Family Of Modular Forms Research Articles

On Shafarevich-Tate groups and analytic ranks in families of modular forms, I. Hida families

Let $f$ be a newform of weight $2$, square-free level and trivial character, let $A_f$ be the abelian variety attached to $f$ and for every good ordinary prime $p$ for $f$ let $\boldsymbol f^{(p)}$ be the $p$-adic Hida family through $f$. We prove that, for all but finitely many primes $p$ as above, if $A_f$ is an elliptic curve such that $A_f(\mathbb Q)$ has rank $1$ and the $p$-primary part of the Shafarevich-Tate group of $A_f$ over $\mathbb Q$ is finite then all specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have finite ($p$-primary) Shafarevich-Tate group and $1$-dimensional image of the relevant $p$-adic \'etale Abel-Jacobi map. Analogous results are obtained also in the rank $0$ case. As a second contribution, with no restriction on the dimension of $A_f$ but assuming the non-degeneracy of certain height pairings \`a la Gillet-Soul\'e between Heegner cycles, we show that if $f$ has analytic rank $1$ then, for all but finitely many $p$, all specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have analytic rank $1$. This result provides some evidence in rank $1$ and weight larger than $2$ for a conjecture of Greenberg predicting that the analytic ranks of even weight modular forms in a Hida family should be as small as allowed by the functional equation, with at most finitely many exceptions.

Supersingular loci from traces of Hecke operators

A classical observation of Deligne shows that, for any prime p ≥ 5 p \geq 5 , the divisor polynomial of the Eisenstein series E p − 1 ( z ) E_{p-1}(z) mod p p is closely related to the supersingular polynomial at p p , S p ( x ) ≔ ∏ E / F ¯ p supersingular ( x − j ( E ) ) ∈ F p [ x ] . \begin{align*} S_p(x) ≔\prod _{E/\overline {\mathbb {F}}_p \text { supersingular}}(x-j(E))\,\, \in \mathbb {F}_p[x]. \end{align*} Deuring, Hasse, and Kaneko and Zagier [Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials, Providence, RI, 1998, pp. 97–126] found other families of modular forms which also give the supersingular polynomial at p p . In a new approach, we prove an analogue of Deligne’s result for the Hecke trace forms T k ( z ) T_k(z) defined by the Hecke action on the space of cusp forms S k S_k . We use the Eichler-Selberg trace formula to identify congruences between trace forms of different weights mod p p , and then relate their divisor polynomials to S p ( x ) S_p(x) using Deligne’s observation.

On interlacing of the zeros of a certain family of modular forms

For s∈{0,4,6,8,10,14}, let k=12m(k)+s≥12 be an even integer and fk be a normalised modular form of weight k with real Fourier coefficients, written asfk=Ek+∑j=1m(k)aj(k)Ek−12jΔj. Under suitable conditions on aj(k) (rectifying an earlier result of Getz), we show that all the zeros of fk, in the standard fundamental domain for the action of SL(2,Z) on the upper half plane, lie on the arc A:={eiθ:π/2≤θ≤2π/3}. Further, we provide a criterion for a family of normalised modular forms {fk}k so that the zeros of fk and fk+12 interlace on A∘:={eiθ:π/2<θ<2π/3}.

A Dimensionally Continued Poisson Summation Formula

We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula, our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, our methods work to establish the (Voronoi) summation formulae associated with functions satisfying (modular) transformations of the Jacobi imaginary type by means of a density argument (as opposed to the usual Mellin transform approach). In particular, we construct a family of generalized theta series from Jacobi theta functions from which these summation formulae can be obtained. This family contains several families of modular forms, but is significantly more general than any of them. Our result also relaxes several of the hypotheses in the standard statements of these summation formulae. The density result we prove for Gaussians in the Schwartz space may be of independent interest.

Automorphic properties of generating functions for generalized odd rank moments and odd Durfee symbols

AbstractWe define two-parameter generalizations of Andrews' (k+1)-marked odd Durfee symbols and 2kth symmetrized odd rank moments, and study the automorphic properties of some of their generating functions. When k = 0 we obtain families of modular forms and mock modular forms. When k ≥ 1, we find quasimodular forms and quasimock modular forms.

On modular forms and the inverse Galois problem

In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL2(Fpn) occurs as the Galois group of some finite extension of the rational nu mbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their resid ual Galois representations are as large as a priori possible. Both results essentially use Khare’s a nd Wintenberger’s notion of gooddihedral primes. Particular care is taken in order to exclud e nontrivial inner twists. 2000 Mathematics Subject Classification: 11F80 (primary); 12F12, 11F11.

Finite groups and families of modular forms associated with them

In the paper, a correspondence between finite groups and q-series which, under specialization, are Fourier expansions of modular forms is studied. The categories thus arising are investigated. The problem of describing groups to which q-series with multiplicative coefficients correspond is considered. Subgroups of this kind are contained in any group. The notion of modular analog of the genetic code of a group is introduced and studied.

Modular forms arising from divisor functions

For an infinite family of modular forms constructed from Klein forms we provide certain explicit formulas for their Fourier coefficients by using the theory of basic hypergeometric series (Theorem 2). By making use of these modular forms we investigate the bases of the vector spaces of modular forms of some levels (Theorem 5) and find its application.

On p-adic families of Hilbert cusp forms of finite slope

Let p be a prime number and F a totally real field. In this article, we obtain a p-adic interpolation of spaces of totally definite quaternionic automorphic forms over F of finite slope, and construct p-adic families of automorphic forms parametrized by affinoid Hecke varieties. Further, as an application to the case where [ F : Q ] is even, we obtain p-adic analytic families of Hilbert eigenforms having fixed finite slope parametrized by weights. This is an analogue of Coleman's analytic families in [R.F. Coleman, p-Adic Banach spaces and families of modular forms, Invent. Math. 127 (1997) 417–479].

