Fundamental Lemma Research Articles

The fundamental lemma for the Bessel and Novodvorsky subgroups of GSp(4) II

Abstract We announce, in the case of the group GSp(4), an equality of two local integrals. One is a Kloosterman integral on the Bessel subgroups of GSp(4) and the other is a Kloosterman integral on the Novodvorsky subgroups of GSp(4). We conjecture that Jacquet's relative trace formula for GL(2) in [7], where Jacquet has given another proof of Waldspurger's result [9], generalizes to GSp(4). We believe that this approach will lead us to a proof and also a precise formulation of a conjecture of Bocherer [1]. Support for this conjecture may be found in the important paper of Bocherer and Schulze-Pillot [2]. Our result serves as the fundamental lemma for our conjectural relative trace formula for the main relevant double cosets.

Integers free of small prime factors in arithmetic progressions

For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfying n ≡ a (mod q). By the fundamental lemma of sieve, it follows that for (a,q) = 1, Φ(x,y;a,q) = φ(q)-1, Φ(x, y){1 + O(exp(-u(log u- log2 3u- 2))) + (u = log x log y) holds uniformly in a wider ranges of x, y and q.Let χ be any character to the modulus q, and L(s, χ) be the corresponding L-function. Let be a (‘exceptional’) real character to the modulus q for which L(s, ) have a (‘exceptional’) real zero satisfying > 1 - c0/log q. In the paper, we prove that in a slightly short range of q the above first error term can be replaced by where ρ(u) is Dickman function, and ρ′(u) = dρ(u)/du.The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that 1 < q < (log q)A.

Faisceaux pervers, homomorphisme de changement de base et lemme fondamental de jacquet et ye

We give a geometric interpretation of the base change homomorphism between the Hecke algebra of GL( n) for an unramified extension of local fields of positive characteristic. For this, we use some results of Ginzburg, Mirkovic and Vilonen related to the geometric Satake isomorphism. We give a new proof of these results in the positive characteristic case. By using that geometric interpretation of the base change homomorphism, we prove the fundamental lemma of Jacquet and Ye for arbitrary Hecke function in the equal characteristic case.

Stable Bi-Period Summation Formula and Transfer Factors

AbstractThis paper starts by introducing a bi-periodic summation formula for automorphic forms on a group G(E), with periods by a subgroup G(F), where E/F is a quadratic extension of number fields. The split case, where E = F ⊕ F, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups H which occur in the case of standard conjugacy.The spectral side of the bi-period summation formula involves periods, namely integrals over the group of F-adele points of G, of cusp forms on the group of E-adele points on the group G. Our stabilization suggests that such cusp forms—with non vanishing periods—and the resulting bi-period distributions associated to “periodic” automorphic forms, are related to analogous bi-period distributions associated to “periodic” automorphic forms on the endoscopic symmetric spaces H(E)/H(F). This offers a sharpening of the theory of liftings, where periods play a key role.The stabilization depends on the “fundamental lemma”, which conjectures that the unit elements of the Hecke algebras on G and H have matching orbital integrals. Even in stating this conjecture, one needs to introduce a “transfer factor”. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case.Finally, the fundamental lemma is verified for SL(2).

The fundamental lemma for the Bessel and Novodvorsky subgroups of GSp(4)

We announce, in the case of the group GSp(4), an equality of two local integrals. One is a Kloosterman integral on the Bessel subgroups of GSp(4) and the other is a Kloosterman integral on the Novodvorsky subgroups of GSp(4). We conjecture that Jacquet's relative trace formula for GL(2) in [7], where Jacquet has given another proof of Waldspurger's result [9], generalizes to GSp(4). We believe that this approach will lead us to a proof and also a precise formulation of a conjecture of Böcherer [1]. Support for this conjecture may be found in the important paper of Böcherer and Schulze-Pillot [2]. Our result serves as the fundamental lemma for our conjectural relative trace formula for the main relevant double cosets.

