Showing papers in "American Journal of Mathematics in 2005"
TL;DR: In this paper, a solution u (t ) of the generalized Korteweg-de Vries equation was constructed for the subcritical and critical cases p = 2, 3, 4 or 5.
Abstract: We consider the generalized Korteweg-de Vries equations
ut+uxx+upx=0t, x ∈ R
[inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]
in the subcritical and critical cases p = 2, 3, 4 or 5. Let R j ( t, x ) = Qc j ( x - c j t - x j ), where j ∈ {1, . . . , N }, be N soliton solutions of this equation, with corresponding speeds 0 c 1 c 2 c N . In this paper, we construct a solution u ( t ) of the generalized Korteweg-de Vries equation such that
[inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] This solution behaves asymptotically as t → +∞ as the sum of N solitons without loss of mass by dispersion. This is an exceptional behavior, indeed, being given the parameters { c j }1≤ j ≤ N , { x j }1≤ j ≤ N , we prove uniqueness of such a solution. In the integrable cases p = 2 and 3, such solutions are explicitly known and their properties were extensively studied in the literature (they are called N -soliton solutions). Therefore, the existence result is new only for the nonintegrable cases. The uniqueness result is new for all cases.
215 citations
TL;DR: In this article, the wave maps equation with values into a Riemannian manifold is considered and it is shown that the Cauchy problem is globally well-posed for initial data which is small in the critical Sobolev spaces.
Abstract: We consider the wave maps equation with values into a Riemannian manifold which is isometrically embedded in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. Our main result asserts that the Cauchy problem is globally well-posed for initial data which is small in the critical Sobolev spaces. This extends and completes recent work of Tao and other authors.
145 citations
TL;DR: In this article, the authors introduced the space P (G ) of abelian p -points of a finite group scheme over an algebraically closed field of characteristic p 0, and constructed a homeomorphism Ψ G : P(G ) → Proj | G | from P( G ) to the projectivization of the cohomology variety for any finite group G.
Abstract: We introduce the space P ( G ) of abelian p -points of a finite group scheme over an algebraically closed field of characteristic p 0. We construct a homeomorphism Ψ G : P ( G ) → Proj | G | from P ( G ) to the projectivization of the cohomology variety for any finite group G . For an elementary abelian p -group (respectively, an infinitesimal group scheme), P ( G ) can be identified with the projectivization of the variety of cyclic shifted subgroups (resp., variety of 1-parameter subgroups). For a finite dimensional G -module M , Ψ G restricts to a homeomorphism P ( G ) M → Proj | G | M , thereby giving a representation-theoretic interpretation of the cohomological support variety.
93 citations
TL;DR: In this article, the authors generalize the construction of the eigencurve by Coleman-Mazur to the setting of real fields and show that a finite slope Hilbert modular eigenform can be deformed into a one parameter family of finite slope eigenforms.
Abstract: We generalize the construction of the eigencurve by Coleman-Mazur to the setting of totally real fields, and show that a finite slope Hilbert modular eigenform can be deformed into a one parameter family of finite slope eigenforms The key point is to show the overconvergence of the canonical subgroup and the complete continuity of the U p operator We deduce this form some general considerations in rigid analytic geometry
87 citations
TL;DR: In this article, it was shown that if the Ricci curvature is uniformly bounded under the flow for all times t ∈ [0, T], then the curvature tensor has to be uniformly bounded as well.
Abstract: Consider the unnormalized Ricci flow (gjj)t = -2R ij for t ∈ [0, T), where T < oc. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times t ∈ [0, T), then the solution can be extended beyond T. We prove that if the Ricci curvature is uniformly bounded under the flow for all times t ∈ [0, T), then the curvature tensor has to be uniformly bounded as well.
87 citations
TL;DR: The set of new modular curves over [inline-graphic xmlns:xlink] of genus g is finite and computable, and the computability result is proved an algorithmic version of the de Franchis-Severi Theorem.
