Journal ArticleAmerican Journal of Mathematics · June 1, 2025
Schubert Eisenstein series are defined by restricting the summation in a degenerate Eisenstein series to a particular Schubert variety. In the case of GL3 over ℚ Bump and the first author proved that these Schubert Eisenstein series have meromor ...
Full textCite
Journal ArticleSelecta Mathematica New Series · April 1, 2025
We prove a Poisson summation formula for the zero locus of a quadratic form in an even number of variables with no assumption on the support of the functions involved. The key novelty in the formula is that all “boundary terms” are given either by constant ...
Full textCite
Journal ArticleScience China Mathematics · June 1, 2021
Braverman and Kazhdan (2000) introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula. Their conjectures should imply that quite general Langlands L-functions have meromorphic continuations and functi ...
Full textCite
Journal ArticleAmerican Journal of Mathematics · October 1, 2020
Let F be a number field and let AF be its ring of adeles. Let B be a quaternion algebra over F and let ν: B → F be the reduced norm. Consider the reductive monoid M over F whose points in an F-algebra R are given by (Formula Presented). Motivate ...
Full textCite
Journal ArticleAdvances in Mathematics · April 30, 2019
Let V 1 ,V 2 ,V 3 be a triple of even dimensional vector spaces over a number field F equipped with nondegenerate quadratic forms Q 1 ,Q 2 ,Q 3 , respectively. Let Y⊂∏i=1V i be th ...
Full textCite
Journal ArticlePacific Journal of Mathematics · January 1, 2018
Braverman and Kazhdan have introduced an influential conjecture on local functional equations for general Langlands L-functions. It is related to L. Lafforgue's equally influential conjectural construction of kernels for functorial transfers. We formulate ...
Full textCite
Journal ArticleResearch in Mathematical Sciences · December 1, 2016
Let F be a number field, let AF be its ring of adeles, and let g1, g2, h1, h2∈ GL 2(AF). We provide an absolutely convergent geometric expression for ∑πKπ(g1,g2)Kπ∨(h1,h2)Ress=1LS ...
Full textCite
Journal ArticleResearch in Mathematical Sciences · December 1, 2015
Let E/F be an everywhere unramified extension of number fields with Gal(E/F) simple and nonabelian. In a recent paper, the first named author suggested an approach to nonsolvable base change and descent of automorphic representations of GL2 alon ...
Full textCite
Journal ArticleProceedings of the American Mathematical Society · January 1, 2015
Let F be a number field and let π be a cuspidal unitary automorphic representation of GLmn(AF) where m and n are integers greater than one. We propose a conjecturally necessary condition for π to be a Rankin-Selberg transfer of an aut ...
Full textCite
Journal ArticlePacific Journal of Mathematics · January 1, 2015
In this paper we prove a relative trace formula for all pairs of connected algebraic groups H ≤ G × G, with G a reductive group and H the direct product of a reductive group and a unipotent group, given that the test function satisfies simplifying hypothes ...
Full textCite
Book · January 1, 2012
In the 1970s Hirzebruch and Zagier produced elliptic modular forms with coefficients in the homology of a Hilbert modular surface. They then computed the Fourier coefficients of these forms in terms of period integrals and L-functions. In this book the aut ...
Full textCite
Chapter · January 1, 2012
Thus far we have ignored classes in (Formula presented.) and their Poincaré duals in intersection homology. We now take up the study of these classes. ...
Full textCite
Chapter · January 1, 2012
In this chapter we collect some known facts on Hilbert modular forms and varieties, mostly for the purpose of fixing our notation. These concepts will be used in Chapter 7, where we will recall the description of the intersection cohomology of Hilbert modu ...
Full textCite
Chapter · January 1, 2012
We use the same setup as in the previous chapter: L/E is a quadratic extension of totally real fields Gal(L/E) = 〈1, ς〉 Gal(L/E)^ = 〈1, η〉, c ⊂ OL is an ideal, and n:=[L:Q], d:=dL/E, D := DL/E cE := c∩OE
Full textCite
Chapter · January 1, 2012
In this section we begin with the general theory of local systems, automorphic vector bundles, and automorphy factors. After describing the finite-dimensional representation theory of GL2 we determine the explicit equations relating modular form ...
Full textCite
Chapter · January 1, 2012
Recall (e.g.,[Hud ]) that a closed convex linear cell is the convex hull of finitely many points in Euclidean space. A convex linear cell complex K is a finite collection of closed convex linear cells in some ℝN such that if σ ∈ K then every fac ...
Full textCite
Chapter · January 1, 2012
In this chapter we use Proposition 6.6 to construct a map Hilbert modular forms ⟶ intersection cohomology which takes weight, nebentypus ⟶ local coefficient system Hecke operator ⟶ action of Hecke correspondence Petersson product ⟶ intersection product. ...
Full textCite
Chapter · January 1, 2012
In this chapter we consider a quadratic extension L/E of totally real fields. The inclusion E → L gives rise to Hilbert modular subvarieties, known as Hirzebruch-Zagier cycles, Z ⊂ Y with dim(Y ) = 2 dim(Z). ...
Full textCite
Book · January 1, 2012
In their seminal paper [Hirz] on the intersection theory of Hilbert modular surfaces, F. Hirzebruch and D. Zagier mentioned that the motivation for their work was to explain an observation of J.-P. Serre. ...
Full textCite
Chapter · January 1, 2012
In this chapter we recall the relation between intersection homology,constructed using (p, i)-allowable chains as in [Gre4],an d intersection cohomology, constructed via sheaf theory. ...
Full textCite
Chapter · January 1, 2012
The goal of this chapter is to prove Theorems 8.4 and 8.5,t he full versions of Theorems 1.1 and 1.2 given in the introduction. We consider a quadratic extension of totally real number fields L/E. ...
Full textCite
Journal ArticleJournal of the Ramanujan Mathematical Society · 2012
We present a collection of conjectural trace identities and explain why they are
equivalent to base change and descent of automorphic representations of GL(n) along
nonsolvable extensions (under some simplifying hypotheses). The case n = 2 is treated in
mo ...
Link to itemCite
Journal ArticleAmerican Journal of Mathematics · December 1, 2007
Let Script O sign be the ring of integers of a totally real number field E and set G := ResE/ℚ( GL2). Fix an ideal c ⊂ Script O sign. For each ideal m ⊂ Script O sign let T(m) denote the mth Hecke operator associated to the standard c ...
Full textCite
Journal ArticleJournal of Mathematical Analysis and Applications · January 1, 2004
Let G-fraktur signp(x) ε double struck F sign p[x] be the polynomial whose zeros are the j-invariants of supersingular elliptic curves over double struck F signp. Generalizing a construction of Atkin described in a recent p ...
Full textCite
Journal ArticleProceedings of the American Mathematical Society · January 1, 2004
Rankin and Swinnerton-Dyer (1970) prove that all zeros of the Eisenstein series Ek in the standard fundamental domain for Γ lie on A:= {eiθ:π/2 ≤ θ ≤ 2π/3}. In this paper we generalize their theorem, providing conditions under which t ...
Full textCite
Journal ArticleJournal of Combinatorial Theory Series A · January 1, 2002
In the study of partition theory and q-series, identities that relate series to infinite products are of great interest (such as the famous Rogers-Ramanujan identities). Using a recent result of Zagier, we obtain an infinite family of such identities that ...
Full textCite
