BY YOUNGJU CHOIE 0. Introduction Automorphic integrals with rational period functions, being gene... more BY YOUNGJU CHOIE 0. Introduction Automorphic integrals with rational period functions, being generalization of automorphic forms on the discrete subgroup of SL(2, R), share properties similar to those of forms. Examples are furnished by the Eichler integrals-automorphic integrals of negative integer weight with polynomial period functions-which have been the object of much attention in recent years [4], [5], [6], [9], [12], [15], [17], [19]. The questional naturally arises whether there exist automorphic integrals with rational period functions which are not polynomials. M. Knopp [13] has constructed modular integrals of weight 2k (k odd) which differ from Eichler integrals. In [14] it has been shown that the poles of any rational period function, qr,2k(z), for the modular group I'(1) must lie in Q(Vr-), N Z /. However, the only previously known quadratic fields containing poles of qr,2k(z) for F(1) were Q(v-), Q(v/-), and Q(vt), and these examples were known only for odd k. The main object of this paper is the construction of qr,2(z) for F(1) with k any integer (even or odd), having poles in an arbitrary real quadratic field, Q(qr-). We have developed three distinct new methods to achieve this goal. First, we have constructed qr,2k(Z) for F(1) by using the coset decomposition of I"(1), the commutator subgroup of I'(1). Since I"(1) is a free group, the necessary and sufficient conditions for the existence of a rational period function qr,2k(Z) of a modular integral on F'(1) reduce to a single condition on rational period functions for I'(1). Then rational period functions of a modular integral on F(1) can be constructed by showing how to satisfy the above condition. This construction can be generalized to incorporate the class of Hecke group. By use of an operator of Bogo-Kuyk [1], qr,2k(z) for I'(1) can be constructed from those on the Hecke groups for , vand vc-. The second method entails the use of Pell's equation to construct qr,2(z). This construction gives qr,2k(z) for I'(1) and any integer k with poles in an
Using the recently found Rankin-Cohen type brackets on the spaces of Jacobi forms, we define gene... more Using the recently found Rankin-Cohen type brackets on the spaces of Jacobi forms, we define generalized Rankin-Cohen algebras. We study their algebraic properties and give examples generalizing the elliptic cases. Recently, the theory of Jacobi forms has been studied extensively and systematically, first by Eichler and Zagier [5] and many others. It turns out that Jacobi forms are connected with modular forms of halfintegral weight as well as integral weight, Siegel modular forms and elliptic curves. It was shown that the heat operator plays an important role connecting Jacobi forms and elliptic modular forms. In [1], [2], [3] the generalization of the Rankin-Cohen brackets, which involves the heat operator, to Jacobi forms has been found.
Given a pseudodifferential operator, as was discussed in Section 1.4, we can obtain the correspon... more Given a pseudodifferential operator, as was discussed in Section 1.4, we can obtain the corresponding formal power series by using some constant multiples of its coefficients in such a way that the correspondence is (formula presented)-equivariant. The space of pseudodifferential operators is a noncommutative algebra over (formula presented) and therefore has a natural structure of a Lie algebra. In this chapter we determine the corresponding Lie algebra structure on the space of formal power series and study some of its properties. We also discuss these results in connection with automorphic pseudodifferential operators, Jacobilike forms, and modular series for a discrete subgroup of (formula presented) (cf. [66])
It is well known that modular forms for a discrete subgroup (formula presented) can be interprete... more It is well known that modular forms for a discrete subgroup (formula presented) can be interpreted geometrically as sections of a sheaf or a line bundle over the Riemann surface
We shall prove a rationality result for a quotient of scalar products involving the Ikeda lift of... more We shall prove a rationality result for a quotient of scalar products involving the Ikeda lift of an elliptic cusp form.
This paper contains three main results: the first one is to derive two "period relations" and the... more This paper contains three main results: the first one is to derive two "period relations" and the second one is a complete characterization of period functions of Jacobi forms in terms of period relations. These are done by introducing a concept of "Jacobi integrals" on the full Jacobi group. The last one is to show, for the given holomorphic function P(τ, z) having two period relations, there exists a unique Jacobi integral, up to Jacobi forms, with a given function P(τ, z) as its period function. This is done by constructing a generalized Jacobi Poincaré series explicitly. This is to say that every holomorphic function with "period relations" is coming from a Jacobi integral. It is an analogy of Eichler cohomology theory studied in Knopp (Bull Am Math Soc 80:607-632, 1974) for the functions with elliptic and modular variables. It explains the functional equations satisfied by the "Mordell integrals" associated with the Lerch sums (Zwegers in Mock theta functions, PhD thesis, Universiteit Utrecht, 2002) or, more generally, with the higher Appell functions (Semikhatov et al. in Commun Math Phys 255(2):469-512, 2005). Developing theories of Jacobi integrals with elliptic and modular variables in this paper is a natural extension of the Eichler integral with modular variable. Period functions can be explained in terms of the parabolic cohomology group as well.