Abstract: A curve X over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] is modular if it is dominated by X 1 ( N ) for some N ; if in addition the image of its jacobian in J 1 ( N ) is contained in the new subvariety of J 1 ( N ), then X is called a new modular curve. We prove that for each g ≥ 2, the set of new modular curves over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] of genus g is finite and computable. For the computability result, we prove an algorithmic version of the de Franchis-Severi Theorem. Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of J 0 ( N ) new with N divisible by a prescribed prime, and for modular curves (new or not) with levels in a restricted set. We study new modular hyperelliptic curves in detail. In particular, we find all new modular curves of genus 2 explicitly, and construct what might be the complete list of all new modular hyperelliptic curves of all genera. Finally we prove that for each field k of characteristic zero and g ≥ 2, the set of genus- g curves over k dominated by a Fermat curve is finite and computable.
76 citations
TL;DR: In this paper, the authors characterize the relationship between the singular values of a Hermitian (resp., real symmetric, complex symmetric) matrix and the singular value of its off-diagonal block.
Abstract: We characterize the relationship between the singular values of a Hermitian (resp., real symmetric, complex symmetric) matrix and the singular values of its off-diagonal block. We also characterize the eigenvalues of a Hermitian (or real symmetric) matrix C = A + B in terms of the combined list of eigenvalues of A and B . The answers are given by Horn-type linear inequalities. The proofs depend on a new inequality among Littlewood-Richardson coefficients.
69 citations
TL;DR: In this paper, a new type of quotient for any triangulated category which generalizes Verdier's construction is introduced, and the derived category of an almost ring is shown to be of this form.
Abstract: The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier's construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S , a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic K -theory and demonstrates the relevance of the telescope conjecture for derived categories. Another application leads to a derived analogue of an almost module category in the sense of Gabber-Ramero. It is shown that the derived category of an almost ring is of this form.
65 citations
TL;DR: In this article, the authors consider the class of Levi non-degenerate hypersurfaces M in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] that admit a local (CR transversal) embedding, near a point p e M, into a standard nondegenerate hyperquadric.
Abstract: We consider the class of Levi nondegenerate hypersurfaces M in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] that admit a local (CR transversal) embedding, near a point p e M , into a standard nondegenerate hyperquadric in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] with codimension k := N - n small compared to the CR dimension n of M . We show that, for hypersurfaces in this class, there is a normal form (which is closely related to the embedding) such that any local equivalence between two hypersurfaces in normal form must be an automorphism of the associated tangent hyperquadric. We also show that if the signature of M and that of the standard hyperquadric in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] are the same, then the embedding is rigid in the sense that any other embedding must be the original embedding composed with an automorphism of the quadric.
62 citations
TL;DR: In this paper, the Saito-Kurokawa liftings from PGL (2) × PGL(2) to PGSp (4) are constructed using the theory of (local and global) theta lifts.
Abstract: Certain nontempered liftings from PGL (2) × PGL (2) to PGSp (4) are constructed using the theory of (local and global) theta lifts. The resulting representations on PGSp (4) are the Saito-Kurokawa representations. The lifting is shown to be functorial under certain reasonable assumptions on the local Langlands correspondence for PGSp (4).
60 citations
TL;DR: It is proved that every member of the Kuranishi family over Mf admits a Lagrangian fibration over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /].
Abstract: Let f : X → S be a Lagrangian fibration between projective varieties. We prove that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] ≅ Ω i S if S is smooth. Suppose that X is an irreducible symplectic manifold or a certain moduli space of semistable torsion free sheaves on a K 3 surface, the Hodge numbers satisfy h p,q ( S ) = h p,q [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /], where n = dim S . If S ≅ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] and X is an irreducible symplectic manifold, there exists a hypersurface M f of the Kuranishi space of X such that every member of the Kuranishi family over M f admits a Lagrangian fibration over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /].
TL;DR: In this article, the mod (p, υ 1 ) homotopy algebra of the topological Hochschild homology spectrum of ku has been computed for the connective complex K -theory spectrum.
