Modular elements over function fields
type: Item
instance of: scholarly article
title: Modular elements over function fields
zbMATH Open document ID: 0802.11026
author: Ki-Seng Tan
published in: Journal of Number Theory
publication date: 1994-12-15T00:00:00Z
review text: This paper deals with the theory of ``modular elements'' for elliptic curves over global fields of characteristic \(p\). The modular elements under consideration are function field versions of the objects considered by B. Mazur and J. Tate in their study of ``refined'' versions of the \(p\)-adic Birch-Swinnerton-Dyer conjecture [see \textit{B. Mazur} and \textit{J. Tate}, Duke Math. J. 54, 771-750 (1987; Zbl 0636.14004)]. The modular elements of this paper, like those of Mazur and Tate, are associated to modular forms and are related to special values of the associated \(L\)- functions. It is possible to express a function field version of Mazur and Tate's refined conjectures using these elements. This paper recalls Mazur's construction of the modular elements in the function field case and proves some basic integrality results for them; it derives a general functional equation satisfied by the modular elements; and it presents some examples where the function field version of the refined conjecture holds true. The paper does not present the details of these examples, but instead cites the author's thesis [``Refined conjecture of the Birch and Swinnerton-Dyer type'', Thesis, Dep. of Math., Harvard Univ. (June 1990)] and another paper with the same title which has not yet appeared. Let \(k\) be a global field of characteristic \(p\) over the field \(\mathbb{F}_ q\), \({\mathbf A}_ k^ \times\) the ideles of \(k\), \(U_ D\) the compact open subgroup of \({\mathbf A}_ k^ \times\) determined by a divisor \(D\) of \(k\), and \[ W_ D= k^* \setminus {\mathbf A}_ k^ \times /U_ D \] the Weil group of conductor \(D\). If \(f\) is a cusp form on \(\text{GL}_ 2 ({\mathbf A}_ k)\), then the modular element \(\Theta_{D,f}\) is the element of the group ring \({\mathbb{Q}} [W_ D]\) obtained as the sum \[ \Theta_{D,f}:= {1\over {I_ D}} \sum_{w\in W_ D} \Biggl(f \begin{pmatrix} dw &w\\ 0 &1 \end{pmatrix}\Biggr)w \] where \(I_ D=1\) if \(D\) is non-trivial and \(I_ D= q-1\) otherwise. …
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 483990
zbMATH Keywords: elliptic curves
zbMATH Keywords: function fields
zbMATH Keywords: Birch-Swinnerton-Dyer conjecture
zbMATH Keywords: modular elements
zbMATH Keywords: modular forms
zbMATH Keywords: special values
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: Mazur and Tate's refined conjectures
zbMATH Keywords: integrality
zbMATH Keywords: functional equation
reviewed by: Jeremy Teitelbaum
MaRDI profile type: Publication
full work available at URL: jnth.1993.1079
OpenAlex ID: W2062292279
DOI: 10.1006/JNTH.1993.1079
description: scientific article; zbMATH DE number 483990