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For an integer      M &gt; 1 M&gt;1       and a prime      p ≥  5 p \geq 5       not dividing      M M      , we study the kernel of the degeneracy map      Φ  M p , p r →  Φ  M p r , p \Phi _{Mp,p}^r \rightarrow \Phi _{Mp^r, p}      , where      Φ  M p , p \Phi _{Mp,p}       and      Φ  M p r , p \Phi _{Mp^r, p}       are the component groups of      J 0 ( M p ) J_0(Mp)       and      J 0 ( M p r ) J_0(Mp^r)      , respectively. This is then used to determine the kernel of the degeneracy map      J 0 ( M p ) 2 →  J 0 ( M p 2 ) J_0(Mp)^2 \rightarrow J_0(Mp^2)       when      J 0 ( M ) = 0 J_0(M) =0      . We also compute the group structure of      Φ  M p 2 , p \Phi _{Mp^2, p}       in some cases.

On component groups of 𝐽₀(𝑁) and degeneracy maps