Multipliers of a Family of Almost Modular Functions

The goal of this paper is to complete an investigation begun by Cohn and Knopp in their 1994 paper, “Application of Dedekind eta-multipliers to modular equations.” The paper concerned Λ k (z), a family of modular forms on Γ0(N) (N a positive integer) with possibly non-trivial multiplier systems. Cohn and Knopp defined new functions Ψ k (z) and a new group containing Γ0(N) and proved that for all S in the larger group and for all k, Λ k (Sz) = M k(S)Ψ k (z), where M k(S)24 = 1. This yielded interesting invariance properties of Λ k , dependent on the values of M k(S). Fixing a constant integer e, independent of k, Cohn and Knopp proved that for all k and all S in the larger group, M k(S) e = (±1) e . They determined the sign of M k(S) e in many, but not all, cases. In this paper, we give a complete determination of the values of M k(S) e in the remaining cases.

ON p-ADIC INTEGRATION IN SPACES OF MODULAR FORMS AND ITS APPLICATIONS

The purpose of this course is to give an introduction to the theory of p-adic integration with values in spaces of modular forms (elliptic modular forms, Siegel modular forms, . . .). We show that very general p-adic families of modular forms can be constructed as moments of certain p-adic measures on a profinite group Y = lim ←− Yi with values in a formal q-expansion ring like Zp[[q ]] where B is an additive semi-group, and q = {q |ξ ∈ B} the corresponding formally written multiplicative semi-group (for example B = Bn = {ξ = ξ ∈ Mn(Q)|ξ ≥ 0, ξ half-integral} is the semi-group, important for the theory of Siegel modular forms). We discuss some applications of this theory to the construction of certain new p-adic families of modular forms (families of Klingen-Eisenstein series, families of theta-series with spherical polynomials. . .). Main sources of this theory are: • Serre’s theory of p-adic forms as certain formal q-expansions (J.-P. Serre, Formes modulaires et fonctions zeta p-adiques, LNM 350 (1973) 191-268) [Se73]. • Hida’s theory of p-adic modular forms and p-adic Hecke algebras (H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge University Press, 1993 [Hi93]). • Construction of p-adic Siegel-Eisenstein series by the author, see [PaSE]. As an application, we describe a solution of a problem of Coleman-Mazur in [PaTV], using the RankinSelberg method and the p-adic integration in a Banach algebra A. An introductory cours given on November 29 in POSTECH (Pohang, Korea) 0 Introduction Let p be a prime number (we often assume p≥ 5). There are two different ways of introducing p-adic modular forms: the first approach uses formal q-expansions with coefficients in a p-adic ring [Se73], and the second approach is the p-adic interpolation of Galois representations attached to classical automorphic forms. The first approach was extensively developped by Katz [Ka78] for the group G = GL2 over a totally real number field, in order to construct p-adic L-functions for CM-fields using p-adic Hilbert-Eisenstein series. In general, in this q-expansion method a typical p-adic family φ of modular (automorphic) forms is an element of the Serre ring: φ ∈ Λ[[q]] where Λ = Zp[[T ]] is the Iwasawa algebra. In the second approach one considers Λ-adic Galois representations of type ρ : Gal(Q/Q) → GLm(Λ) (“Big Galois representations”, see [Hi86], [Til-U]). These two theories are essentially equivalent if we start from holomorphic automorphic forms on the group G = GL2 over a totally real field, but in other cases there is no direct link between φ and ρ. On the other hand there exist interesting examples of p-adic L-functions Lφ,p and Lρ,p attached to φ and to ρ. In general Lφ,p and Lρ,p should belong to the quotient field L = QuotΛ or to its finite extensions. If ρ interpolates a p-adic family of motives then there are conjectural general definitions of Lρ,p (see [Co-PeRi], [Colm98], [PaAdm]). It would be very interesting to formulate a general Langlands-type conjecture relating Λ-adic automorphic forms and Λ-adic Galois representations. As an application, we describe a solution of a problem of Coleman-Mazur, using the Rankin-Selberg method and the theory of p-adic integration with values in a p-adic algebra A. This problem was stated in "The Eigencurve" (1998), R.Coleman and B.Mazur stated the following as follows: Given a prime p and Coleman’s family {fk′} of cusp eigenforms of a fixed positive slope σ = ordp(αp(k )) > 0, to construct a two variable p-adic L-function interpolating on k the Amice-Velu p-adic L-functions Lp(fk′ ). Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. Such measures come from Eisenstein distributions with values in certain Banach A-modules M = M(N ;A) of families of overconvergent forms over A.

P-adic Banach spaces and families of modular forms

Let p be a prime, Cp the completion of an algebraic closure of the p-adic numbers Qp and K a nite extension of Qp contained in Cp. Let v be the valuation on Cp such that v(p) = 1 and let | | be the absolute value on Cp such that |x| = p−v(x) for x ∈ Cp. Suppose N is a positive integer prime to p. Let X1(Np) denote the modular curve over K which represents elliptic curves with 1(Np)-structure and let Up be the Hecke operator on modular forms on X1(Np) which takes a form with q-expansion ∑ n anq n to the modular form with q-expansion ∑ n anpq n. A modular form F is said to have slope ∈ Q if there is a polynomial R(T ) over Cp such that R(Up)F = 0 and such that the Newton polygon of R(T ) has only one side and its slope is − .