Relation of orbital integrals on SO(5) and PGL(2)

We relate the “Fourier” orbital integrals of corresponding spherical functions on thep-adic groups SO(5) and PGL(2). The correspondence is defined by a “lifting” of representations of these groups. This is a local “fundamental lemma” needed to compare the geometric sides of the global Fourier summation formulae (or relative trace formulae) on these two groups. This comparison leads to conclusions about a well known lifting of representations from PGL(2) to PGSp(4). This lifting produces counter examples to the Ramanujan conjecture.

Elementary Proof of the Fundamental Lemma For a Unitary Group

AbstractThe fundamental lemma in the theory of automorphic forms is proven for the (quasi-split) unitary group U(3) in three variables associatedwith a quadratic extension of p-adic fields, and its endoscopic group U(2), by means of a new, elementary technique. This lemma is a prerequisite for an application of the trace formula to classify the automorphic and admissible representations of U(3) in terms of those of U(2) and base change to GL(3). It compares the (unstable) orbital integral of the characteristic function of the standard maximal compact subgroup K of U(3) at a regular element (whose centralizer T is a torus), with an analogous (stable) orbital integral on the endoscopic group U(2). The technique is based on computing the sum over the double coset space T\G/K which describes the integral, by means of an intermediate double coset space H\G/K for a subgroup H of G= U(3) containing T. Such an argument originates from Weissauer's work on the symplectic group. The lemma is proven for both ramified and unramified regular elements, for which endoscopy occurs (the stable conjugacy class is not a single orbit).

Le lemme fondamental de Jacquet et Ye en caractéristiques égales

We prove in the case of equal characteristics a fundamental lemma conjectured by Jacquet and Ye for GL(r). We use the étale cohomology interpretation of exponential sums, perverse sheaves and Fourier-Deligne's transformation.

Comparative Statics of Contests and Rent-Seeking Games

This paper develops the comparative statics properties of a contest encountered in patent races and rent-seeking games. The author shows that symmetric equilibrium expenditure and profit per player decrease with the number of competitors and the discount rate but increase with the value of the prize. A fundamental lemma established in the paper shows that a stability condition usually imposed on the model is automatically satisfied. The author identifies a class of likelihood or hazard rate functions for which aggregate expenditure increases with the number of competitors and total profits converge to zero. Copyright 1997 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

On the symmetric square. Unit elements

A twisted analogue of Kazhdan's decomposition of compact elements into a commuting product of topologically unipotent and absolutely semi-simple elements, is developed and used to give a direct and elementary proof of the Langlands' fundamental lemma for the symmetric square lifting from SL(2) to PGL(3) and the unit element of the Hecke algebra. Thus we give a simple proof that the stable twisted orbital integral of the unit element of the Hecke algebra of PGL(S) is suitably related to the stable orbital integral of the unit element of the Hecke algebra of SX(2), while the unstable twisted orbital integral of the unit element on FGL(3) is matched with the orbital integral of the unit element on PGL(2). An Appendix examines the implications of Waldspurger's fundamental lemma in the case of endolifting to the theory of endolifting and that of the metaplectic correspondence for GL(n). Let F be a p-adic field (p Φ 2), and F a separable closure of F. Put

On a Generalization of a Result of Waldspurger

AbstractWe consider a generalization of a trace formula identity of Jacquet, in the context of the symmetric spaces GL(2n)/GL(/n) × GL(n) and G′/H′. Here G′ is an inner form of GL(2n) over F with a subgroup H′ isomorphic to GL(n, E) where E/F is a quadratic extension of number field attached to a quadratic idele class character η of F. A consequence of this identity would be the following conjecture: Let π be an automorphic cuspidal representation of GL(2n). If there exists an automorphic representation π′ of G′ which is related to π by the Jacquet-Langlands correspondence, and a vector ø in the space of π′ whose integral over H′ is nonzero, then both L(1/2, π) and L(1/2,π ⊗ η) are nonvanishing. Moreover, we have L(1/2, π)L(1/2, π ⊗ η) > 0. Here the nonvanishing part of the conjecture is a generalization of a result of Waldspurger for GL(2) and the nonnegativity of the product is predicted from the generalized Riemann Hypothesis. In this article, we study the corresponding local orbital integrals for the symmetric spaces. We prove the "fundamental lemma for the unit Hecke functions" which says that unit Hecke functions have "matching" orbital integrals. This serves as the first step toward establishing the trace formula identity and in the same time it provides strong evidence for what we proposed.

