Period Polynomials Research Articles

Period functions associated to real-analytic modular forms

We define L-functions for the class of real-analytic modular forms recently introduced by F. Brown. We establish their main properties and construct the analogue of period polynomial in cases of special interest, including those of modular iterated integrals of length one.

Zeta-polynomials, Hilbert polynomials, and the Eichler–Shimura identities

In 2017, Ono, Rolen, and Sprung [ORS17] answered problems of Manin [Man16] by defining zeta-polynomials $Z_f(s)$ for even weight newforms $f\in S_k(\Gamma_0(N)$; these polynomials can be defined by applying the "Rodriguez-Villegas transform" to the period polynomial of $f$. It is known that these zeta-polynomials satisfy a functional equation $Z_f(s) = \pm Z_f(1-s)$ and they have a conjectural arithmetic-geometric interpretation. Here, we give analogous results for a slightly larger class of polynomials which are also defined using the Rodriguez-Villegas transform.

An algebraic characterization of the Kronecker function

We characterize Zagier’s generating series of extended period polynomials of normalized Hecke eigenforms for {{,mathrm{PSL},}}_2(mathbb {Z}) in terms of the period relations and existence of a suitable factorization. For this, we prove a characterization of the Kronecker function as the “fundamental solution” of the Fay identity.

Periods of modular forms on $\Gamma_0(N)$ and products of Jacobi theta functions

Generalizing a result of [15] for modular forms of level one, we give a closed formula for the sum of all Hecke eigenforms on \Gamma_0(N) , multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level N . We also show that for N = 2, 3 and 5 this formula completely determines the Fourier expansions all Hecke eigenforms of all weights on \Gamma_0(N) .

An elementary proof of the Eichler–Selberg trace formula

Abstract We give a purely algebraic proof of the trace formula for Hecke operators on modular forms for the full modular group SL 2 ⁢ ( ℤ ) {\mathrm{SL}_{2}(\mathbb{Z})} , using the action of Hecke operators on the space of period polynomials. This approach, which can also be applied for congruence subgroups, is more elementary than the classical ones using kernel functions, and avoids the analytic difficulties inherent in the latter (especially in weight two). Our main result is an algebraic property of a special Hecke element that involves neither period polynomials nor modular forms, yet immediately implies both the trace formula and the classical Kronecker–Hurwitz class number relation. This key property can be seen as providing a bridge between the conjugacy classes and the right cosets contained in a given double coset of the modular group.

A generalization of Ramanujan's congruence to modular forms of prime level

We prove congruences between cuspidal newforms and Eisenstein series of prime level, which generalize Ramanujan's congruence. Such congruences were recently found by Billerey and Menares, and we refine them by specifying the Atkin–Lehner eigenvalue of the newform involved. We show that similar refinements hold for the level raising congruences between cuspidal newforms of different levels, due to Ribet and Diamond. The proof relies on studying the new subspace and the Eisenstein subspace of the space of period polynomials for the congruence subgroup Γ0(N), and on a version of Ihara's lemma.

Quantum modular forms and Hecke operators

It is known that there are one-to-one correspondences among the space of cusp forms, the space of homogeneous period polynomials and the space of Dedekind symbols with polynomial reciprocity laws. We add one more space, the space of quantum modular forms with polynomial period functions, to extend results from Fukuhara. Also, we consider Hecke operators on the space of quantum modular forms and construct new quantum modular forms.

On the trace formula for Hecke operators on congruence subgroups

We give a new, simple proof of the trace formula for Hecke operators on modular forms for finite index subgroups of the modular group. The proof uses algebraic properties of certain universal Hecke operators acting on period polynomials of modular forms, and it generalizes an approach developed by Don Zagier and the author for the modular group. This approach leads to a very simple formula for the trace on the space of cusp forms plus the trace on the space of modular forms. As applications, we investigate what happens when one varies the weight or the level in the trace formula.

Period polynomials, derivatives of L-functions, and zeros of polynomials

Period polynomials have long been fruitful tools for the study of values of L-functions in the context of major outstanding conjectures. In this paper, we survey some facets of this study from the perspective of Eichler cohomology. We discuss ways to incorporate non-cuspidal modular forms and values of derivatives of L-functions into the same framework. We further review investigations of the location of zeros of the period polynomial as well as of its analogue for L-derivatives.

Modular forms of real weights and generalised Dedekind symbols

In a previous paper, I have defined non-commutative generalised Dedekind symbols for classical PSL(2,mathbf {Z})-cusp forms using iterated period polynomials. Here I generalise this construction to forms of real weights using their iterated period functions introduced and studied in a recent article by R. Bruggeman and Y. Choie.

The number of maximal period polynomial mappings over the Galois fields of odd characteristics

Пусть $R = GR(q^n, p^n)$ - кольцо Галуа мощности $q^n$ и характеристики $p^n,$ где $q = p^m$, $m, n > 1$. Пусть последовательность $U = \{u_i\}$ определяется соотношением $u_{i+1} = f(u_i)$, $i \in \mathbb N_0$ и $f$ - полиномиальное преобразование кольца $R$. Ранее было показано, что максимально возможный период последовательности $U$ равен $q(q-1)p^{n-2}$. Здесь найдено число полиномиальных преобразований максимального периода над кольцами Галуа при $p \not= 2$.

On period polynomials of degree 2m and weight distributions of certain irreducible cyclic codes

We explicitly determine the values of reduced cyclotomic periods of order 2m, m≥4, for finite fields of characteristic p≡3 or 5(mod8). These evaluations are applied to obtain explicit factorizations of the corresponding reduced period polynomials. As another application, the weight distributions of certain irreducible cyclic codes are described.

