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Modular forms and converse theorems for Dirichlet series
Linköping University, Department of Mathematics, Applied Mathematics.
2009 (English)Independent thesis Advanced level (degree of Master (One Year)), 30 credits / 45 HE creditsStudent thesis
Abstract [en]

This thesis makes a survey of converse theorems for Dirichlet series. A converse theo-rem gives sufficient conditions for a Dirichlet series to be the Dirichlet series attachedto a modular form. Such Dirichlet series have special properties, such as a functionalequation and an Euler product. Sometimes these properties characterize the modularform completely, i.e. they are sufficient to prove the proper transformation behaviourunder some discrete group. The problem dates back to Hecke and Weil, and has morerecently been treated by Conrey The articles surveyed are:

  • "An extension of Hecke's converse theorem", by B. Conrey and D. Farmer
  • "Converse theorems assuming a partial Euler product", by D. Farmer and K.Wilson
  • "A converse theorem for ¡0(13)", by B. Conrey, D. Farmer, B. Odgers and N.Snaith

The results and the proofs are described. The second article is found to contain anerror. Finally an alternative proof strategy is proposed.

Place, publisher, year, edition, pages
2009. , 80 p.
Keyword [en]
Modular forms, Dirichlet series, converse theorems, Hecke groups, Euler products, elliptic curves
National Category
URN: urn:nbn:se:liu:diva-19446ISRN: LiTH-MAT-EX--2009/05--SEOAI: diva2:278038
Physics, Chemistry, Mathematics
Available from: 2009-11-25 Created: 2009-06-23 Last updated: 2010-02-01Bibliographically approved

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