On the automorphism groups of some modular curves

Abstract

Let $N$ be a prime number, and $X$ a curve that is intermediate to the cover $X_1(N)\rightarrow X_0(N)$. We study the automorphism group of $X$, and prove that in most cases it is generated by the Galois group of the cover $X\rightarrow X_0(N)$, and any lift of the Atkin–Lehner involution to $X$.

Bielliptic intermediate modular curves

“…And on the other hand, after eliminating the mistake in [Mo] one would have a result that is much more general than the currently established ones. The paper [Ka2] determines the automorphism group of X ∆ (N ) for N a prime bigger than 311 (which is too big to be of any help to us). Both, [Ka2] itself and its review in MathSciNet suggest that it should be possible to generalize to the case of square-free N .…”

“…The paper [Ka2] determines the automorphism group of X ∆ (N ) for N a prime bigger than 311 (which is too big to be of any help to us). Both, [Ka2] itself and its review in MathSciNet suggest that it should be possible to generalize to the case of square-free N .…”

The group of automorphisms of the Heisenberg curve

“…Remark 4. Deciding if there are extra non-modular automorphism is a difficult classical question for the case of modular and Shimura curves, see [1], [13], [12], [8], [16], [21], [17] for some related results.…”