This work is devoted to the algebraic and arithmetic properties of Rankin-Cohen brackets allowing... more This work is devoted to the algebraic and arithmetic properties of Rankin-Cohen brackets allowing to define and study them in several natural situations of number theory. It focuses on the property of these brackets to be formal deformations of the algebras on which they are defined, with related questions on restriction-extension methods. The general algebraic results developed here are applied to the study of formal deformations of the algebra of weak Jacobi forms and their relation with the Rankin-Cohen brackets on modular and quasimodular forms.
Generalizing a result of [15] for modular forms of level one, we give a closed formula for the su... more Generalizing a result of [15] for modular forms of level one, we give a closed formula for the sum of all Hecke eigenforms on Γ0(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 of all Hecke eigenforms of all weights on Γ0(N).
Let f be a non-zero cusp form with real Fourier coefficients a(n) (n ≥ 1) of positive real weight... more Let f be a non-zero cusp form with real Fourier coefficients a(n) (n ≥ 1) of positive real weight k and a unitary multiplier system v on a subgroup ⊂ SL 2 )ޒ( that is finitely generated and of Fuchsian type of the first kind. Then, it is known that the sequence (a(n))(n ≥ 1) has infinitely many sign changes. In this short note, we generalise the above result to the case of entire modular integrals of non-positive integral weight k on the group * 0 (N) (N ∈ )ގ generated by the Hecke congruence subgroup 0 (N) and the Fricke involution W N :=
In spite of the good security of the cryptosystem on an elliptic curve defined over finite field,... more In spite of the good security of the cryptosystem on an elliptic curve defined over finite field, the cryptosystem on an elliptic curve is slower than that on a finite field. To be practical, we need a better method to improve a speed of the cryptosystem on an elliptic curve defined over a finite field. In 1991, Koblitz suggested to use an anomalous curve over , which is an elliptic curve with Frobenious map whose trace is 1, and reduced a speed of computation of mP. In this paper, we consider an elliptic curve defined over with Frobenious map whose trace is 3 and suggest an efficient algorithm to compute mP. On the proposed elliptic curve, we can compute multiples mP with +1 addition in worst case.
Journal of Mathematical Analysis and Applications, Jun 1, 2020
A generalized Riemann hypothesis states that all zeros of the completed Hecke L-function L * (f, ... more A generalized Riemann hypothesis states that all zeros of the completed Hecke L-function L * (f, s) of a normalized Hecke eigenform f on the full modular group should lie on the vertical line Re(s) = k 2. It was shown in [7] that there exists a Hecke eigenform f of weight k such that L * (f, s) = 0 for sufficiently large k and any point on the line segments Im(s) = t 0 , k−1 2 < Re(s) < k 2 − ǫ, k 2 + ǫ < Re(s) < k+1 2 , for any given real number t 0 and a positive real number ǫ. This paper concerns the non-vanishing of the product L * (f, s)L * (f, w) (s, w ∈ C) on average.
Since the 70's there are isomorphisms between ring of polynomial invariants of certain finite... more Since the 70's there are isomorphisms between ring of polynomial invariants of certain finite groups and certain rings of modular forms. These rings of polynomial invariants contain the weight enumerators of self dual codes. The theta series of unimodular lattices are amongst these modular forms. We survey known results on Jacobi and Siegel modular forms, and give recent results on modular forms for special levels.
We derive various identities among the special values of multiple Hecke Lseries. We show that lin... more We derive various identities among the special values of multiple Hecke Lseries. 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.
We study various properties of quasimodular forms by using their connections with Jacobi-like for... more We study various properties of quasimodular forms by using their connections with Jacobi-like forms. Such connections are made by identifying quasimodular forms for a discrete subgroup $$\Gamma $$Γ of $$SL(2, {\mathbb {R}})$$SL(2,R) with certain polynomials over the ring of holomorphic functions of the Poincaré upper half plane that are $$\Gamma $$Γ-invariant. We consider a surjective map from Jacobi-like forms to quasimodular forms and prove that it has a right inverse, which may be regarded as a lifting from quasimodular forms to Jacobi-like forms.
This series publishes advanced monographs giving well-written presentations of the "state-of-the-... more This series publishes advanced monographs giving well-written presentations of the "state-of-the-art" in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.
