Critical slope p-adic L-functions of CM modular forms

For ordinary modular forms, there are two constructions of a p-adic L-function attached to the non-unit root of the Hecke polynomial, which are conjectured but not known to coincide. We prove this conjecture for modular forms of CM type, by calculating the critical-slope L-function arising from Kato’s Euler system and comparing this with results of Bellaïche on the critical-slope L-function defined using overconvergent modular symbols.     

Some Siegel modular standard L-values, and Shafarevich-Tate groups

We explain how the Bloch-Kato conjecture leads us to the following conclusion: a large prime dividing a critical value of the L-function of a classical Hecke eigenform f of level 1, should often also divide certain ratios of critical values for the standard L-function of a related genus two (and in general vector-valued) Hecke eigenform F. The relation between f and F (Harder?s conjecture in the vector-valued case) is a congruence involving Hecke eigenvalues, modulo the large prime. In the scalar-valued case we prove the divisibility, subject to weak conditions. In two instances in the vector-valued case, we confirm the divisibility using elaborate computations involving special differential operators. These computations do not depend for their validity on any unproved conjecture.     

Special values of multiple Hecke L-functions

The purpose of this study is to clarify the structure of the space generated by the special values of multiple Hecke L-functions for the full modular group SL ₂(Z). In this paper we describe the explicit relations among the values by using Manin??s results, and show the numerical example in case of the Ramanujan ??-function.     

On Wronskians of weight one Eisenstein series

We describe the span of Hecke eigenforms of weight four with nonzero central value of L-function in terms of Wronskians of certain weight one Eisenstein series.     

Congruences for Hecke eigenvalues of Siegel modular forms

We prove some congruences for Hecke eigenvalues of Klingen-Eisenstein series and those of cusp forms for Siegel modular groups modulo special values of automorphic L-functions.     

Image of Λ-adic Galois representations modulo p

Let p≥5 be a prime. If an irreducible component of the spectrum of the ‘big’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its mod p Galois representation contains an open subgroup of for the canonical “weight” variable T. This fact appears to be deep, as it is almost equivalent to the vanishing of the μ-invariant of the Kubota–Leopoldt p-adic L-function and the anticyclotomic Katz p-adic L-function. Another key ingredient of the proof is the anticyclotomic main conjecture proven by Rubin/Mazur–Tilouine.     

An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials

In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an arithmetic formula for these coefficients using the “explicit formula” of prime number theory. In this paper, the author obtains an arithmetic formula for corresponding coefficients associated with the Euler product of Hecke polynomials, which is essentially a product of L-functions attached to weight 2 cusp forms (both newforms and oldforms) over Hecke congruence subgroups Γ₀(N). The nonnegativity of these coefficients gives a criterion for the Riemann hypothesis for all these L-functions at once.     

Jacobi–Hecke algebras and a rationality theorem for a formal Hecke series

It is well known that the L-function associated to a Siegel eigenform f is equal to a Rankin-Selberg type zeta-integral involving f and a restricted Eisenstein series ([3], [14]). At some point in the proof one has to show the equality of a certain Dirichlet series and the L-function, which follows from a rationality theorem for a certain formal power series over the Hecke algebra. The main purpose of this paper is to develop a Hecke theory for the Jacobi group and to prove such a rationality theorem. Received: 17 August 1998 / Revised version: 17 February 1999     

Shintani lifts and fractional derivatives for harmonic weak Maass forms

In this paper, we construct Shintani lifts from integral weight weakly holomorphic modular forms to half-integral weight weakly holomorphic modular forms. Although defined by different methods, these coincide with the classical Shintani lifts when restricted to the space of cusp forms. As a side effect, this gives the coefficients of the classical Shintani lifts as new cycle integrals. This yields new formulas for the L-values of Hecke eigenforms. When restricted to the space of weakly holomorphic modular forms orthogonal to cusp forms, the Shintani lifts introduce a definition of weakly holomorphic Hecke eigenforms. Along the way, auxiliary lifts are constructed from the space of harmonic weak Maass forms which yield a “fractional derivative” from the space of half-integral weight harmonic weak Maass forms to half-integral weight weakly holomorphic modular forms. This fractional derivative complements the usual ξ-operator introduced by Bruinier and Funke.     

A Rankin-Selberg integral for the adjointL-function of Sp4

In this paper we study the AdjointL-function for Sp₄. For generic cusp forms of Sp₄(A) we construct a global Rankin-Selberg integral which represents thisL-function.     

A new zero-density result of <Emphasis Type="Italic">L</Emphasis>-functions attached to Maass forms

Let f denote a normalized Maass cusp form for SL(2, ℤ), which is an eigenfunction of all the Hecke operators T(n) as well as the reflection operator $ T_{ - 1} :z \to - \bar z $ T_{ - 1} :z \to - \bar z . We obtain a zero-density result of the L-function attached to f near σ = 1. This improves substantially the previous results in this direction.     

