Eisenstein series attached to lattices¶and modular forms on orthogonal groups

We study certain vector valued Eisenstein series on the metaplectic cover of SL₂(ℝ), which transform with the Weil representation associated with the discriminant group of an even lattice L. We find a closed formula for the Fourier coefficients in terms of Dirichlet L-series and representation numbers of L modulo “bad” primes. Such Eisenstein series naturally occur in the context of Borcherds' theory of automorphic products. We indicate some applications to modular forms on the orthogonal group of L with zeros on Heegner divisors. Received: 27 September 2001     

Symmetric square L-values and dihedral congruences for cusp forms

Let be a prime, and k=(p+1)/2. In this paper we prove that two things happen if and only if the class number . One is the non-integrality at p of a certain trace of normalised critical values of symmetric square L-functions, of cuspidal Hecke eigenforms of level one and weight k. The other is the existence of such a form g whose Hecke eigenvalues satisfy “dihedral” congruences modulo a divisor of p (e.g. p=23, k=12, g=Δ). We use the Bloch-Kato conjecture to link these two phenomena, using the Galois interpretation of the congruences to produce global torsion elements which contribute to the denominator of the conjectural formula for an L-value. When , the trace turns out always to be a p-adic unit.     

On simultaneous non-vanishing of modular <Emphasis Type="Italic">L</Emphasis>-functions

For i = 1, , r, let f _i be newforms of weight 2k _i for Γ₀(N _i ) with trivial character. We consider the simultaneous non-vanishing problem for the central values of twisted L-functions of f _i . By using the Shimura correspondence, we give a certain relation between this problem and the kernel fields of 2-adic Galois representations associated to modular forms. Received: 28 January 2006     

On Atkin and Swinnerton-Dyer congruence relations (2)

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigenforms. We also show that there is an automorphic L-function over whose local factors agree with those of the l-adic Scholl representations attached to the space of noncongruence cusp forms. The research of the second author was supported in part by an NSA grant #MDA904-03-1-0069 and an NSF grant #DMS-0457574. Part of the research was done when she was visiting the National Center for Theoretical Sciences in Hsinchu, Taiwan. She would like to thank the Center for its support and hospitality. The third author was supported in part by an NSF-AWM mentoring travel grant for women. She would further thank the Pennsylvania State University and the Institut des Hautes études Scientifiques for their hospitality.     

Some complete metrics on spaces of fuzzy subsets

The classes of the L_p,∞- and L_p-metrics play an important role to develop a probability theory in fuzzy sample spaces. All of these metrics are known to be separable, but not complete. The classes are closely related as for each L_p,∞-metric there exists some L_p-metric which induces the same topology. This paper deals with the completion of the L_p,∞- and L_p-metrics. We can also show that the relationship between the classes of L_p,∞- and L_p-metrics still holds for the obtained respective classes of their completions.     

(Almost) primitivity of Hecke <Emphasis Type="Italic">L</Emphasis>-functions

Let f be a cusp form of the Hecke space and let L _f be the normalized L-function associated to f. Recently it has been proved that L _f belongs to an axiomatically defined class of functions . We prove that when λ ≤ 2, L _f is always almost primitive, i.e., that if L _f is written as product of functions in , then one factor, at least, has degree zeros and hence is a Dirichlet polynomial. Moreover, we prove that if then L _f is also primitive, i.e., that if L _f = F ₁ F ₂ then F ₁ (or F ₂) is constant; for the factorization of non-primitive functions is studied and examples of non-primitive functions are given. At last, the subset of functions f for which L _f belongs to the more familiar extended Selberg class is characterized and for these functions we obtain analogous conclusions about their (almost) primitivity in .     

Rankin-Selberg L-functions on the critical line

Let f and g be two primitive (holomorphic or Maass) cusp forms of arbitrary level, character and infinity parameter by which we mean the weight in the holomorphic case and the spectral parameter in the Maass case. Let L(s,f × g) be the associated Rankin-Selberg L-function.If g is fixed and the infinity parameter _f of f varies, then for s on the critical line, the subconvex estimate is any admissible value for the Ramanujan-Petersson-conjecture.     

An explicit formula of the Koecher-Maass series associated with Hermitian modular forms belonging to the Maass space

We give an explicit form of the Koecher-Maass series for Hermitian modular forms belonging to the Maass space. We express the Koecher-Maass series as a finite sum of products of two L-functions associated with automorphic forms of one variable. In particular the Koecher-Maass series associated with the Hermitian-Eisenstein series of degree two can be described by a finite sum of products of four shifted Dirichlet L-functions associated with some quadratic characters under the assumption that the class number of imaginary quadratic fields is one.     

An estimate for symmetric square <Emphasis Type="Italic">L</Emphasis>-functions on weight aspect

We establish a mean square estimate on the weight aspect for symmetric square L-functions at every point on the critical line. Received: 15 February 2002     

Construction of entire modular forms of weights 5 and 6 for the congruence group Γ0(4N)

Two classes of entire modular forms of weight 5 and two of weight 6 are constructed for the congruence subgroup ₀(4N). The constructed modular forms as well as the modular forms from [1] will be helpful in the theory of representation of numbers by the quadratic forms in 10 and 12 variables.     