On the Fundamental Lemma for Standard Endoscopy: Reduction to Unit Elements

AbstractThe fundamental lemma for standard endoscopy follows from the matching of unit elements in Hecke algebras. A simple form of the stable trace formula, based on the matching of unit elements, shows the fundamental lemma to be equivalent to a collection of character identities. These character identities are established by comparing them to a compact-character expansion of orbital integrals.

Unbiased Stereological Estimation of d-Dimensional Volume in ℝn from an Isotropic Random Slice Through a Fixed Point

Unbiased stereological estimators of d-dimensional volume in ℝ n are derived, based on information from an isotropic random r-slice through a specified point. The content of the slice can be subsampled by means of a spatial grid. The estimators depend only on spatial distances. As a fundamental lemma, an explicit formula for the probability that an isotropic random r-slice in ℝ n through O hits a fixed point in ℝ n is given.

Unbiased Stereological Estimation of d-Dimensional Volume in ℝn from an Isotropic Random Slice Through a Fixed Point

Unbiased stereological estimators of d-dimensional volume in ℝn are derived, based on information from an isotropic random r-slice through a specified point. The content of the slice can be subsampled by means of a spatial grid. The estimators depend only on spatial distances. As a fundamental lemma, an explicit formula for the probability that an isotropic random r-slice in ℝn through O hits a fixed point in ℝn is given.

Unipotent Representations and Unipotent Classes SL(N)

To achieve the stabilization of the trace formula for a reductive group [18], there are two related problems that must first be solved. One is the matching of the sorbital integrals of smooth functions on a reductive group G (over a p-adic field of characteristic zero) with stable orbital integrals on its endoscopic groups H. Igusa theory clarifies the nature of this first problem [11], [12], [13], [19], [22], [23]. The second problem is the fundamental lemma, which asserts the compatibility of the matching of smooth functions with the Hecke algebras on G and H. Buildings have been used to study this problem [25]. The statements of the problems are closely related, and the main results of this paper give a method of reducing one problem to the other. If the first problem (matching smooth functions) is solved for SL(n), then the second problem (the fundamental lemma for the Hecke algebras) follows as a consequence, at least for the identity of the algebra. The solutions of the two problems are linked by making a detailed study of the representations and the classes of SL(n). Representations with Iwahori fixed vectors on SL(n, F) are called unipotent in this paper to emphasize the close relationship between conjugacy classes and these representations. Not long after this paper was written, Waldspurger refined, reformulated and greatly extended the results of this paper. This paper made a modest contribution to this subsequent work of his. He eliminated the hypothesis that functions on SL(n) have matching smooth functions, by making clever use of a strengthened version of Kazhdan's lemma. The opinion of the experts is still divided, will the solution of the fundamental lemma for other reductive groups come from the matching of smooth functions, or will it come from even stronger versions of Kazhdan's lemma. This much is clear: the fundamental lemma no longer needs to be viewed as an isolated combinatorial problem in buildings. Howe's conjecture, uniform germ expansions, homogeneity of germs, compact traces, orbits, and Kazhdan's density theorem give us a framework through which the fundamental lemma may be understood. In the final portion of the introduction, I would like to state a theorem that has considerable interest beyond its usefulness for the fundamental lemma. This theorem will be used in Section 1 to produce a uniform germ expansion. The

Relative Kloosterman Integrals For Gl(3): III

AbstractLet E be a quadratic extension of a number field F with Galois conjugation σ, G’ the quasi-split unitary group in three variables, G the group GL(3, E). We let S be the space of the matrices s in G such that σ(s)s = e. One conjectures a comparison identity between the relative Kuznietsov trace formula for the symmetric space S and the ordinary Kuznietsov trace formula for the group G’ (See [10]). We prove the corresponding “fundamental lemmas”.

