Bounded \(p\)-adic \(L\)-functions of motives at supersingular primes
For a prime number \(p\) let \(X_p:=\text{Hom}_{\text{cont}}({\mathbb Z}_p^\times,{\mathbb C}_p^\times)\). Let \(M\) be a pure motive over \({\mathbb Q}\) of weight \(w=w(M)\) and rank \(d=d(M)\). For the Betti realization \(H_B(M)\) there is a Hodge decomposition into \({\mathbb C}\)-vector spaces \(H_B(M) = \bigoplus_{i+j=w}H^{i,j}(M) \). Let \(h(i,j)=\dim H^{i,j}(M)\), and let \( d^\pm=d^\pm(M)\) be the \( {\mathbb Q}\)-dimension of the \(\pm\)-subspace of \(\rho_B\), the involution of \(H_B(M)\). Let \(L(M,s)=\prod_pL_p(M,p^{-s})\) be the \(L\)-function of \(M\), \(\Lambda(M,s)=L_\infty(M,s)L(M,s) \), where \(L_\infty(M,s)\) is the factor at infinity. Fix a sign \(\varepsilon_0=\pm\). The author formulates \textbf{Conjecture 1.} There exists a \({\mathbb C}_p\)-meromorphic function \(L_p^{(\varepsilon_0)}:X_p\to {\mathbb C}_p\) such that {\parindent=8mm \begin{itemize}\item[(i)]for all but a finite number of pairs \((m,\chi)\in {\mathbb Z}\times X_p^{\text{tors}}\) such that \(M(\chi)(m)\) is critical and \(\varepsilon_0=\text{sgn}((-1)^m\varepsilon(\chi))\), we have \[ L_p^{(\varepsilon_0)}(\chi x_p^m) =G(\chi)^{-d(M)^{\varepsilon_0}}A_p(M(\chi)(m),m)\frac{\Lambda(M(\chi),m)}{\Omega(\varepsilon_0,M)}, \] with \(x_p:{\mathbb Z}_p^\times\to{\mathbb C}_p^\times\) the inclusion, \(G(\chi)\) the Gauss sum, \(\Omega(\varepsilon_0,M) \) one of the modified periods of \(M\), and \(A_p(M(\chi)(m),m)\) an explicit \(p\)-factor.; \item[(ii)]if \(h(\frac w2, \frac w2)=0\), then \(L_p^{(\varepsilon_0)}\) is holomorphic; otherwise the function \(\prod_\xi(x(g_0)-\xi(g_0))^{n(\xi)}L_p^{(\epsilon_0)}(x)\) is holomorphic, where \(\xi\) runs over a finite set of \(p\)-adic characters, \(n(\xi)\) are positive integers, and \(g_0\in {\mathbb Z}_p^\times\); \item[(iii)]if the generalized Hasse invariant of \(M\) \(h_p(M)=0\), then the holomorphic function in (ii) is bounded; \item[(iv)]the function from (ii) is holomorphic of the type \(O(\log_p^{h_p(M)}) \) and can be represented as the Mellin transform of an \(h_p(M)\)-admissible measure. \end{itemize}} Suppose that \(p\) is good for \(M\), and that the inverse roots of \(L_p(M,X)^{-1}\) are indexed in such a way that \(\text{ord}_p\alpha_p^{(1)}\leq \text{ord}_p\alpha_p^{(2)}\leq\ldots\leq \text{ord}_p\alpha_p^{(d)}\). Define \(L_p(M,\psi,T)\) as \(L_p^{(\varepsilon_0)}(\psi\chi_{(1+T)}\), where \(L_p^{(\epsilon_0)}\) is the \(p\)-adic \(L\)-function given by Conjecture 1, \(\psi\) is a fixed tame character such that \(\psi(-1)=\varepsilon_0\). \textbf{Conjecture 2.} Assume that \(p\) is very supersingular for \(M\). Then \[ L_p(M,\psi,T)=L_p^+(M,\psi,T)\cdot\log^+_p(M,T)+ \prod_{i=1}^{d^+}\alpha_p^{(i)}\cdot L_p^-(M,\psi,T)\cdot\log^-_p(M,T), \] where \(L_p^{\pm}(M,\psi,T)\) are bounded. \textbf{Theorem 1.} Conjecture 1 implies Conjecture 2.
scientific article; zbMATH DE number 5886315
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