Bounded \(p\)-adic \(L\)-functions of motives at supersingular primes
type: Item
instance of: scholarly article
title: Bounded \(p\)-adic \(L\)-functions of motives at supersingular primes
zbMATH Open document ID: 1219.11075
author: Andrzej Dąbrowski
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2011-05-10T00:00:00Z
review text: 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 …
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 5886315
zbMATH Keywords: motives
zbMATH Keywords: \(p\)-adic \(L\)-function
zbMATH Keywords: supersingular primes
MaRDI profile type: Publication
full work available at URL: j.crma.2011.03.015
OpenAlex ID: W2123218197
cites work: Q4097036
cites work: Admissible \(p\)-adic measures attached to triple products of elliptic cusp forms
cites work: Q3487506
cites work: Q3972986
cites work: \(p\)-adic \(L\)-functions of Hilbert modular forms
cites work: S -adic L -Functions Attached to the Symmetric Square of a Newform
cites work: Admissible \(p\)-adic \(L\)-functions of automorphic forms
cites work: Q3895546
cites work: The existence of infinitely many supersingular primes for every elliptic curve over \(\mathbb Q\).
cites work: Wach modules and Iwasawa theory for modular forms
cites work: On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer
cites work: Q4033549
cites work: Q4297083
cites work: Motives over totally real fields and \(p\)-adic \(L\)-functions
cites work: Plus/minus \(p\)-adic \(L\)-functions for Hilbert modular forms
cites work: Q4840145
cites work: On the \(p\)-adic \(L\)-function of a modular form at a supersingular prime
cites work: Q4133740
cites work: On the \(p\)-adic \(L\)-function of Hilbert modular forms at supersingular primes
DOI: 10.1016/J.CRMA.2011.03.015
description: scientific article; zbMATH DE number 5886315