Abstract: Let ku be the connective complex K -theory spectrum, completed at an odd prime p . We present a computation of the mod ( p, υ 1 ) homotopy algebra of the topological Hochschild homology spectrum of ku .
TL;DR: In this paper, the displacement map associated to small one-parameter polynomial unfoldings of Hamiltonian vector fields on the plane was studied and sufficient conditions for M ( t ) to satisfy a differential equation of Fuchs or Picard-Fuchs type were derived.
Abstract: We study the displacement map associated to small one-parameter polynomial unfoldings of polynomial Hamiltonian vector fields on the plane. Its leading term, the generating function M ( t ), has an analytic continuation in the complex plane and the real zeroes of M ( t ) correspond to the limit cycles bifurcating from the periodic orbits of the Hamiltonian flow. We give a geometric description of the monodromy group of M ( t ) and use it to formulate sufficient conditions for M ( t ) to satisfy a differential equation of Fuchs or Picard-Fuchs type. As examples, we consider in more detail the Hamiltonian vector fields [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] and [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /], possessing a rotational symmetry of order two and three, respectively. In both cases M ( t ) satisfies a Fuchs-type equation but in the first example M ( t ) is always an Abelian integral (that is to say, the corresponding equation is of Picard-Fuchs type) while in the second one this is not necessarily true. We derive an explicit formula of M ( t ) and estimate the number of its real zeroes.
TL;DR: In this paper, it was shown that the deformation space of convex real projective structures, that is, projectively flat torsion-free connections with the geodesic convexity property on a compact 2-orbifold of negative Euler characteristic, is homeomorphic to a cell of certain dimension.
Abstract: We determine that the deformation space of convex real projective structures, that is, projectively flat torsion-free connections with the geodesic convexity property on a compact 2-orbifold of negative Euler characteristic is homeomorphic to a cell of certain dimension. The basic techniques are from Thurston's lecture notes on hyperbolic 2-orbifolds, the previous work of Goldman on convex real projective structures on surfaces, and some classical geometry.
Abstract: The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction f corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign( f ) with respect to the surface area. It is measured as follows: take any disc centered at the nodal line { f = 0}, and pick at random a point in this disc. What is the probability that the function assumes a positive value at the chosen point? We show that this quantity may decay logarithmically as the eigenvalue goes to infinity, but never faster than that. In other words, only a mild local asymmetry may appear. The proof combines methods due to Donnelly-Fefferman and Nadirashvili with a new result on harmonic functions in the unit disc.
TL;DR: In this paper, the Tian-Yau-Zelditch expansion of the Szego kernel on polarized Kahler metrics was applied to approximate almost plurisubharmonic functions and derived a formula to compute the α-invariant on toric Fano manifolds.
Abstract: The global holomorphic α-invariant plays an important role in the study of the existence of Kahler-Einstein metrics on complex manifolds with positive first Chern class. In this paper, we apply the Tian-Yau-Zelditch expansion of the Szego kernel on polarized Kahler metrics to approximate almost plurisubharmonic functions and derive a formula to compute the α-invariant on toric Fano manifolds.
TL;DR: In this paper, the distribution of algebraic points on K3 surfaces is studied and connections between geometric properties of algebraIC varieties and their arithmetic properties over k, over its finite extensions k'/k or over k.
Abstract: We study the distribution of algebraic points on K3 surfaces. 1. Introduction. Let k be a field and k a fixed algebraic closure of k. We are interested in connections between geometric properties of algebraic varieties and their arithmetic properties over k, over its finite extensions k ' /k or over k. Here we study certain varieties of intermediate type, namely K3 surfaces and their higher dimensional generalizations, Calabi-Yau varieties. To motivate the following discussion, let 5 be a K3 surface over k. In positive characteristic, 5 may be unirational and covered by rational curves. Examples are supersingular K3 surfaces over fields of characteristic two or the surface
TL;DR: In this article, the authors describe the dynamics of a family of birational mappings of the plane and give an essentially complete account of the behavior of both wandering and nonwandering orbits.
