Conjectures Relating to a Generalization of the Ramanujan Tau Function

Jesse Ira Deutsch

Chapter 5 in Number Theory, 1991, pp 105-118 from Springer

Abstract: Abstract While working with Hecke transforms of weight 12 modular forms over Q(√2) and Q(√3) a peculiar phenomenom was observed. The characteristic polynomial for the Hecke transform factored into a linear and an irreducible part. The linear factor was related to the Ramanujan tau function in an intrinsic fashion that implied this phenomenom may be more than an accident. For instance, over Q(√2) the Hecke transform of index 2 + √2 had -24 as an eigenvalue, while the Hecke transform of index 3 + √2 had eigenvalue -16744. One notes τ ( N ( 2 + √ 2 ) ) = τ ( 2 ) = − 24 τ ( N ( 3 + √ 2 ) ) = τ ( 7 ) = − 16744 $$ \begin{array}{*{20}{c}} {\tau \left( {N\left( {2 + \surd 2} \right)} \right) = \tau \left( 2 \right) = - 24} \\ {\tau \left( {N\left( {3 + \surd 2} \right)} \right) = \tau \left( 7 \right) = - 16744} \end{array} $$ where N is the field norm of Q(√2) over Q. Unfortunately some of the eigenvalues of the corresponding modular eigenform are not Ramanujan tau function values. An explanation of this phenomenom is given by Doi-Naganuma lifting of modular forms. The author would like to thank Harvey Cohn and Carlos Moreno for illuminating discussions.

Date: 1991
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4757-4158-2_5

Ordering information: This item can be ordered from
http://www.springer.com/9781475741582

DOI: 10.1007/978-1-4757-4158-2_5

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().