ITERATION ALGEBRAS

We assume the reader has some familiarity with theories and iteration theories. The main topic of the paper is properties of varieties of iteration algebras. After a preliminary section which contains all of the necessary definitions, we spend some time on a coproduct construction which is needed to prove a fundamental lemma: for each iteration theory T, any T-algebra is a retract of a T-iteration algebra. The proof of the lemma shows that only one property of iteration theories is used: the parameter identity. Hence, the lemma applies to any preiteration theory in which this identity is valid. It follows from this fact that the variety of all T-iteration algebras has “nice” properties only when every T-algebra is an iteration algebra. Some of the possible pathology in varieties of iteration algebras is demonstrated. It is shown that for each set Z of non-negative integers there is a variety of iteration algebras having an n-generated free algebra iff n∈Z. Also given is a theorem characterizing certain functors between varieties of iteration theories which are induced by iteration theory morphisms. We find an explicit description of all of the theory congruences on theories of partial functions. The last section is connected with the strong iteration algebras introduced in a paper by Ésik (1983).

A compactness Lemma for quasilinear problems: Application to parabolic equations

In this paper, we solve the problem u t + Au + F( u, ▽ u) = μ and u(0) = u 0 where μ and u 0 are bounded Radon measures. The method is based upon three fundamental lemmas: A compactness result (Lemma 3), a regularity result (Lemma 7), and an integration by parts (Lemma 2).

ALGEBRAIC CONNECTIVITY

Let [Formula: see text] be a type of algebra in the sense of universal algebra. By defining singular simplices in algebras and emulating singular [co] homology, we introduce for each variety, pseudo-variety, and divisional class V of type [Formula: see text], a homology and cohomology theory which measure the V-connectivity of type-[Formula: see text] algebras. Intuitively, if we were to think of an algebra as a space and subalgebras which lie in V as simplices, then V-connectivity describes the failure of subalgebras to lie in V, i.e., it describes the "holes" in this space. These [co]homologies are functorial on the class of type-[Formula: see text] algebras and are characterized by a natural topological interpretation. All these notions extend to subsets of algebras. One obtains for this algebraic connectivity, the long exact sequences, relative [co]homologies, and the analogues of the usual [co]homological notions of the algebraic topologists. In fact, we show that the [co]homologies are actually the same as the simplicial [co]homology of simplicial complexes that depend functorially on the algebras. Thus the connectivities in question have a natural geometric meaning. This allows the wholesale import into algebra of the concepts, results, and techniques of algebraic topology. In particular, functoriality implies that the [co]homology of a pair of algebras A ⊆ B is an invariant of the position of A in B. When one V contains another, we obtain relationships between the [co] homology theories in the form of long exact sequences. Furthermore for finite algebras, V-[co]homology is effectively computable if membership in V is. We obtain an analogue of the Poincaré lemma (stating that subsets of an algebra in V are V-homologically trivial), extremely general guarantees of the existence of subsets with non-trivial V-homology for algebras not in V, long exact V-homotopy sequences, as well as analogues of the powerful Eilenberg-Zilber theorems and Kunneth theorems in the setting of V-connectivity for V a variety or pseudo-variety. Also in the more general case of any divisionally closed V, we construct the long exact Mayer-Vietoris sequences for V-homology. Results for homomorphisms include an algebraic version of contiguity for homomorphisms (which implies they are V-homotopic) and a proof that V-surmorphisms are V-homotopy equivalences. If we allow the divisional classes to vary, then algebraic connectivity may be viewed as a functor from the category of pairs W ⊆ V of divisional classes of [Formula: see text]-algebras with inclusions as morphisms' to the category of functors from pairs of [Formula: see text]-algebras to pairs of simplicial complexes. Examples show the non-triviality of this theory (e.g. "associativity tori"), and two preliminary applications to semigroups are given: 1) a proof that the group connectivity of a torsion semigroup S is homotopy equivalent to a space whose points are the maximal subgroups of S, and 2) an aperiodic connectivity analogue of the fundamental lemma of complexity.

A characterization of L-fuzzy prime ideals

In this paper the concepts of a L-fuzzy prime and a L-fuzzy completely prime ideal are given and some fundamental lemmas are proved. In particular, bijections among some of the L-fuzzy (prime) ideals of two homomorphic rings are constructed. Also, a characterization of a L-fuzzy prime ideal of a ring is given.