NT↗

arXiv2023

Crystalline representations and Wach modules in the imperfect residue field case

For an absolutely unramified field extension $L/\mathbb{Q}_p$ with imperfect residue field, we define and study Wach modules in the setting of $(\varphi,\Gamma)$-modules for $L$. Our main result establishes a direct equivalence between the category of lattices inside crystalline representations of the absolute Galois group of $L$ and the category of integral Wach modules for $L$. Moreover, we provide a direct relation between a rational Wach module equipped with the Nygaard filtration and the filtered $\varphi$-module of its associated crystalline representation.

math.NT↗

arXiv2023

Syntomic complex and $p$-adic nearby cycles

In local relative $p$-adic Hodge theory, we show that the Galois cohomology of a finite height crystalline representation (up to a twist) is essentially computed via the (Fontaine--Messing) syntomic complex with coefficients in the associated $F$-isocrystal. In global applications, for smooth ($p$-adic formal) schemes, we establish a comparison between the syntomic complex with coefficients in a locally free Fontaine--Laffaille module and the $p$-adic nearby cycles of the associated \'etale local system on the (rigid) generic fibre.

math.NT↗

arXiv2021

Crystalline representations and Wach modules in the relative case

We study the notion of Wach modules in relative setting, generalizing the arithmetic case. Over an unramified base, for a $p$-adic representation admitting such structure, we examine the relationship between its relative Wach module and filtered $(\varphi, \partial)$-module. Moreover, we show that such a representation is crystalline (in the sense of Brinon), and one can recover its filtered $(\varphi, \partial)$-module from the relative Wach module. Conversely, for low Hodge-Tate weights $[0, p-2]$, we construct relative Wach modules from free relative Fontaine-Laffaille modules (in the sense of Faltings).

math.NT↗