On the \(p\)-adic \(L\)-function of a modular form at a supersingular prime
type: Item
instance of: scholarly article
title: On the \(p\)-adic \(L\)-function of a modular form at a supersingular prime
zbMATH Open document ID: 1074.11061
DOI: 10.1215/S0012-7094-03-11835-9
author: Robert Pollack
published in: Duke Mathematical Journal
publication date: 2003-10-13T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.dmj/1082744678
review text: In this important paper the author shows what the ``arithmetically significant'' part of the \(p\)-adic \(L\)-function attached to a modular form \(f\) of weight \(k\) should be in the supersingular case. Actually one has to deal with two functions indexed by + and \(-\). For instance when \(f\) comes from a (modular) elliptic curve \(E\) with \(a_p=0\) (which is, for \(p>3\), tantamount to supersingularity at \(p\)), and \(\alpha\) is a root of \(x^2+p\), then \(L_p(E,\alpha,T) = G^+(T) + G^-(T)\alpha\) where \(G^\pm(T)\) has \(p\)-adic rational coefficients. From the interpolation approach to \(p\)-adic \(L\)-functions it follows that \(G^\pm(T)\) has an infinite series of trivial zeros (in the supersingular case which we assume from now on), and this actually implies that \(L_p(E,\alpha,T)\) has infinitely many zeros as well, even though we don't see them immediately. (These observations are due to B.~Perrin-Riou.) A similar result holds for \(L\)-functions involving twists. The task is to eliminate these predictable zeros. The paper carefully reviews the construction of \(p\)-adic \(L\)-functions, following Vishik and others, in \S2. Then the author proceeds to modify \(G^\pm(T)\) so as to eliminate the trivial zeros. To make this beautiful idea work, he defines two power series (of necessity, with unbounded denominators in their coefficients) \(\log^+_p\) and \(\log^-_p\), the so-called half-logarithms, whose zeros are exactly the trivial zeros of \(G^\pm(T)\). For \(k=2\), these half-logs live in \({\mathbb Q}_pT\). In general, they involve an extra integral parameter \(j=0,\ldots,k-2\). The definition of the half-logs is quite elegant: one takes the product of all \(p^n\)-th cyclotomic polynomials with \(n\) of a fixed parity, each polynomial divided by \(p\). Recall that these cyclotomic polynomials are Eisenstein in \(T\) with constant coefficient exactly \(p\); so the division by \(p\) changes this into 1, and one sees at least that the constant term o…
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 1990939
zbMATH Keywords: L-functions
zbMATH Keywords: elliptic curves
zbMATH Keywords: trivial zeros
zbMATH Keywords: Iwasawa invariants
reviewed by: Cornelius Greither
describes a project that uses: ecdata
MaRDI profile type: Publication
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OpenAlex ID: W2034350303
description: scientific article; zbMATH DE number 1990939