Research -- J. Hundley, SIUC

Work in Progress

• The egut project (ongoing programming project).
  
• Descent Construction for GSpin Groups (Joint with E. Sayag):
  • Odd cuspidal case(Last Updated 5/29/09)
    
  • Draft version of the general odd case.(Last Updated 5/29/09)
    
  • Draft version of the even case.(Last Updated 6/4/09)
  
  
• Holomorphicity of adjoint L functions and ratios of Dedekind zeta functions. Show that the adjoint L function of a cuspidal representation of GL(3) or U(2,1) can not have a pole using the integral representation in $G_2$ given below. Show the same about a ratio of Dedekind zeta functions attached to a cubic extension of number fields using the integral representation due to Jiang and Rallis. Methods should extend to GL(4) and GL(5).
  
• Doubling integral for $G_2$ (joint with D. Ginzburg)
  
• Survey of integral representations in the exceptional groups (long term, joint with D. Ginzburg)
  

Recently completed

• Fourier expansions of GL(2) newforms at various cusps(with D. Goldfeld and M. Lee)
  
• On multiplicativity of Fourier coefficients at cusps other than infinity
  
• Constructions of global integrals in the exceptional groups(with D. Ginzburg)
  
• On some results of Bump-Choie and Choie-Kim
  
• Adjoint L function of SU(2,1) Modifies this construction of Ginzburg for SL(3) to accomodate quasisplit unitary groups. Unfolding and unramified computations only. To appear in the proceedings of the Edinburgh multiple Dirichlet series conference.
  

Published Work

• (with E. Sayag) Descent construction for GSpin groups: main results and applications. Electron. Res. Announc. Math. Sci. 16 (2009), 30Ð36.
  
• (with D. Ginzburg) On certain Rankin-Selberg integrals on $GE_6.$ . Nagoya Math. J. 191 (2008), 21Ð78.
  
• Spin $L$-functions for ${\rm GSO}\sb {10}$ and ${\rm GSO}\sb {12}$. Israel J. Math. 165 (2008), 103--132.
  
• (with D. Ginzburg) The adjoint $L$-function for ${\rm GL}\sb 5$. Electron. Res. Announc. Math. Sci. 15 (2008), 24--32.
  
• (with D. Ginzburg) On Spin $L$-functions for ${\rm GSO}\sb {10}$. J. Reine Angew. Math. 603 (2007), 183--213.
  
• Siegel zeros of Eisenstein series. Acta Arith. 126 (2007), no. 4, 341--356.
  
• (with. W.T. Gan) The spin $L$-function of quasi-split $D\sb 4$. IMRP Int. Math. Res. Pap. 2006, Art. ID 68213, 74 pp.
  
• (with D. Ginzburg) A new tower of Rankin-Selberg integrals. Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 56--62 (electronic).
  
• (with D. Ginzburg) Multivariable Rankin-Selberg integrals for orthogonal groups. Int. Math. Res. Not. 2004, no. 58, 3097--3119.
  

"To-Do" List:

• The third integral announced in "A new tower of Rankin-Selberg integrals."
  
• The two integrals announced in "The adjoint $L$-function for ${\rm GL}\sb 5$."