Abstract: We describe the (real) dynamics of a family of birational mappings of the plane. By combining complex intersection theory and techniques from smooth dynamical systems, we are able to give an essentially complete account of the behavior of both wandering and nonwandering orbits. In particular, the golden mean subshift provides a topological model for the dynamics on the nonwandering set. While the mappings are not hyperbolic, they are shown to possess many of the structures associated with hyperbolicity.
TL;DR: In this article, sharp L 2 estimates for general local one-dimensional integral operators with smooth phase were proved by a coarse resolution of singularities in conjunction with a stopping time argument and extensions of the analytic methods of D. Phong and E. M. Stein.
Abstract: In this paper, sharp L 2 estimates are proved for general local one-dimensional oscillatory integral operators with smooth phase. The proof involves a coarse resolution of singularities in conjunction with a stopping time argument and extensions of the analytic methods of D. H. Phong and E. M. Stein.
TL;DR: In this paper, the volume rigidity results of Besson-Courtois-Gallot were extended to finite volume manifolds, and a lower bound on the minimal volume of finite-volume manifolds possessing a proper map to a finite volume manifold of pinched negative curvature was derived.
Abstract: We extend to finite volume manifolds some volume rigidity results of Besson-Courtois-Gallot. Our main result is a Volume Theorem which shows that proper maps from finite volume manifolds with Ricci curvature bounded from below to finite volume manifolds of pinched negative curvature decrease volume. As a consequence we deduce a lower bound on the minimal volume of finite volume manifolds possessing a proper map to a finite volume manifold of pinched negative curvature, and we show asymptotic isometry holds when the minimal volume is attained. Finally, we prove that finite, rank one, locally symmetric manifolds minimize normalized entropy rigidity in their homotopy class.
TL;DR: In this article, it was shown that the classifying space for algebraic K-theory of the integers Z(1/2) can be expressed as a fiber product of well-understood spaces BO and BGL(F3)+ over BU.
Abstract: Rognes and Weibel used Voevodsky's work on the Milnor conjecture to deduce the strong Dwyer-Friedlander form of the Lichtenbaum-Quillen conjecture at the prime 2. In consequence (the 2-completion of) the classifying space for algebraic K-theory of the integers Z(1/2) can be expressed as a fiber product of well-understood spaces BO and BGL(F3)+ over BU. Similar results are now obtained for Hermitian K-theory and the classifying spaces of the integral symplectic and orthogonal groups. For the integers Z(1/2), this leads to computations of the 2-primary Hermitian K-groups and affirmation of the Lichtenbaum-Quillen conjecture in the framework of Hermitian K-theory.
TL;DR: In this paper, the Selberg orthogonality conjecture for automorphic L -functions L ( s, π) and L( s, π'), under the Ramanujan conjecture on π and π', was proved.
Abstract: Let π and π' be automorphic irreducible unitary cuspidal representations of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] and [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /], respectively. Assume that either π or π' is self contragredient. Under the Ramanujan conjecture on π and π', we deduce a prime number theorem for L ( s , π × [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]), which can be used to asymptotically describe whether π' ≅ π, or π' ≅ π ⊗ |det(·)| i τ0 for some nonzero τ0 ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /], or π' ≅ π ⊗|det(·)| it for any t ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /]. As a consequence, we prove the Selberg orthogonality conjecture, in a more precise form, for automorphic L -functions L ( s , π) and L ( s , π'), under the Ramanujan conjecture. When m = m ' = 2 and π and π' are representations corresponding to holomorphic cusp forms, our results are unconditional.
TL;DR: The K-theoretic quiver formula of Buch expresses the structure sheaves of degeneracy loci as integral linear combinations of products of stable Grothendieck polynomials as mentioned in this paper.
Abstract: Fulton's universal Schubert polynomials give cohomology formulas for a class of degeneracy loci, which generalize Schubert varieties. The K-theoretic quiver formula of Buch expresses the structure sheaves of these loci as integral linear combinations of products of stable Grothendieck polynomials. We prove an explicit combinatorial formula for the coefficients, which shows that they have alternating signs. Our result is applied to obtain new expansions for the Grothendieck polynomials of Lascoux and Schutzenberger.
