On congruences between the coefficients of two \(L\)-series which are related to a hyperelliptic curve over \(\mathbb Q\) (Q5937555)

Let \(f(x)\in \mathbb{Q}[X]\) be a polynomial of degree \(n\) with no multiple roots. Let \(C\) be a hyperelliptic curve defined by \(y^2=f(x)\), of genus \(g\geq 1\) and with at least one \(\mathbb{Q}\)-rational point. Let \(J= \text{Jac}(C)\) be its Jacobian variety, defined over \(\mathbb{Q}\). Let \(K/\mathbb{Q}\) denote the Galois extension generated by the coordinates of 2-divisible points of \(J\); put \(G= \text{Gal}(K/\mathbb{Q})\). Let \(\rho_2\) be the 2-adic representation of \(\text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\) on the 2-adic Tate module of \(J\), and \(\pi\) the restriction of the standard representation of \(G(GS_n)\), where \(n=2g+1\) or \(2g+2\). For any odd good prime \(p\) (for \(J\)) we put NEWLINE\[NEWLINE\begin{aligned} P_p(u) &:= \det(I_{n-1}- \pi(\sigma_{\mathfrak p})u),\\ Q_p(u) &:= \det(I_{2g}- \rho_2(\sigma_\wp)u), \end{aligned}NEWLINE\]NEWLINE where \(\sigma_\wp\) is the Frobenius automorphism at \(\wp/p\) (in \(\overline{\mathbb{Q}}\)) and \(\sigma_{\mathfrak p}\) is its restriction to \(K\). The main result is the following congruence: NEWLINE\[NEWLINE\begin{alignedat}{2} P_p(u)&\equiv Q_p(u)\bmod 2 &\quad &(1\neq n\text{ odd})\\ \text{and} P_p(u)&\equiv (1-u) Q_p(u)\bmod 2 &\quad &(n\text{ even},\;\neq 2,3). \end{alignedat}NEWLINE\]NEWLINE The result generalizes the theorem of \textit{M. Koike} \((n=3)\) [Nagoya Math. J. 98, 109-115 (1985; Zbl 0569.12007)]. NEWLINENEWLINENEWLINENote that \(1/P_p(p^{-s})\) (resp. \(1/Q_p(p^{-s})\)) is the \(p\)-factor of the Artin \(L\)-series \(L(\pi, K/\mathbb{Q},s)\) (resp. the \(L\)-series \(L(J/\mathbb{Q},s)\)).