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Questions tagged [characteristic-2]

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10
votes
1answer
150 views

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 ...
14
votes
1answer
313 views

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 ...
4
votes
1answer
85 views

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 ...
4
votes
1answer
239 views

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 ...
5
votes
2answers
202 views

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 ...
2
votes
1answer
110 views

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 ...
2
votes
0answers
243 views

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 ...
9
votes
2answers
237 views

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 ...
7
votes
3answers
267 views

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 ...
6
votes
0answers
315 views

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 $$ ...