BY YOUNGJU CHOIE 0. Introduction Automorphic integrals with rational period functions, being gene... more BY YOUNGJU CHOIE 0. Introduction Automorphic integrals with rational period functions, being generalization of automorphic forms on the discrete subgroup of SL(2, R), share properties similar to those of forms. Examples are furnished by the Eichler integrals-automorphic integrals of negative integer weight with polynomial period functions-which have been the object of much attention in recent years [4], [5], [6], [9], [12], [15], [17], [19]. The questional naturally arises whether there exist automorphic integrals with rational period functions which are not polynomials. M. Knopp [13] has constructed modular integrals of weight 2k (k odd) which differ from Eichler integrals. In [14] it has been shown that the poles of any rational period function, qr,2k(z), for the modular group I'(1) must lie in Q(Vr-), N Z /. However, the only previously known quadratic fields containing poles of qr,2k(z) for F(1) were Q(v-), Q(v/-), and Q(vt), and these examples were known only for odd k. The main object of this paper is the construction of qr,2(z) for F(1) with k any integer (even or odd), having poles in an arbitrary real quadratic field, Q(qr-). We have developed three distinct new methods to achieve this goal. First, we have constructed qr,2k(Z) for F(1) by using the coset decomposition of I"(1), the commutator subgroup of I'(1). Since I"(1) is a free group, the necessary and sufficient conditions for the existence of a rational period function qr,2k(Z) of a modular integral on F'(1) reduce to a single condition on rational period functions for I'(1). Then rational period functions of a modular integral on F(1) can be constructed by showing how to satisfy the above condition. This construction can be generalized to incorporate the class of Hecke group. By use of an operator of Bogo-Kuyk [1], qr,2k(z) for I'(1) can be constructed from those on the Hecke groups for , vand vc-. The second method entails the use of Pell's equation to construct qr,2(z). This construction gives qr,2k(z) for I'(1) and any integer k with poles in an
Using the recently found Rankin-Cohen type brackets on the spaces of Jacobi forms, we define gene... more Using the recently found Rankin-Cohen type brackets on the spaces of Jacobi forms, we define generalized Rankin-Cohen algebras. We study their algebraic properties and give examples generalizing the elliptic cases. Recently, the theory of Jacobi forms has been studied extensively and systematically, first by Eichler and Zagier [5] and many others. It turns out that Jacobi forms are connected with modular forms of halfintegral weight as well as integral weight, Siegel modular forms and elliptic curves. It was shown that the heat operator plays an important role connecting Jacobi forms and elliptic modular forms. In [1], [2], [3] the generalization of the Rankin-Cohen brackets, which involves the heat operator, to Jacobi forms has been found.
Given a pseudodifferential operator, as was discussed in Section 1.4, we can obtain the correspon... more Given a pseudodifferential operator, as was discussed in Section 1.4, we can obtain the corresponding formal power series by using some constant multiples of its coefficients in such a way that the correspondence is (formula presented)-equivariant. The space of pseudodifferential operators is a noncommutative algebra over (formula presented) and therefore has a natural structure of a Lie algebra. In this chapter we determine the corresponding Lie algebra structure on the space of formal power series and study some of its properties. We also discuss these results in connection with automorphic pseudodifferential operators, Jacobilike forms, and modular series for a discrete subgroup of (formula presented) (cf. [66])
It is well known that modular forms for a discrete subgroup (formula presented) can be interprete... more It is well known that modular forms for a discrete subgroup (formula presented) can be interpreted geometrically as sections of a sheaf or a line bundle over the Riemann surface
We shall prove a rationality result for a quotient of scalar products involving the Ikeda lift of... more We shall prove a rationality result for a quotient of scalar products involving the Ikeda lift of an elliptic cusp form.
This paper contains three main results: the first one is to derive two "period relations" and the... more This paper contains three main results: the first one is to derive two "period relations" and the second one is a complete characterization of period functions of Jacobi forms in terms of period relations. These are done by introducing a concept of "Jacobi integrals" on the full Jacobi group. The last one is to show, for the given holomorphic function P(τ, z) having two period relations, there exists a unique Jacobi integral, up to Jacobi forms, with a given function P(τ, z) as its period function. This is done by constructing a generalized Jacobi Poincaré series explicitly. This is to say that every holomorphic function with "period relations" is coming from a Jacobi integral. It is an analogy of Eichler cohomology theory studied in Knopp (Bull Am Math Soc 80:607-632, 1974) for the functions with elliptic and modular variables. It explains the functional equations satisfied by the "Mordell integrals" associated with the Lerch sums (Zwegers in Mock theta functions, PhD thesis, Universiteit Utrecht, 2002) or, more generally, with the higher Appell functions (Semikhatov et al. in Commun Math Phys 255(2):469-512, 2005). Developing theories of Jacobi integrals with elliptic and modular variables in this paper is a natural extension of the Eichler integral with modular variable. Period functions can be explained in terms of the parabolic cohomology group as well.