The meta-abelian elliptic KZB associator and periods of Eisenstein series

We compute the image of Enriquez’ elliptic KZB associator in the (maximal) meta-abelian quotient of the fundamental Lie algebra of a once-punctured elliptic curve. Our main result is an explicit formula for this image in terms of Eichler integrals of Eisenstein series, and is analogous to Deligne’s computation of the depth one quotient of the Drinfeld associator. We also show how to retrieve Zagier’s extended period polynomials of Eisenstein series, as well as the values at zero of Beilinson–Levin’s elliptic polylogarithms from the meta-abelian elliptic KZB associator.

Period polynomial relations of binomial coefficients and binomial realization of formal double zeta space

In this paper, we give a realization of the formal double zeta space by using binomial coefficients. Along with the results in [Period polynomial relations between formal double zeta values of odd weight, Math. Ann. 365 (2016) 345–362], this gives us two families of period polynomial relations among binomial coefficients. We also give another family of period polynomial relations among binomial coefficients which cannot be obtained from our binomial realization. At the end, some higher depth observation is provided.

The “Riemann Hypothesis” is true for period polynomials of almost all newforms

The period polynomial \(r_f(z)\) for a weight \(k \ge 3\) and level N newform \(f \in S_k(\varGamma _0(N),\chi )\) is the generating function for special values of L(s, f). The functional equation for L(s, f) induces a functional equation on \(r_f(z)\). Jin, Ma, Ono, and Soundararajan proved that for all newforms f of even weight \(k \ge 4\) and trivial nebentypus, the “Riemann Hypothesis” holds for \(r_f(z)\): that is, all roots of \(r_f(z)\) lie on the circle of symmetry \(|z| =1/\sqrt{N}\). We generalize their methods to prove that this phenomenon holds for all but possibly finitely many newforms f of weight \(k \ge 3\) with any nebentypus. We also show that the roots of \(r_f(z)\) are equidistributed if N or k is sufficiently large.

Modelling the shrinking generator in terms of linear CA

This work analyses the output sequence from a cryptographic non-linear generator, the so-called shrinking generator.This sequence, known as the shrunken sequence, can be built by interleaving a unique PN-sequence whose characteristic polynomial serves as basis for the shrunken sequence's characteristic polynomial.In addition, the shrunken sequence can be also generated from a linear model based on cellular automata.The cellular automata here proposed generate a family of sequences with the same properties, period and characteristic polynomial, as those of the shrunken sequence.Moreover, such sequences appear several times along the cellular automata shifted a fixed number.The use of discrete logarithms allows the computation of such a number.The linearity of these cellular automata can be advantageously employed to launch a cryptanalysis against the shrinking generator and recover its output sequence.

Series expansion of the period function and representations of Hecke operators

The period polynomial of a cusp form of an integral weight plays an important role in the number theory. In this paper, we study the period function of a cusp form of real weight. We obtain a series expansion of the period function of a cusp form of real weight for SL(2,Z) by using the binomial expansion. Furthermore, we study two kinds of Hecke operators acting on cusp forms and period functions, respectively. With these Hecke operators we show that there is a Hecke-equivariant isomorphism between the space of cusp forms and the space of period functions. As an application, we obtain a formula for a certain L-value of a Hecke eigenform by using the series expansion of its period function.

Parabolic cohomology and multiple Hecke L-values

We derive various identities among the special values of multiple Hecke L-series. We show that linear combinations of multiple Hecke L-values can be expressed as linear combinations of products of the usual Hecke L-series evaluated at the critical points. The period polynomials introduced here are values of 2-cocycles, whereas the classical period polynomials of elliptic modular forms come from the 1-cocycles. We derive the 2-cycle and the 3-cycle relations among them.

$L$-series for Vector-valued Modular Forms

Motivated by the recent works of Bringmann, Guerzhoy, Kent, and Ono [4] and Bringmann, Fricke, and Kent [3], we introduce $L$-series for vector-valued weakly holomorphic cusp forms, and mock modular period polynomials for vector-valued harmonic weak Maass forms. In particular, we will discuss an integral representation of this new $L$-series and the limiting behavior of special values. Moreover, we also give relations between mock modular periods and $L$-series for vector-valued harmonic weak Maass forms.

On a secant Dirichlet series and Eichler integrals of Eisenstein series

We consider the secant Dirichlet series $$\psi _s (\tau ) = \sum _{n = 1}^{\infty } \frac{\sec (\pi n \tau )}{n^s}$$ , recently introduced and studied by Lalin, Rodrigue and Rogers. In particular, we show, as conjectured and partially proven by Lalin, Rodrigue and Rogers, that the values $$\psi _{2 m} (\sqrt{r})$$ , with $$r > 0$$ rational, are rational multiples of $$\pi ^{2 m}$$ . We then put the properties of the secant Dirichlet series into context by showing that, for even s, they are Eichler integrals of odd weight Eisenstein series of level 4. This leads us to consider Eichler integrals of general Eisenstein series and to determine their period polynomials. In the level 1 case, these polynomials were recently shown by Murty, Smyth and Wang to have most of their roots on the unit circle. We provide evidence that this phenomenon extends to the higher level case. This observation complements recent results by Conrey, Farmer and Imamoglu as well as El-Guindy and Raji on zeros of period polynomials of Hecke eigenforms in the level 1 case. Finally, we briefly revisit results of a similar type in the works of Ramanujan.