2005/06/30 by Eduardo Duenez, Eduardo Dueñez, Steven J. Miller · 35 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #Conjecture #Lying #Moment (physics) #Symmetry (geometry) #Symplectic geometry #Type (biology) #math-ph #math.MP #math.NT #msc:11M26 #msc:11M41 #msc:15A52
paper · pdf · doi:10.1112/s0010437x0600220x
published in Compositio Mathematica 142(06), 1403-1425 (Cambridge University Press) · 26 pages: revised draft: fixed some typos, added an appendix with the calculations for the signs of the functional equation and Gamma factors
arxiv created 2006/03/20 · openalex publication_date 2006/11/01 · arxiv updated 2010/11/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We investigate the large weight (, where they find evidence of orthogonal and symplectic symmetry, respectively. The present examples suggest a relation between the symmetry type of a family and that of its twistings, which will be further studied in a subsequent paper. Both the GL(4) and the GL(6) families above have all even functional equations, and neither is naturally split from an orthogonal family. A folklore conjecture states that such families must be symplectic, which is true for the first family but false for the second. Thus, the theory of low lying zeros is more than just a theory of signs of functional equations. An analysis of these families suggest that it is the second moment of the Satake parameters that determines the symmetry group.