TL;DR: In this paper, a characteristic p representation of the absolute Galois group of rational numbers is shown to deform to the p -adics while guaranteeing that the characteristic polynomials of one set of primes are algebraic and pure of specified weight.
Abstract: Consider a characteristic p representation of the absolute Galois group of the rational numbers. In this paper we show how to deform this representation to the p -adics while guaranteeing that the characteristic polynomials of Frobenius at a density one set of primes are algebraic and pure of specified weight. The resulting representation is ramified at an infinite (density zero) set of primes. As a consequence of the technique of proof we show that one can compatibly lift a mod pq representation. Again, the resulting representation is ramified at infinitely many primes.
TL;DR: For a number field F and an odd prime p, the Capitulation Cokernels as mentioned in this paper stabilize and characterize their direct limit in Iwasawa theoretic terms, thus generalizing previous partial results obtained by H. Ichimura.
Abstract: For a number field F and an odd prime p , we study the "capitulation cokernels" coker ( A ' n → A ' Γ n ∞ ) associated with the ( p )-class groups of the cyclotomic [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-extension of F . We prove that these cokernels stabilize and we characterize their direct limit in Iwasawa theoretic terms, thus generalizing previous partial results obtained by H. Ichimura. This problem is intimately related to Greenberg's Conjecture.
TL;DR: In this paper, it was shown that the weight spectral sequences of Rapoport-Zink degenerate at E 2 in any characteristic without using log geometry, up to shift for proper smooth varieties over equal characteristic local fields.
Abstract: The aim of this paper is to study certain properties of the weight spectral sequences of Rapoport-Zink by a specialization argument. By reducing to the case over finite fields previously treated by Deligne, we prove that the weight filtration and the monodromy filtration defined on the l -adic etale cohomology coincide, up to shift, for proper smooth varieties over equal characteristic local fields. We also prove that the weight spectral sequences degenerate at E 2 in any characteristic without using log geometry. Moreover, as an application, we give a modulo p 0 reduction proof of a Hodge analogue previously considered by Steenbrink.
TL;DR: In this paper, the authors construct one parameter families of n dimensional Calabi-Yau manifolds, which are complete intersections in toric varieties and have a monodromy operator T such that (T N - id ) n + 1 = 0 but (T n - id n ≠ 0, i.e., the monodrome operator is maximal unipotent.
Abstract: The computations that are suggested by String Theory in the B model requires the existence of degenerations of CY manifolds with maximum unipotent monodromy. In String Theory such a point in the moduli space is called a large radius limit (or large complex structure limit). In this paper we are going to construct one parameter families of n dimensional Calabi-Yau manifolds, which are complete intersections in toric varieties and which have a monodromy operator T such that (T N - id ) n +1 = 0 but (T N - id ) n ≠ 0, i.e., the monodromy operator is maximal unipotent.
TL;DR: In this article, the eta invariants of Dirac operators and regularized determinants over hyperbolic manifolds with cusps and their relations with Selberg zeta functions were studied.
Abstract: We study eta invariants of Dirac operators and regularized determinants of Dirac Lapla- cians over hyperbolic manifolds with cusps and their relations with Selberg zeta functions. Using the Selberg trace formula and a detailed analysis of the unipotent orbital integral, we show that the eta and zeta functions defined by the relative traces are regular at the origin so that we can define the eta invariant and the regularized determinant. We also show that the Selberg zeta function of odd type has a meromorphic extension over C, prove a relation of the eta invariant and a certain value of the Selberg zeta function of odd type, and derive a corresponding functional equation. These results generalize the earlier work of John Millson to hyperbolic manifolds with cusps. We also prove that the Selberg zeta function of even type has a meromorphic extension over C, relate it to the regularized determinant, and obtain a corresponding functional equation.
TL;DR: In this article, a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields is presented, by a "weighted count" on rational points over the corresponding moduli spaces of semi-stable vector bundles using moduli interpretation of these points.
