# Questions tagged [characteristic-2]

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### Is there any point in considering Form Rings when 2 admits an inverse?

In the study of quadratic spaces over general rings, there is a type of scalar which people consider called a
Form ring $(R,\Lambda)$ relative to some anti-automorphism denoted $(-)^J:R\to R$ and ...

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### Proof of Theorem Concerning Conway's "Nim Field"

I have a question about the proof of theorem 44 in "On Numbers and Games" on page 58, concerning the "Nim field" ${ON}_2$. As background, ${ON}_2$ turns the ordinals into a field ...

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### Solving efficiently a quadratic equation in a large finite field of characteristic two

I'm trying to solve efficiently a quadratic equation in the finite field $\text{GF}(2^{128})$ represented as $(\mathbb{Z}/2\mathbb{Z})[x] / (x^{128} + x^7 + x^2 + x + 1)$.
Until now, I came across ...

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### Generators of the orthogonal group of a quadratic form in odd dimension in characteristic 2

In characteristic not $2$, the Theorem of Cartan-Dieudonné states:
[Grove, Theorem 6.6]: Let $q$ be a nondegenerate symmetric quadratic form of dimension $n$ in characteristic not $2$. Then every ...

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### Surfaces of general type with $h^1(-K_X)\neq 0$

By a result of Ekedahl, in characteristic 2 one may have minimal surfaces of general type such that $h^1(X,-K_X)\neq 0$ and $X$ is birational to an inseparable double cover of a rational surface. How ...

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### Do regular (but non-smooth) conics over a discretely valued field of characteristic $2$ admit a regular model over the valuation ring?

Let $K$ be a non-perfect field of characteristic $2$. Let $T \subseteq K$ be a discrete valuation ring.
Assume there exist $a,b \in K^{\times}$ such that the projective conic $C$ defined by $$aX^2 + ...

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### The adjoint representation of the symplectic group in characteristic 2

For a prime $p$ and some $g \geq 2$, consider the adjoint representation $\mathfrak{sp}_{2g}(\mathbb{F}_p)$ of the symplectic group $\text{Sp}_{2g}(\mathbb{F}_p)$. For $p \geq 3$, it is not hard to ...

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### Conceptual explanation for curious linear-algebra fact in characteristic $2$

All matrices and vectors in this post have entries in the field $\mathbb{F}_2$.
Fix some $n \geq 1$. For an $n \times n$ matrix $X$, write $X_0$ for the column vector whose entries are the diagonal ...

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### A generalization of Witt's theorem for quaternion algebra isomorphism

Let $Q$ be a quaternion $k$-algebra (namely, a dimension 4 $k$-central simple algebra).
Then it is possible to (canonically) attach a smooth projective conic $C_Q\subseteq \mathbf{P}_k^2$ to $Q$: if ...

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### A strange (possible) fact about the Hecke operator T_3 in level 13 and characteristic 2

delta(z) + delta (13z) is a weight 12 modular form of level Gamma_0 (13). Let A in Z/2[[q]] be the mod 2 reduction of the Fourier expansion of this form. (The exponents appearing in A are the odd ...

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### Mod 2 eigensystems not defined over Z/2--looking for simple examples

Consider the weight 2 newform 67.2.1 b in the LMFDB table. It is defined over Q(root 5), and reducing modulo the inert prime (2) we get a mod 2 eigensystem defined over an extension of Z/2 but not ...

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### How to enumerate the extended affine equivalence classes of bent functions of degree 4 in 8 variables?

"There are 536 class of quartic forms Q (header) [in 8 boolean variables] providing bent functions of the form Q+f where f is a cubic functions."
Philippe Langevin, 2008.
What is the current ...

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### A problem in Galois Geometry

Given a prime $p$, out of $N$ vectors of length $p^k$ over $\Bbb F_2$ of Hamming weight $w^{k}$ that are chosen, how many vectors can there be with pairwise Hamming distance at least $2w^{k}$ given ...

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### Effectiveness of the distinguished theta characteristic in characteristic 2

Let $k$ be an algebraically closed field of characteristic 2. Let $C$ be a (smooth projective connected) curve over $k$. Can there exist a rational function on $C$ whose differential is holomorphic ...

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### Sets from $(F_2)^n$ which are not fixed by any non-identity isomorphism

This is a followup question to the discussion in the comments of
Sets which are not fixed by any non-identity isomorphism
So consider a finite $n$-dimensional vector space $V$ over $F_2$. For which ...

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### Are these two subspaces of $\mathbb{Z}/2[[x]]$ the same?

The following questions arise from modular form theory. But this theory isn't needed to formulate or understand them, and I'm not using the modular-forms tag.
NOTATION
Fix an odd prime $N$. Let
$$
...