Existence of hyperelliptic curve with specific number of points in a family - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T12:08:52Z http://mathoverflow.net/feeds/question/4894 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4894/existence-of-hyperelliptic-curve-with-specific-number-of-points-in-a-family Existence of hyperelliptic curve with specific number of points in a family Dan Petersen 2009-11-10T17:29:32Z 2010-02-01T21:11:44Z <p>Hi,</p> <p>the following question was posed to me, it apparently has applications for linear codes. Let <em>n>1</em>, and $K = \rm{GF}(2^n)$. Let $k$ be coprime to $2^n-1$. Does there always exist $a \neq 0$ in $K$ such that the curve</p> <p>$y^2+y = x^k+ax$</p> <p>has exactly $2^n$ affine solutions? (I ran some computer checks [although only for quite small <em>n</em> and <em>k</em>] without finding a counterexample.)</p> http://mathoverflow.net/questions/4894/existence-of-hyperelliptic-curve-with-specific-number-of-points-in-a-family/13630#13630 Answer by unknown (google) for Existence of hyperelliptic curve with specific number of points in a family unknown (google) 2010-02-01T06:13:00Z 2010-02-01T21:11:44Z <p>I think that this is equivalent to a known open question. Here are the details. For $K:=\mathbb{F}_{2^n}$, the function $f:y\mapsto y+y^2:K\to K$ is $\mathbb{F}_2$-linear, and its kernel $\{0,1\}$ has dimension 1. The image is therefore of dimension $n-1$, and for $z$ in the image, the fiber $f^{-1}(z)$ has exactly 2 elements. </p> <p>Hence, to prove that $y^2+y=x^k+ax$ has exactly $2^n$ solutions for some fixed $a\in K$, we have to show that $|\{x\in K|x^k+ax\in \mathrm{Im}(f)\}|=2^{n-1}$. </p> <p>Since $\sigma:y\mapsto y^2$ is a generator of the Galois group of $K/\mathbb{F}_2$, Hilbert's Theorem 90 (in additive form) says that $z\in \mathrm{Im}(f)$ if and only if $\mathrm{Tr}(z)=0$, where $\mathrm{Tr}$ stands for the trace map from $K$ to $\mathbb{F}_2$. </p> <p>So the problem is equivalent to showing that there exists an $a\neq 0$ in $K$ such that $|\{x\in K|\mathrm{Tr}(x^k+ax)=0\}|=2^{n-1}$. In other words, we would like to show that there exists a nonzero $a\in K$ such that $$S_k(a):=\sum_{x\in K}(-1)^{\mathrm{Tr}(x^k+ax)}=0.$$</p> <p>Apparently, this question was addressed in the coding community. In detail, in [1, p. 258], the following conjecture (of Helleseth) is mentioned:</p> <p>Conjecture 3. For any $m$ and $k$ such that $\mathrm{gcd}(2^m-1,k)=1$, the sum $\sum_{x\in\mathbb{F}_{2^m}}(-1)^{\mathrm{Tr}(x^k+ax)}$ is null for at least one nonzero $a$.'' (Note that $n$ in the current question is $m$ in [1]).</p> <p>It seems that in [1, Corollary 1, p. 253], Conjecture 3 is proved for even $m$ and for certain values of $k$ (the Niho exponents,'' defined on p. 252 of [1]).</p> <p>Interestingly, at least at a first glance it seems that [1] has nothing to say on $k\in\{1,\ldots,2^{n-1}\}$, but to me it seems that this case is trivial (am I missing something?): Consider a normal basis for $K/\mathbb{F}_2$, that is, a basis $B$ consisting of an orbit of an element $\gamma\in K$ under the Galois group of $K/\mathbb{F}_2$ (the $i$th element of $B$ is $b_i:=\gamma^{2^i}$ for $i\in\{0,\ldots,n-1\}$). </p> <p>From the linearity of the trace and the fact that the trace is onto, we must have $\mathrm{Tr}(b)=1$ for at least one element $b\in B$, and from $\mathrm{Tr}(b^2)=\mathrm{Tr}(b)$ we then have $\mathrm{Tr}(b)=1$ for all $b\in B$. So the trace of an element in $K$ is just the modulo-2 sum of the coefficients in its decomposition according to the basis $B$. </p> <p>Let $a$ be any element in the trace-dual basis of $B$, say $\mathrm{Tr}(ab_i)=\delta_{i,0}$. Then for $k=2^j$, if we write $x=\sum_i \alpha_i b_i$, we get: $\mathrm{Tr}(ax)=\alpha_0$, $\mathrm{Tr}(x^k)=\mathrm{Tr}(x)=\sum \alpha_i$ (sum in $\mathbb{F}_2$). These agree for half of the $x\in K$, as required. </p> <p>That's about it. I hope at least some of this makes sense :) I also hope that the original person asking this question didn't actually want to solve the above open question by converting it to a question about curves, for then this answer is useless.</p> <p>[1] P. Charpin, <a href="http://www-rocq.inria.fr/secret/Pascale.Charpin/Charpin-jct04.pdf" rel="nofollow">Cyclic codes with few weights and Niho exponents</a>,'' Journal of Combinatorial Theory, Series A 108 (2004) 247--259.</p>