Abstract: In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields. More precisely, we first introduce new yet genuine non-abelian zeta functions for curves defined over finite fields, by a "weighted count" on rational points over the corresponding moduli spaces of semi-stable vector bundles using moduli interpretation of these points. Then we define non-abelian L-functions for curves over finite fields using integrations of Eisenstein series associated to L2-automorphic forms over certain generalized moduli spaces. Introduction. In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields. It consists of two chapters. More precisely, in Chapter I, we first introduce new yet genuine non-abelian zeta functions for curves defined over finite fields. This is achieved by a "weighted count" on rational points over the corresponding moduli spaces of semi-stable vector bundles using moduli interpretation of these points. We justify our con- struction by establishing basic properties for these new zetas such as functional equation and rationality, and show that if only line bundles are involved, our newly defined zetas coincide with Artin's Zeta. All this, in particular, the ratio- nality, then leads naturally to our definition of (global) non-abelian zeta functions (for curves defined over number fields), which themselves are justified by a con- vergence result. We end this chapter with a detailed study on rank two non-abelian zeta functions for genus two curves, based on what we call infinitesimal structures of Brill-Noether loci (and Weierstrass points). In Chapter II, we begin with a similar construction for the field of rationals to motivate what follows. In particular, we show that there is an intrinsic relation between our non-abelian zeta functions and Eisenstein series. Due to this, instead of introducing general non-abelian L-functions for curves defined over finite fields with more general test functions (as what Tate did in his Thesis for abelian L- functions), we then define non-abelian L-functions for curves over finite fields as integrations of Eisenstein series associated to L2-automorphic forms over certain generalized moduli spaces. Here geometric truncations play a key role. Basic properties for these non-abelian L-functions, such as meromorphic continuation,
TL;DR: In this paper, DeBacker and Kazhdan gave an integral formula for the supercuspidal characters of SL� (F), whereis an odd prime. But this formula is not applicable to SL�(F) in the sense that the characters of these characters are bounded on regular elliptic elements near the singular set.
Abstract: Let F be a p-adic field of characteristic 0. For odd primeswith certain tameness restric- tions, techniques due to Assem allow us to write supercuspidal characters of SL� (F) in terms of a family of distributions defined by Kazhdan. Following an argument of Harish-Chandra, we derive an integral formula for these distributions. Results of DeBacker and Kazhdan allow us to evaluate this integral formula. This in turn gives supercuspidal character formulas for SL� (F), explicit up to the constants appearing in the GL� (F) local character expansion. Our result also describes the explicit local Langlands correspondence for those positive-depth supercuspidal representations of GL� (F) which do not restrict irreducibly to SL� (F). 1. Introduction. In this note, using techniques due to Assem (see (3)), we derive formulas for the supercuspidal characters of SL� (F), whereis an odd prime. Here, F is a p-adic field of characteristic 0 (i.e., a finite extension of Qp). We assume p > 2� , although most of the time we will only need p >� . Our results illustrate some facts about supercuspidal characters which are not immediately obvious from previous work. In general, there is an infinite class of supercuspidal representations of GL� (F) which decompose intoirreducible components upon restriction to SL� (F). The character of each component then blows up near the singular set, despite the fact that supercuspidal characters of GL� (F) are bounded on regular elliptic elements near the singular set. On the other hand, under a certain simple condition on the order of the residue class field of F, there is another infinite class of supercuspidal representations of GL� (F) which remain irreducible upon restriction to SL� (F). This means that the charac- ters of the restrictions have the same behaviour near the singular set on SL� (F) as on GL� (F). In particular, the characters of these supercuspidal representations of SL� (F) are bounded on regular elliptic elements near the singular set. This has also been noted in (27). The ultimate effect of this phenomenon on harmonic analysis on SL� (F) is not yet fully understood and will be explored in future work. We briefly outline the method we will use here. The construction of the supercuspidal representations of GL� (F) is completely understood. It is known that all such representations are induced from open compact (modulo the centre)