The automorphism group of a hyperelliptic curve - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:10:45Z http://mathoverflow.net/feeds/question/47353 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47353/the-automorphism-group-of-a-hyperelliptic-curve The automorphism group of a hyperelliptic curve shenghao 2010-11-25T18:05:32Z 2012-07-01T04:19:59Z <p>Let $C$ be the projective smooth genus 2 curve defined by $y^2=x^5-x$ over $\mathbb F_5.$ What is the order of its automorphism group (automorphisms over $\mathbb F_5$)? </p> <p>I have seen different answers. In Hartshorne's Algebraic Geometry, p. 306, the answer is $2p(p^2-1)=240.$ In INFORMATION Volume 8, Number 6, pp. 837-844, Isomorphism classes of genus-2 hyperelliptic curves over finite fields $\mathbb F_{5^m},$ by L. Hernández Encinas and J. Muñoz Masqué, theorem 2, the answer is $|A_{4221}|=20$ (using notations there). A professor (let me not to mention the name for now) told me the order is 120. </p> <p>Maybe I misunderstand some of the references above?</p> http://mathoverflow.net/questions/47353/the-automorphism-group-of-a-hyperelliptic-curve/47358#47358 Answer by Tim Dokchitser for The automorphism group of a hyperelliptic curve Tim Dokchitser 2010-11-25T18:13:30Z 2010-11-25T20:51:50Z <p>According to Magma it is 120, and it is an extension of $A_5$ by $C_2$ (A:=AutomorphismGroup(HyperellipticCurve(Polynomial(GF(5),[0,-1,0,0,0,1])))), and over $F_{25}$ or over $\bar F_5$ it is 240.</p> <hr> <p><strong>Edit:</strong> Hartshorne works over an algebraically closed field, so Exc. 2.5 on p.305 proves that over $\bar F_5$ the automorphism group has order 240. Explicitly, it is generated by</p> <p>$\alpha: x\mapsto x+1, y\mapsto y$ of order 5,</p> <p>$\beta: x\mapsto \frac{1}{x+1}, y\mapsto \frac{y}{(1+x)^3}$ of order 6,</p> <p>$\gamma: x\mapsto 2x, y\mapsto \sqrt{2}y$ of order 8.</p> <p>Actually, it is clear that the group they generate has order 240 and not less, because $\beta^3$ is not the hyperelliptic involution and $\gamma^4$ is. On the other hand, as Dan explains, you cannot get more than a double cover of $PGL(2,F_5)$, so this is the whole group. Over $F_5$ however, the automorphism group is generated by $\alpha, \beta$ and $\gamma^2$, and it has order 120.</p> http://mathoverflow.net/questions/47353/the-automorphism-group-of-a-hyperelliptic-curve/47359#47359 Answer by Dan Petersen for The automorphism group of a hyperelliptic curve Dan Petersen 2010-11-25T18:13:44Z 2010-11-25T20:07:32Z <p><strong>The following is wrong</strong>:</p> <p>The correct answer is 240. The unnamed professor was probably thinking about the reduced automorphism group (NB. The reduced automorphism group of a hyperelliptic curve is the quotient of the automorphism group by the normal subgroup generated by the hyperelliptic involution.)</p> <p>It is quite easy to describe these explicitly as well. Your curve is the unique double cover of $\mathbb P^1_{\mathbb F_5}$ which is branched over all six points which are rational over $\mathbb F_5$. The reduced automorphism group always consists of the automorphisms which preserve the unordered set of branch points, so in this case it is $PGL(2,\mathbb F_5)$. This group has order $120$.</p> <hr> <p>I think I've got it now. There is a short exact sequence $1 \to \mu_2 \to \mathrm{Aut} C \to PGL(2,\mathbb F_5) \to 1$, where $\mathrm{Aut} C$ denotes the group of automorphisms defined over $\mathbb F_{25}$ or equivalently $\overline{\mathbb F}_5$. </p> <p>Here is what I think is the most natural description of $\mathrm{Aut} C$ and the maps above. Let $G$ be the product $GL(2,\mathbb F_5) \times \overline{\mathbb F}_5^\ast$. Let $\Delta$ denote the normal subgroup $\{(z\cdot \mathrm{id},z^{3}) | z \in \mathbb F_5^\ast \}$ of $G$. We can let $G$ act on $x$ and $y$ via $(\gamma,\rho) \ast (x,y) \mapsto (\frac{ax+b}{cx+d},\frac{\rho y}{(cx+d)^3})$. Then $\Delta$ acts trivially, so we get an action of $G/\Delta$. </p> <p>An explicit computation shows that an element $(\gamma,\rho)$ of $G$ preserves the curve $y^2 = x^5 + x$ if and only if $\det\gamma =\rho^2$. If $\Gamma$ is the subgroup of $G$ defined by this condition, then $\Delta \lhd \Gamma$ and $\mathrm{Aut} C = \Gamma/\Delta$. Now we can see in a very explicit way the short exact sequence above: the inclusion $\mu_2 \to \overline{\mathbb F}_5^\ast$ gives the first map in the short exact sequence, and the projection to the first factor gives the second map. </p> <p>Moreover, it is now clear that exactly half of the automorphisms will be defined over $\mathbb F_5$, namely those for which $\det \gamma$ is a square in $\mathbb F_5$ (and this depends only on the class of $\gamma$ in $PGL(2,\mathbb F_5)$. This recovers Tim's statement that the automorphism group is an extension of $C_2$ by $A_5 \cong PSL(2,\mathbb F_5)$. </p> http://mathoverflow.net/questions/47353/the-automorphism-group-of-a-hyperelliptic-curve/47781#47781 Answer by Steven Galbraith for The automorphism group of a hyperelliptic curve Steven Galbraith 2010-11-30T10:25:59Z 2010-11-30T10:25:59Z <p>There are quite a few papers counting "isomorphism classes" of hyperelliptic curves following on from a paper of Lockhart. They all restrict to isomorphisms which fix a point at infinity. </p> <p>If I recall correctly, the paper by L. Hernández Encinas and J. Muñoz Masqué is in that line of work, which is why their result on Aut( C ) will be different.</p> http://mathoverflow.net/questions/47353/the-automorphism-group-of-a-hyperelliptic-curve/101040#101040 Answer by Tony for The automorphism group of a hyperelliptic curve Tony 2012-07-01T04:19:59Z 2012-07-01T04:19:59Z <p>This paper should provide the answer for all automorphism groups of genus 2 curves defined over an algebraically closed field of characteristic not equal to 2. </p> <p>Shaska, T.; Voelklein, H, <a href="https://0ac73f36-a-62cb3a1a-s-sites.googlegroups.com/site/tshaska/home/papers/d2.pdf?attachauth=ANoY7coDMZVW_Z8yqwa2ggzAY3Y3HGt7C75-_2-AQwY0CCCCHw_q1pLFd2v9o1lEleIutUZMj5tYYVe3ZEKsQddn3x7Mkv1DNJjb_XNWAhKnQSDXoSFWpL1FnYeaZM7gnwsk6bBOfrcVcw4i0ZoUpngkzo_z8Gm-uGHUOmjb22rPhmfoc4SjAEXPqwdveNn0x01GPDCsCSOsfH0Wy8YMemzjbtR-wf_oHQ%253D%253D&amp;attredirects=0" rel="nofollow">Elliptic subfields and automorphisms of genus 2 function fields.</a> Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000),703--723, Springer, Berlin, 2004.</p>