The classification of irreducible admissible mod <Emphasis Type="Italic">p</Emphasis> representations of a <Emphasis Type="Italic">p</Emphasis>-adic GL<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

Let F be a finite extension of ℚ _p . Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over  [`( \mathbbF)]<sub>p</sub>\overline{ \mathbb{F}}_{p} to be supersingular. We then give the classification of irreducible admissible smooth GL <sub>n</sub> (F)-representations over  [`( \mathbbF)]_p\overline{ \mathbb{F}}_{p} in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel–Livné for n=2. For general split reductive groups we obtain similar results under stronger hypotheses.     

Local Jacquet-Langlands correspondence and parametric degrees

Let F be a non-Archimedean local field with finite residue field. Let n be a positive integer, let G = GL_n(F), and let D be a central F-division algebra of dimension n². The Jacquet-Langlands correspondence gives a canonical bijection ^D from the set of equivalence classes of irreducible, smooth, essentially square-integrable representations of G to the set of equivalence classes of irreducible smooth representations of D^![![times;. We give a necessary and sufficient condition, in terms of dim, for an irreducible smooth representation of D^× to be of the form ^D, for an irreducible supercuspidal representation of G, thereby solving an old problem. This relies on the explicit classification of the irreducible smooth representations of G and the parallel classification of the irreducible representations of D^×.This paper was written while the first-named author was visiting, and partly supported by, Université de Paris-Sud. At that time, the second-named author was enjoying the hospitality of the IHES, during a stay at the CNRS granted by Université de Paris-Sud; he would like to thank all those institutions. The work was also partially supported by EU network Arithmetical Algebraic Geometry.Mathematics Subject Classification (2000): 22E50     

Parabolic bundles and representations¶of the fundamental group

Let X be a smooth complex projective variety with Neron–Severi group isomorphic to ℤ, and D an irreducible divisor with normal crossing singularities. Assume 1<r≤ 3. We prove that if π₁(X) doesn't have irreducible PU(r) representations, then π₁(X- D) doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth. Received: 20 December 1999 / Revised version: 7 May 2000     

Friabilité des valeurs d’un polynôme

Let F(X) be an absolutely irreducible polynomial in \mathbbZ [X₁,..., X_n]{\mathbb{Z} [X_{1},\dots, X_{n}]}, with degree d. We prove that, for any δ < 4/3, for any sufficiently large x, there exists a positive density of integral n-tuples m = (m ₁, . . . , m _n ) in the hypercube max |m _i | ≤ x such that every prime divisor of F(m) is smaller than x ^d–δ . This result is improved when F satisfies some geometrical hypotheses.     

Euler–Poincaré formula in equal characteristic¶under ordinariness assumptions

Let X be an irreducible smooth projective curve over an algebraically closed field of characteristic p>0. Let ? be either a finite field of characteristic p or a local field of residue characteristic p. Let F be a constructible étale sheaf of $\BF$-vector spaces on X. Suppose that there exists a finite Galois covering π:Y→X such that the generic monodromy of π^* F is pro-p and Y is ordinary. Under these assumptions we derive an explicit formula for the Euler–Poincaré characteristic χ(X,F) in terms of easy local and global numerical invariants, much like the formula of Grothendieck–Ogg–Shafarevich in the case of different characteristic. Although the ordinariness assumption imposes severe restrictions on the local ramification of the covering π, it is satisfied in interesting cases such as Drinfeld modular curves. Received: 7 December 1999 / Revised version: 28 January 2000     

On the abelianizations of congruence subgroups of Aut(<Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>)

Let F _n be the free group of rank n, and let Aut⁺(F _n ) be its special automorphism group. For an epimorphism π : F _n → G of the free group F _n onto a finite group G we call the standard congruence subgroup of Aut⁺(F _n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ⁺(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ⁺(G, π) ≤ Aut⁺(F ₂) has infinite abelianization.     