This work is devoted to the algebraic and arithmetic properties of Rankin-Cohen brackets allowing... more This work is devoted to the algebraic and arithmetic properties of Rankin-Cohen brackets allowing to define and study them in several natural situations of number theory. It focuses on the property of these brackets to be formal deformations of the algebras on which they are defined, with related questions on restriction-extension methods. The general algebraic results developed here are applied to the study of formal deformations of the algebra of weak Jacobi forms and their relation with the Rankin-Cohen brackets on modular and quasimodular forms.
Generalizing a result of [15] for modular forms of level one, we give a closed formula for the su... more Generalizing a result of [15] for modular forms of level one, we give a closed formula for the sum of all Hecke eigenforms on Γ0(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 of all Hecke eigenforms of all weights on Γ0(N).
Let f be a non-zero cusp form with real Fourier coefficients a(n) (n ≥ 1) of positive real weight... more Let f be a non-zero cusp form with real Fourier coefficients a(n) (n ≥ 1) of positive real weight k and a unitary multiplier system v on a subgroup ⊂ SL 2 )ޒ( that is finitely generated and of Fuchsian type of the first kind. Then, it is known that the sequence (a(n))(n ≥ 1) has infinitely many sign changes. In this short note, we generalise the above result to the case of entire modular integrals of non-positive integral weight k on the group * 0 (N) (N ∈ )ގ generated by the Hecke congruence subgroup 0 (N) and the Fricke involution W N :=
In spite of the good security of the cryptosystem on an elliptic curve defined over finite field,... more In spite of the good security of the cryptosystem on an elliptic curve defined over finite field, the cryptosystem on an elliptic curve is slower than that on a finite field. To be practical, we need a better method to improve a speed of the cryptosystem on an elliptic curve defined over a finite field. In 1991, Koblitz suggested to use an anomalous curve over , which is an elliptic curve with Frobenious map whose trace is 1, and reduced a speed of computation of mP. In this paper, we consider an elliptic curve defined over with Frobenious map whose trace is 3 and suggest an efficient algorithm to compute mP. On the proposed elliptic curve, we can compute multiples mP with +1 addition in worst case.
Journal of Mathematical Analysis and Applications, Jun 1, 2020
A generalized Riemann hypothesis states that all zeros of the completed Hecke L-function L * (f, ... more A generalized Riemann hypothesis states that all zeros of the completed Hecke L-function L * (f, s) of a normalized Hecke eigenform f on the full modular group should lie on the vertical line Re(s) = k 2. It was shown in [7] that there exists a Hecke eigenform f of weight k such that L * (f, s) = 0 for sufficiently large k and any point on the line segments Im(s) = t 0 , k−1 2 < Re(s) < k 2 − ǫ, k 2 + ǫ < Re(s) < k+1 2 , for any given real number t 0 and a positive real number ǫ. This paper concerns the non-vanishing of the product L * (f, s)L * (f, w) (s, w ∈ C) on average.
Since the 70's there are isomorphisms between ring of polynomial invariants of certain finite... more Since the 70's there are isomorphisms between ring of polynomial invariants of certain finite groups and certain rings of modular forms. These rings of polynomial invariants contain the weight enumerators of self dual codes. The theta series of unimodular lattices are amongst these modular forms. We survey known results on Jacobi and Siegel modular forms, and give recent results on modular forms for special levels.
We derive various identities among the special values of multiple Hecke Lseries. We show that lin... more We derive various identities among the special values of multiple Hecke Lseries. 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.
We study various properties of quasimodular forms by using their connections with Jacobi-like for... more We study various properties of quasimodular forms by using their connections with Jacobi-like forms. Such connections are made by identifying quasimodular forms for a discrete subgroup $$\Gamma $$Γ of $$SL(2, {\mathbb {R}})$$SL(2,R) with certain polynomials over the ring of holomorphic functions of the Poincaré upper half plane that are $$\Gamma $$Γ-invariant. We consider a surjective map from Jacobi-like forms to quasimodular forms and prove that it has a right inverse, which may be regarded as a lifting from quasimodular forms to Jacobi-like forms.
This series publishes advanced monographs giving well-written presentations of the "state-of-the-... more This series publishes advanced monographs giving well-written presentations of the "state-of-the-art" in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.
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