Second order average estimates on local data of cusp forms

We specify sufficient conditions for the square modulus of the local parameters of a family of GL_n cusp forms to be bounded on average. These conditions are global in nature and are satisfied for n ≦ 4. As an application, we show that Rankin-Selberg L-functions on , for n_i ≦ 4, satisfy the standard convexity bound. Received: 21 June 2005     

Special values of L-functions by a Siegel-Weil-Kudla-Rallis formula

We study the arithmeticity of special values of L-functions attached to cuspforms which are Hecke eigenfunctions on hermitian quaternion groups Sp^∗(m,0) which form a reductive dual pair with G=O^∗(4n). For f₁ and f₂ two cuspforms on H, consider their theta liftings θf₁ and θf₂ on G. Then we compute a Rankin-Selberg type integral and obtain an integral representation of the standard L-function:

G〈θf₁⋅Es,θf₂〉=H〈f₁,f₂〉⋅Lstd(f₁,s).     

Siegel modular forms and representations

This paper explicitly describes the procedure of associating an automorphic representation of PGSp(2n,?) with a Siegel modular form of degree n for the full modular group Γ _n =Sp(2n,ℤ), generalizing the well-known procedure for n=1. This will show that the so-called “standard” and ldquo;spinor”L-functions associated with such forms are obtained as Langlands L-functions. The theory of Euler products, developed by Langlands, applied to a Levi subgroup of the exceptional group of type F _<4, is then used to establish meromorphic continuation for the spinor L-function when n=3. Received: 28 March 2000 / Revised version: 25 October 2000     

Central values of L-functions over CM fields

We prove an explicit formula for the central values of certain Rankin L-functions. These L-functions are the L-functions attached to Hilbert newforms over a totally real field F, twisted by unitary Hecke characters of a totally imaginary quadratic extension of F. This formula generalizes our former result on L-functions twisted by finite CM characters.     

On the oscillatory behavior of certain arithmetic functions associated with automorphic forms

We establish the oscillatory behavior of several significant classes of arithmetic functions that arise (at least presumably) in the study of automorphic forms. Specifically, we examine general L-functions conjectured to satisfy the Grand Riemann Hypothesis, Dirichlet series associated with classical entire forms of real weight and multiplier system, Rankin-Selberg convolutions (both “naive” and “modified”), and spinor zeta-functions of Hecke eigenforms on the Siegel modular group of genus two. For the second class we extend results obtained previously and jointly by M. Knopp, W. Kohnen, and the author, whereas for the fourth class we provide a new proof of a relatively recent result of W. Kohnen.     

Finiteness obstructions and Euler characteristics of categories

We introduce notions of finiteness obstruction, Euler characteristic, L²-Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Γ of type (FPR) is a class in the projective class group K₀(RΓ); the functorial Euler characteristic and functorial L²-Euler characteristic are respectively its RΓ-rank and L²-rank. We also extend the second author's K-theoretic Möbius inversion from finite categories to quasi-finite categories. Our main example is the proper orbit category, for which these invariants are established notions in the geometry and topology of classifying spaces for proper group actions. Baez and Dolan's groupoid cardinality and Leinster's Euler characteristic are special cases of the L²-Euler characteristic. Some of Leinster's results on Möbius–Rota inversion are special cases of the K-theoretic Möbius inversion.     

Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique

Let F be a finite extension of ℚ _p . For each integer n≥1, we construct a bijection from the set ?_F ⁰ (n) of isomorphism classes of irreducible degree n representations of the (absolute) Weil group of F, onto the set ? _F ⁰ (n) of isomorphism classes of smooth irreducible supercuspidal representations of GL _n (F). Those bijections preserve epsilon factors for pairs and hence we obtain a proof of the Langlands conjectures for GL _n over F, which is more direct than Harris and Taylor’s. Our approach is global, and analogous to the derivation of local class field theory from global class field theory. We start with a result of Kottwitz and Clozel on the good reduction of some Shimura varieties and we use a trick of Harris, who constructs non-Galois automorphic induction in certain cases. Oblatum 1-III-1999 & 21-VII-1999 / Published online: 29 November 1999     

Twisted Hecke L-values and period polynomials

Let f₁,…,fd be an orthogonal basis for the space of cusp forms of even weight 2k on Γ₀(N). Let L(fi,s) and L(fi,χ,s) denote the L-function of fi and its twist by a Dirichlet character χ, respectively. In this note, we obtain a “trace formula” for the values at integers m and n with 0<m,n<2k and proper parity. In the case N=1 or N=2, the formula gives us a convenient way to evaluate precisely the value of the ratio L(f,χ,m)/L(f,n) for a Hecke eigenform f.     

A relation between <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions and the Tamagawa number conjecture for Hecke characters

We prove that the submodule in K-theory which gives the exact value of the L-function by the Beilinson regulator map at non-critical values for Hecke characters of imaginary quadratic fields K with cl (K) = 1(p-local Tamagawa number conjecture) satisfies that the length of its coimage under the local Soulé regulator map is the p-adic valuation of certain special values of p-adic L-functions associated to the Hecke characters. This result yields immediately, up to Jannsens conjecture, an upper bound for in terms of the valuation of these p-adic L-functions, where V_p denotes the p-adic realization of a Hecke motive.Received: 4 June 2003     

Distributions of class Lα

L_α (0 α 1) is a class of infinitely divisible distributions defined by restricting the measure in the Levy-Khinchin formula to a special form. When α = 1, L_α is just the classical class L. Several properties for L_α classes, which are similar to the most important properties for the class L, are established. Also, a conjecture of Wolfe about unimodality of some L_α distributions is disproved by giving a counterexample.