Fractional Moments of Automorphic L-Functions on GL（m）

Let π be an irreducible unitary cuspidal representation of GLm(AQ), m ≥ 2. Assume that π is self-contragredient. The author gets upper and lower bounds of the same order for fractional moments of automorphic L-function L(s, π) on the critical line under Generalized Ramanujan Conjecture; the upper bound being conditionally subject to the truth of Generalized Riemann Hypothesis.     

Connected components of spaces of Morse functions with fixed critical points

Let M be a smooth closed orientable surface and F = F _p,q,r be the space of Morse functions on M having exactly p points of local minimum, q ≥ 1 saddle critical points, and r points of local maximum, moreover, all the points are fixed. Let F _f be a connected component of a function f ∈ F in F.We construct a surjection π ₀(F) → ℤ ^p+r−1 by means of the winding number introduced by Reinhart (1960). In particular, |π₀(F)| = ∞, and the component F _f is not preserved under the Dehn twist about the boundary of any disk containing exactly two critical points, exactly one of which is a saddle point. Let D be the group of orientation preserving diffeomorphisms of M leaving fixed the critical points, D ⁰ be the connected component of id _M in D, and D _f ⊂ D be the set of diffeomorphisms preserving F _f . Let H _f be the subgroup of D _f generated by D ⁰ and all diffeomorphisms h ∈ D preserving some function f ₁ ∈ F _f , and let H _f ^abs be its subgroup generated by D ⁰ and the Dehn twists about the components of level curves of the functions f ₁ ∈ F _f . We prove thatH _f ^abs ⊊ D _f for q ≥ 2 and construct an epimorphism D _f /H _f ^abs → ℤ₂ ^q−1 by means of the winding number. A finite polyhedral complex K = K _p,q,r associated with the space F is defined. An epimorphism μ: π ₁(K) → D _f /H _f and finite generating sets for the groups D _f /D ⁰ and D _f /H _f in terms of the 2-skeleton of the complex K are constructed.     

Conductors and newforms for U(1,1)

Let F be a non-Archimedean local field whose residue characteristic is odd. In this paper we develop a theory of newforms forU (1, 1)(F), building on previous work onSL ₂(F). This theory is analogous to the results of Casselman forGL ₂(F) and Jacquet, Piatetski-Shapiro, and Shalika forGL _n(F). To a representation π ofU(1, 1)(F), we attach an integer c(π) called the conductor of π, which depends only on theL-packet π containing π. A newform is a vector in π which is essentially fixed by a congruence subgroup of level c(π). We show that our newforms are always test vectors for some standard Whittaker functionals, and, in doing so, we give various explicit formulae for newforms.     

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

On almost-divided domains

Let B be a domain, Q a maximal ideal of B, π: B → B/Q the canonical surjection, D a subring of B/Q, and A:=π ⁻¹(D). If both B and D are almost-divided domains (resp., n-divided domains), then A = B × _B/Q D is an almost-divided domain (resp., an n-divided domain); the converse holds if B is quasilocal. If 2 ≤ d ≤ ∞, an example is given of an almost-divided domain of Krull dimension d which is not a divided domain.      

Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups III. Restriction of Harish-Chandra modules and associated varieties

Let H⊂G be real reductive Lie groups and π an irreducible unitary representation of G. We introduce an algebraic formulation (discretely decomposable restriction) to single out the nice class of the branching problem (breaking symmetry in physics) in the sense that there is no continuous spectrum in the irreducible decomposition of the restriction π| _H . This paper offers basic algebraic properties of discretely decomposable restrictions, especially for a reductive symmetric pair (G,H) and for the Zuckerman-Vogan derived functor module , and proves that the sufficient condition [Invent. Math. '94] is in fact necessary. A finite multiplicity theorem is established for discretely decomposable modules which is in sharp contrast to known examples of the continuous spectrum. An application to the restriction π| _H of discrete series π for a symmetric space G/H is also given. Oblatum 2-X-1996 & 10-III-1997     

On the convergence to equilibrium of Brownian motion on compact simple Lie groups

Let M =G/H be an irreducible homogeneous compact manifold of dimension n equipped with its canonical Riemannian metric. Let γ be the lowest nonzero eigenvalue of the Laplace operator. Let μ be the normalized Haar measure and μ _t be the heat diffusion measure, i.e., the law of Brownian motion started at a fixed origin in M. We show that the total variation distance between μ_t and μ is not small for t ≪λ ⁻¹ logn.This is sharp, up to a factor of two, in the case of compact irreducible simply connected symmetric spaces.     