On the zeros on the critical line of L-functions corresponding to automorphic cusp forms

We consider an automorphic cusp form of integer weight k ≥ 1, which is the eigenfunction of all Hecke operators. It is proved that, for the L-series whose coefficients correspond to the Fourier coefficients of such an automorphic form, the positive fraction of nontrivial zeros lie on the critical line.     

CM values and central L-values of elliptic modular forms

Let f be a holomorphic cusp form of weight l on SL₂(Z) and Ω an algebraic Hecke character of an imaginary quadratic field K with Ω((α)) = (α/|α|) ^l for ${\alpha\in K^{\times}}Let f be a holomorphic cusp form of weight l on SL₂(Z) and Ω an algebraic Hecke character of an imaginary quadratic field K with Ω((α)) = (α/|α|) ^l for a ? K^×{\alpha\in K^{\times}}. Let L(f, Ω; s) be the Rankin-Selberg L-function attached to (f, Ω) and P(f, Ω) an “Ω-averaged” sum of CM values of f. In this paper, we give a formula expressing the central L-values L(f, Ω; 1/2) in terms of the square of P(f, Ω).     

Determining modular forms on $${S L_2(\mathbb{Z})}$$ by central values of convolution L-functions

We show that a cuspidal normalized Hecke eigenform g of level one and even weight is uniquely determined by the central values of the family of Rankin– Selberg L-functions \({L(s, f\otimes g)}\) , where f runs over the Hecke basis of the space of cusp forms of level one and weight k with k varying over an infinite set of even integers.     

Period Integrals and Rankin–Selberg L-functions on GL(n)

We compute the second moment of a certain family of Rankin–Selberg L-functions L(f ×?g, 1/2) where f and g are Hecke–Maass cusp forms on GL(n). Our bound is as strong as the Lindel?f hypothesis on average, and recovers individually the convexity bound. This result is new even in the classical case n?=?2.     

Values of <Emphasis Type="Italic">L</Emphasis>-functions at integers outside the critical strip

We incorporate the non-critical values of L-functions of cusp forms into a cohomological set-up analogous to the one of Eichler, Manin and Shimura. We use the 1-cocycles we associate in this way to non-critical values to prove an expression for such values which is similar in structure to Manin’s formula for the critical value of the L-function of a weight 2 cusp form. YoungJu Choie is partially supported by KOSEF R01-2003-00011596-0 and by ITRC Research Fund. N. Diamantis is partially supported by EPSRC grant EP/D032350/1.     

Hecke operators on Drinfeld cusp forms

In this paper, we study the Drinfeld cusp forms for Γ₁(T) and Γ(T) using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for Γ₁(T) of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that the Hecke operators are not diagonalizable in general for Γ₁(T) of large weights, and not for Γ(T) even of small weights. The Hecke eigenvalues on cusp forms for Γ(T) with small weights are determined and the eigenspaces characterized.     

Modular-type relations associated to the Rankin–Selberg <Emphasis Type="Italic">L</Emphasis>-function

Hafner and Stopple proved a conjecture of Zagier relating to the asymptotic behaviour of the inverse Mellin transform of the symmetric square L-function associated with the Ramanujan tau function. In this paper, we prove a similar result for any cusp form over the full modular group.     

A new Bartholdi zeta function of a digraph II

We introduce a new type of the Bartholdi zeta function of a digraph D. Furthermore, we define a new type of the Bartholdi L-function of D, and give a determinant expression of it. We show that this L-function of D is equal to the L-function of D defined in [H. Mizuno, I. Sato, A new Bartholdi zeta function of a digraph, Linear Algebra Appl. 423 (2007) 498-511]. As a corollary, we obtain a decomposition formula for a new type of the Bartholdi zeta function of a group covering of D by new Bartholdi L-functions of D.     

Automorphic forms for elliptic function fields

Let F be the function field of an elliptic curve X over ${\mathbb{F}_q}$ . In this paper, we calculate explicit formulas for unramified Hecke operators acting on automorphic forms over F. We determine these formulas in the language of the graph of an Hecke operator, for which we use its interpretation in terms of ${\mathbb{P}^1}$ -bundles on X. This allows a purely geometric approach, which involves, amongst others, a classification of the ${\mathbb{P}^1}$ -bundles on X. We apply the computed formulas to calculate the dimension of the space of unramified cusp forms and the support of a cusp form. We show that a cuspidal Hecke eigenform does not vanish in the trivial ${\mathbb{P}^1}$ -bundle. Further, we determine the space of unramified F′-toroidal automorphic forms where F′ is the quadratic constant field extension of F. It does not contain non-trivial cusp forms. An investigation of zeros of certain Hecke L-series leads to the conclusion that the space of unramified toroidal automorphic forms is spanned by the Eisenstein series E( · , s) where s?+?1/2 is a zero of the zeta function of X—with one possible exception in the case that q is even and the class number h equals q?+?1.