arXiv subjects

Viet-Cuong Pham

Publications and source records attributed to Viet-Cuong Pham.

5 recordsLinked to original sources

arXiv2026

The Exotic $K(2)$-Local Picard Group at the Prime $2$

We calculate the group $κ_2$ of exotic elements in the $K(2)$-local Picard group at the prime $2$ and find it is a group of order $2^9$ isomorphic to $(\mathbb{Z}/8)^2 \times (\mathbb{Z}/2)^3$. In order to do this we must define and exploit a variety of different ways of constructing elements in the Picard group, and this requires a significant exploration of the theory. The most innovative technique, which so far has worked best at the prime $2$, is the use of a $J$-homomorphism from the group of real representations of finite quotients of the Morava stabilizer group to the $K(n)$-local Picard group.

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arXiv2022

Cohomology of the Morava stabilizer group through the duality resolution at $n=p=2$

We compute the continuous cohomology of the Morava stabilizer group with coefficients in Morava $E$-theory, $H^*(\mathbb{G}_2, E_t)$, at $p=2$, for $0\leq t < 12$, using the Algebraic Duality Spectral Sequence. Furthermore, in that same range, we compute the $d_3$-differentials in the homotopy fixed point spectral sequence for the $K(2)$-local sphere spectrum. These cohomology groups and differentials play a central role in $K(2)$-local stable homotopy theory.

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arXiv2021

The topological modular forms of $\mathbb{R}P^2$ and $\mathbb{R}P^2 \wedge \mathbb{C}P^2$

In this paper, we study the elliptic spectral sequence computing $tmf_*(\mathbb{R} P^2)$ and $tmf_* (\mathbb{R} P^2 \wedge \mathbb{C} P^2)$. Specifically, we compute all differentials and resolve exotic extensions by 2, $η$, and $ν$. For $tmf_* (\mathbb{R} P^2 \wedge \mathbb{C} P^2)$, we also compute the effect of the $v_1$-self maps of $\mathbb{R} P^2 \wedge \mathbb{C} P^2$ on $tmf$-homology.

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arXiv2019

On the tmf-Hurewicz image of $A_1$

Let $A_1$ be any spectrum in the class of finite spectra whose mod-2 cohomology is isomorphic to $\mathcal{A}(1)$ as a module over the subalgebra $\mathcal{A}(1)$ of the Steenrod algebra; let $tmf$ be the connective spectrum of topological modular forms. In this paper, we prove that the $tmf$-Hurewicz image of $A_1$ is surjective.

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arXiv2019

Homotopy groups of $E_{C}^{hG_{24}}\wedge A_1$

Let $A_1$ be any spectrum in a class of finite spectra whose mod $2$ cohomology is isomorphic to a free module of rank one over the subalgebra $\mathcal{A}(1)$ of the Steenrod algebra. Let $E_{C}$ be the second Morava-$E$ theory associated to a universal deformation of the formal completion of the supersingular elliptic curve $(C) : y^{2}+y = x^{3}$ defined over $\mathbb{F}_{4}$ and $G_{24}$ a maximal finite subgroup of automorphism group $\mathbb{S}_{C}$ of the formal completion of $C$. In this paper, we compute the homotopy groups of $E_{C}^{hG_{24}}\wedge A_1$ by means of the homotopy fixed point spectral sequence.

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