Size of automorphism group of element of Y 0 ( 5 )The modular curve Y 0 ( 5 ) parametrizes elliptic curves E with an isogeny of degree five. So an element of Y 0 ( 5 ) can be interpreted as E → ϕ E ′ Suppose we are working over a finite field. An automorphism of these curves is an automorphism of the curve E, that sends the kernel G of the map ϕ to itself, over some algebraic closure. The elements of Y 0 ( 5 ) do admit automorphisms, because any curve has the hyperelliptic involution.Because G is of order five, it can have at most four automorphisms. If the curve E has six automorphisms, it is clear, by comparing the sizes of the groups, that only the identity and the hyperelliptic involution respect the subgroup G. If E has four automorphisms, this kind of reasoning doesn't work. Is there a similar result if E has four automorphisms? To me it is not clear if there are always only two automorphisms that map G to itself, or that there could be four.

Question

Size of automorphism group of element of       Y    0    (  5  )The modular curve       Y    0    (  5  ) parametrizes elliptic curves E with an isogeny of degree five. So an element of       Y    0    (  5  ) can be interpreted as  E      →          ϕ            E    ′  Suppose we are working over a finite field. An automorphism of these curves is an automorphism of the curve E, that sends the kernel G of the map   ϕ to itself, over some algebraic closure. The elements of       Y    0    (  5  ) do admit automorphisms, because any curve has the hyperelliptic involution.Because G is of order five, it can have at most four automorphisms. If the curve E has six automorphisms, it is clear, by comparing the sizes of the groups, that only the identity and the hyperelliptic involution respect the subgroup G. If E has four automorphisms, this kind of reasoning doesn&#039;t work. Is there a similar result if E has four automorphisms? To me it is not clear if there are always only two automorphisms that map G to itself, or that there could be four.

kappastud98u · Accepted Answer

Step 1Suppose that E has CM by Z[i]. The prime 5 splits in Z[i], as   5  =  π      π    ¯    ,, where   π  =  2  +  i.Step 2Let G be the subgroup of   π-torsion in E, or the subgroup of       π    ¯  -torsion. These are each of order 5, and are preserved under mult. by all of Z[i], and in particular by the automorphism group   {  ±  1  ,  ±  i  }. Conversely, if G is preserved under   {  ±  1  ,  ±  i  }, then in fact is preserved under all of Z[i], and so is easily checked to be either the   π or       π    ¯   torsion subgroup.So of the 6 subgroups of order 5, two of them are preserved the full automorphism group, and the other four aren&#039;t.

Size of automorphism group of element of Y_0(5).

Answered question

Answer & Explanation

New Questions in College Statistics