Newforms of half-integral weight: the minus space of S_k+1/2(Γ₀(8M))

2018/08/16 by Ehud Moshe Baruch, Baruch, Ehud Moshe, Soma Purkait +1
Mathematics · #11F70 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary: 11F37 #Secondary: 11F12

paper · pdf · doi:10.48550/arxiv.1808.05526

openalex publication_date 2018/08/16 · openalex created_date 2018/08/22 · openalex updated_date 2026/07/28

Abstract

We compute generators and relations for a certain 2-adic Hecke algebra of level 8 associated with the double cover of SL₂ and a 2-adic Hecke algebra of level 4 associated with PGL₂. We show that these two Hecke algebras are isomorphic as expected from the Shimura correspondence. We use the 2-adic generators to define classical Hecke operators on the space of holomorphic modular forms of weight k+1/2 and level 8M where M is odd and square-free. Using these operators and our previous results on half-integral weight forms of level 4M we define a subspace of the space of half-integral weight forms as a common -1 eigenspace of certain Hecke operators. Using the relations and a result of Ueda we show that this subspace which we call the minus space is isomorphic as a Hecke module under the Ueda correspondence to the space of new forms of weight 2k and level 4M. We observe that the forms in the minus space satisfy a Fourier coefficient condition that gives the complement of the plus space but does not define the minus space.

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