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DENSITY OF INTEGER POINTS ON AFFINE HOMOGENEOUS VARIETIES
 DUKE MATHEMATICAL JOURNAL
, 1993
"... Let F be an affine variety defined over Z by integral polynomials x,]: (1.1) V {x e C": f(x) O, j 1,..., v} A basic problem of diophantine analysis is to investigate the asymptotics as T of (1.2) N(T, V) = {m V(Z): Ilmll T} where we denote by V(A), for any ring A, the set of Apoints of V. Henc ..."
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Cited by 104 (4 self)
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Let F be an affine variety defined over Z by integral polynomials x,]: (1.1) V {x e C": f(x) O, j 1,..., v} A basic problem of diophantine analysis is to investigate the asymptotics as T of (1.2) N(T, V) = {m V(Z): Ilmll T} where we denote by V(A), for any ring A, the set of Apoints of V. Hence I1 " is some Euclidean norm on R". The only general method available for such problems is the HardyLittlewood circle method, which however has certain limitations, requiring roughly that the codimension of V in the ambient space A", as well as the degree of the equations (1.1), be small relative to n. Furthermore, there are restrictions on the size of the singular sets of the related varieties: V u {x e C": f(x) &, j 1,..., v}, u () e c". We refer to [Bi] and [Sch] for a discussion of the restriction. Regardless of these restrictions, one hopes that for many more cases N(T, V) can be given in the form predicted by the HardyLittlewood method, that is, as a product of local densities: (,) N(T, V) lI l,(V)lUoo ( T, v), p <oo where the "singular series " II,< #(V) #,(V) is given by padic densities: lira k # v(z/pz) and/(T, V) is a real densitymthe "singular integral. " Following Schmidt [Sch], we say that V is a HardyLittlewood system if the above asymptotics (,) is valid. pk dim V
Modularity Of The RankinSelberg LSeries, And Multiplicity One For SL(2)
"... Contents 1. Introduction 1 2. Notations and Preliminaries 5 3. Construction of # : A(GL(2)) A(GL(2)) # A(GL(4)) 8 3.1. Relevant objects and the strategy 9 3.2. Weak to strong lifting, and the cuspidality criterion 13 3.3. Triple product Lfunctions: local factors and holomorphy 15 3.4. Boundednes ..."
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Cited by 88 (15 self)
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Contents 1. Introduction 1 2. Notations and Preliminaries 5 3. Construction of # : A(GL(2)) A(GL(2)) # A(GL(4)) 8 3.1. Relevant objects and the strategy 9 3.2. Weak to strong lifting, and the cuspidality criterion 13 3.3. Triple product Lfunctions: local factors and holomorphy 15 3.4. Boundedness in vertical strips 18 3.5. Modularity in the good case 30 3.6. A descent criterion 32 3.7. Modularity in the general case 35 4. Applications 37 4.1. A multiplicity one theorem for SL(2) 37 4.2. Some new functional equations 40 4.3. Root numbers and representations of orthogonal type 42 4.4. Triple product Lfunctions revisited 44 4.5. The Tate conjecture for 4fold products of modular curves 47 Bibliography 52 1. Introduction Let f, g be primitive cusp forms, holomorphic or otherwise, on
Eisenstein series for higherrank groups and string theory amplitudes
 Commun. Num. Theor. Phys
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On the spectral decomposition of affine Hecke algebras
 J. Inst. Math. Jussieu
"... Abstract. An affine Hecke algebra H contains a large abelian subalgebra A spanned by the BernsteinZelevinskiLusztig basis elements θx, where x runs over (an extension of) the root lattice. The center Z of H is the subalgebra of Weyl group invariant elements in A. The natural trace (“evaluation at ..."
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Cited by 39 (13 self)
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Abstract. An affine Hecke algebra H contains a large abelian subalgebra A spanned by the BernsteinZelevinskiLusztig basis elements θx, where x runs over (an extension of) the root lattice. The center Z of H is the subalgebra of Weyl group invariant elements in A. The natural trace (“evaluation at the identity”) of the affine Hecke algebra can be written as integral of a certain rational nform (with values in the linear dual of H) over a cycle in the algebraic torus T = spec(A). This cycle is homologous to a union of “local cycles”. We show that this gives rise to a decomposition of the trace as an integral of positive local traces against an explicit probability measure on the spectrum W0\T of Z. From this result we derive the Plancherel formula of the affine Hecke algebra.
The Iwasawa main conjectures for GL2
, 2010
"... In this paper we prove the IwasawaGreenberg Main Conjecture for a large class of elliptic curves and modular forms. 1.1. The IwasawaGreenberg Main Conjecture. Let p be an odd prime. Let Q ⊂ C be the algebraic closure of Q in C. We fix an embedding Q ↩ → Q p. For simplicity we ..."
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Cited by 29 (1 self)
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In this paper we prove the IwasawaGreenberg Main Conjecture for a large class of elliptic curves and modular forms. 1.1. The IwasawaGreenberg Main Conjecture. Let p be an odd prime. Let Q ⊂ C be the algebraic closure of Q in C. We fix an embedding Q ↩ → Q p. For simplicity we
Nonvanishing of the central value of the RankinSelberg Lfunctions
, 2004
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String theory dualities and supergravity divergences
 JHEP 1006 (2010) 075
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ON A FAMILY OF DISTRIBUTIONS OBTAINED FROM EISENSTEIN SERIES I: APPLICATION OF THE PALEYWIENER THEOREM
, 1982
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Tilings and finite energy retractions of locally symmetric spaces
 COMMENTARII MATHEMATICI HELVETICI
, 1997
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The Plancherel decomposition for a reductive symmetric space II. Representation theory
"... We obtain the Plancherel decomposition for a reductive symmetric space in the sense of representation theory. Our starting point is the Plancherel formula for spherical Schwartz functions, obtained in part I. ..."
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Cited by 24 (8 self)
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We obtain the Plancherel decomposition for a reductive symmetric space in the sense of representation theory. Our starting point is the Plancherel formula for spherical Schwartz functions, obtained in part I.