On the number of Galois points for a plane curve in positive characteristic,III

We consider the following problem: For a smooth plane curve C of degree d ≥ 4 in characteristic p > 0, determine the number δ(C) of inner Galois points with respect to C. This problem seems to be open in the case where d ≡ 1 mod p and C is not a Fermat curve F(p ^e  + 1) of degree p ^e  + 1. When p ≠ 2, we completely determine δ(C). If p = 2 (and C is in the open case), then we prove that δ(C) = 0, 1 or d and δ(C) = d only if d−1 is a power of 2, and give an example with δ(C) = d when d = 5. As an application, we characterize a smooth plane curve having both inner and outer Galois points. On the other hand, for Klein quartic curve with suitable coordinates in characteristic two, we prove that the set of outer Galois points coincides with the one of \mathbbF₂{\mathbb{F}_{2}} -rational points in \mathbbP²{\mathbb{P}^{2}}.     

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Let f be an endomorphism of \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and ν be an f-invariant measure with positive Lyapunov exponents (λ ₁, . . . , λ _k ). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by dim_Hm 3 [(log d)/(l₁)] + [(log d)/(l₂)]{{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of f ⁿ in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and a normalization theorem for the ν-generic inverse branches of f ⁿ .     

Vanishing theorem for irreducible symmetric spaces of noncompact type

We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then any E-valued L2 harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type.     

Involutions whose top dimensional component of the fixed point set is indecomposable

Let (M ^m , T) be a smooth involution on a closed smooth m-dimensional manifold and F = è_j=0ⁿ F^j{F = \bigcup_{j=0}^{n} F^j} (n ≤ m) its fixed point set, where F ^j denotes the union of those components of F having dimension j. In this paper we show that, if the top dimensional component F ⁿ is indecomposable, then m ≤ 2n + 1. We also give examples to show that this bound is best possible. This gives an improvement for the famous Five Halves Theorem of J. Boardman when the top dimensional component of the fixed point set is indecomposable.     

Comparison of germ expansion¶on inner forms of GL(n)

We prove that the germ expansion of a discrete series representation π^′ on GL _n (D) where D is a division algebra over k of index m and the germ expansion of the representation π of GL _mn (k) associated to π^′ by the Deligne–Kazhdan–Vigneras correspondence are closely related, and therefore certain coefficients in the germ expansion of a discrete series representation of GL _mn (k) can be interpreted (and therefore sometimes calculated) in terms of the dimension of a certain space of (degenerate) Whittaker models on GL _n (D). Received: 30 September 1999 / Revised version: 11 February 2000     

Existence and analyticity of eigenvalues of a two-channel molecular resonance model

We consider a family of operators H_γμ(k), k ∈ \mathbbT^d \mathbb{T}^d := (−π,π]^d, associated with the Hamiltonian of a system consisting of at most two particles on a d-dimensional lattice ℤ^d, interacting via both a pair contact potential (μ > 0) and creation and annihilation operators (γ > 0). We prove the existence of a unique eigenvalue of H_γμ(k), k ∈ \mathbbT^d \mathbb{T}^d , or its absence depending on both the interaction parameters γ,μ ≥ 0 and the system quasimomentum k ∈ \mathbbT^d \mathbb{T}^d . We show that the corresponding eigenvector is analytic. We establish that the eigenvalue and eigenvector are analytic functions of the quasimomentum k ∈ \mathbbT^d \mathbb{T}^d in the existence domain G ⊂ \mathbbT^d \mathbb{T}^d .