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Root numbers for twisted Fermat quotient curves

Ryosuke Yanagihara

TL;DR

The paper addresses root numbers for twisted Fermat quotient curves $X^{\ell^N}+Y^{\ell^N}=\delta$ by decomposing the Jacobian $J_N$ into CM pieces and associating Hecke characters $\varphi_{\delta}^{(i)}$ to the new parts. It develops an explicit local–global framework for the root number $W(\phi_{\delta}^{(N)})$ via the factorization $L(s, J_N)=\prod_i L(s, \mathrm{Jac}(C_i)^{\rm new})=\prod_i L(s,\varphi_{\delta}^{(i)})$, applies Rohrlich's formula and Hilbert symbol calculus at ramified primes, and expresses the final root number in terms of local data, including a Fleck-number–type sum $J$ arising from combinatorial sums. The global conductor $N_{r,s,t,\delta}$ is given explicitly, tying the analytic side to the arithmetic of ramification and CM structure; the results generalize prior work of Stoll and Shu and illuminate how root numbers encode Mordell–Weil rank information via BSD. The paper also offers a conjectural refinement involving Fleck congruences, supported by numerical checks for several $(N,\ell)$.

Abstract

Let $\ell$ be an odd prime, $N \geq 1$ be an integer, and $δ\geq 1$ be a $\ell^N$-th power free integer such that ${\rm ord}_{\ell}(δ) = 0$ or $\ell \nmid {\rm ord}_{\ell}(δ)$. In this paper, we give an explicit formula for the root number of the Hecke character associated with a certain quotient curve of the twisted Fermat curve $X^{\ell^N} + Y^{\ell^N} = δ$. This result gives a generalization of Stoll (2002) and Shu (2021).

Root numbers for twisted Fermat quotient curves

TL;DR

The paper addresses root numbers for twisted Fermat quotient curves by decomposing the Jacobian into CM pieces and associating Hecke characters to the new parts. It develops an explicit local–global framework for the root number via the factorization , applies Rohrlich's formula and Hilbert symbol calculus at ramified primes, and expresses the final root number in terms of local data, including a Fleck-number–type sum arising from combinatorial sums. The global conductor is given explicitly, tying the analytic side to the arithmetic of ramification and CM structure; the results generalize prior work of Stoll and Shu and illuminate how root numbers encode Mordell–Weil rank information via BSD. The paper also offers a conjectural refinement involving Fleck congruences, supported by numerical checks for several .

Abstract

Let be an odd prime, be an integer, and be a -th power free integer such that or . In this paper, we give an explicit formula for the root number of the Hecke character associated with a certain quotient curve of the twisted Fermat curve . This result gives a generalization of Stoll (2002) and Shu (2021).

Paper Structure

This paper contains 14 sections, 18 theorems, 110 equations, 5 tables.

Table of Contents

  1. Introduction
  2. $L$-function for the curve $C_N$
  3. $L$-functions for Abelian varieties
  4. $L$-function for nonsingular projective algebraic curves
  5. Computation of the $L$-polynomial for the curve $C_N$
  6. A Hecke character associated to the new part of $\mathrm{Jac} (C_N)$
  7. Root numbers for the Hecke characters
  8. Preparation for Root Numbers
  9. Calculation of $W_p(\phi_{\delta})$ when $p\neq \ell$
  10. Calculation of $W_{\ell}(\phi_{\delta})$
  11. Calculation of $W(\eta_{\pi_F})$.
  12. Calculation of the relative root number $W(\phi_{\delta,\pi}, \eta_{\pi_F})$.
  13. Global root number $W(\phi_{\delta}^{(i)})$ and global conductor $N_{r,s,t,\delta}$
  14. Some speculation

Key Result

Theorem 1.2

Let $r, s, t > 0$ be integers such that $r + s + t = \ell^N$ and $\ell \nmid rst$. Let $\epsilon'_N \in \mu_{\ell-1}(\mathbb{Q}_{\ell})$, $b'_N \in \mathbb{Z}$, and $c_N' \in \ell \mathbb{Z}_{\ell}$ such that $r^r s^s (\ell^N-t)^t \delta^{r+s} = \epsilon'_N \ell^{b'_N}(1 + c_N')$. Then the global ro where for primes $p \neq \ell$, and, where Since $J$ is originally defined as $J=\frac{I}{\ell^{

Theorems & Definitions (37)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 4.1
  • ...and 27 more