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The modular automorphisms of quotient modular curves

Francesc Bars, Tarun Dalal

TL;DR

This work determines the modular automorphism groups of quotient modular curves $X_0(N)/W_N$ for $4,9\nmid N$ by describing the full normalizers $\mathcal{N}(\langle \Gamma_0(N),W_N\rangle)$. Employing Conway's Big Picture, the authors first bound and then exactly compute these normalizers, distinguishing cases where Atkin–Lehner involutions are square vs non-square, and treating the special role of $w_{25}$. A key outcome is that, in general, $\mathcal{N}(\langle \Gamma_0(N),W_N\rangle)/\langle \Gamma_0(N),W_N\rangle$ contains only the abelian group $B(N)/W_N$, but in the exceptional $w_{25}$-involved cases there are extra elements, including order-$3$ automorphisms that may be defined over $\mathbb{Q}(\sqrt{5})$. These results yield the full automorphism group of $X_0^*(N^2)$ for sufficiently large $N$ and explain the occurrence of nontrivial automorphisms of order $3$ on certain quotient curves (e.g., involving $w_{25}$). Together, the paper advances understanding of how modular symmetries of quotient curves relate to arithmetic properties of the corresponding Jacobians and rational points.

Abstract

We obtain the modular automorphism group of any quotient modular curve of level $N$, with $4,9\nmid N$. In particular, we obtain some non-expected automorphisms of order 3 that appear for the quotient modular curves when the Atkin-Lehner involution $w_{25}$ belongs to the quotient modular group, such automorphisms are not necessarily defined over $\mathbb{Q}$. As a consequence of the results, we obtain the full automorphism group of the quotient modular curve $X_0^*(N^2)$, for sufficiently large $N$.

The modular automorphisms of quotient modular curves

TL;DR

This work determines the modular automorphism groups of quotient modular curves for by describing the full normalizers . Employing Conway's Big Picture, the authors first bound and then exactly compute these normalizers, distinguishing cases where Atkin–Lehner involutions are square vs non-square, and treating the special role of . A key outcome is that, in general, contains only the abelian group , but in the exceptional -involved cases there are extra elements, including order- automorphisms that may be defined over . These results yield the full automorphism group of for sufficiently large and explain the occurrence of nontrivial automorphisms of order on certain quotient curves (e.g., involving ). Together, the paper advances understanding of how modular symmetries of quotient curves relate to arithmetic properties of the corresponding Jacobians and rational points.

Abstract

We obtain the modular automorphism group of any quotient modular curve of level , with . In particular, we obtain some non-expected automorphisms of order 3 that appear for the quotient modular curves when the Atkin-Lehner involution belongs to the quotient modular group, such automorphisms are not necessarily defined over . As a consequence of the results, we obtain the full automorphism group of the quotient modular curve , for sufficiently large .
Paper Structure (10 sections, 31 theorems, 96 equations)

This paper contains 10 sections, 31 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. The Conway Big Picture for quotient modular groups
  3. The Big Picture
  4. The normalizer of $\Gamma:=\Gamma_0(N)+W_N$ with $w_{u^2,N}\notin\Gamma$.
  5. Towards the normalizer of $\Gamma:=\Gamma_0(N)+W_N$ with $w_{u^2,N}\in\Gamma$.
  6. Exact normalizer of $\langle \Gamma_0(N),W\rangle$
  7. Exact normalizer of $\langle \Gamma_0(N), w_{u_1^2}, w_{u_2^2}, \ldots, w_{u_k^2}\rangle$ with $w_{25}\notin \langle \Gamma_0(N), w_{u_1^2}, w_{u_2^2}, \ldots, w_{u_k^2}\rangle$
  8. Exact normalizer of $\langle \Gamma_0(N), w_{25}, w_{u_2^2}, \ldots, w_{u_k^2}\rangle$
  9. Exact normalizer of $\langle \Gamma_0(N), W\rangle$ for arbitrary subgroup $W$
  10. On the modular automorphisms of order 3 of $X_0(N)/W_N$

Key Result

Theorem 1.1

[Theorem Complete normalizer without 25 theorem in text] Let $N\in \mathbb{N}$ and $W_N$ be a subgroup generated by the Atkin-Lehner involutions such that $4,9\nmid N$ and $w_{25}\notin W_N$. Then $\mathcal{N}(\langle \Gamma_0(N), W_N\rangle) = \langle \Gamma_0(N),{ w_{d}}: d||N\rangle.$

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1: Conway
  • Theorem 2.2: Conway
  • Example 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 42 more