A Generalization of a Theorem of Rankin and Swinnerton-dyer on Zeros of Modular Forms
نویسندگان
چکیده
Rankin and Swinnerton-Dyer [R, S-D] prove that all zeros of the Eisenstein series Ek in the standard fundamental domain for Γ lie on A := {eiθ : π 2 ≤ θ ≤ 2π 3 }. In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc A. Using this result we prove a speculation of Ono, namely that the zeros of the unique “gap function” in Mk, the modular form with the maximal number of consecutive zero coefficients in its q-expansion following the constant 1, has zeros only on A. In addition, we show that the j-invariant maps these zeros to totally real algebraic integers of degree bounded by a simple function of the weight k.
منابع مشابه
On the Zeros of Certain Poincaré
We locate all of the zeros of certain Poincaré series associated with the Fricke groups Γ0(2) and Γ ∗ 0(3) in their fundamental domains by applying and extending the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (“On the zeros of Eisenstein series”, 1970).
متن کاملOn the Zeros of Certain Poincaré Series
We locate all of the zeros of certain Poincaré series associated with the Fricke groups Γ0(2) and Γ ∗ 0(3) in their fundamental domains by applying and extending the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (“On the zeros of Eisenstein series”, 1970).
متن کاملOn the Zeros of Eisenstein Series for Γ
We locate all of the zeros of the Eisenstein series associated with the Fricke groups Γ0(2) and Γ ∗ 0(3) in their fundamental domains by applying and expanding the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (“On the zeros of Eisenstein series”, 1970).
متن کاملOn the Zeros of the Eisenstein Series
We locate almost all the zeros of the Eisenstein series associated with the Fricke groups of level 5 and 7 in their fundamental domains by applying and extending the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (1970). We also use the arguments of some terms of the Eisenstein series in order to improve existing error bounds.
متن کاملExperimental finding of modular forms for noncongruence subgroups
In this paper we will use experimental and computational methods to find modular forms for non-congruence subgroups, and the modular forms for congruence subgroups that they are associated with via the Atkin–Swinnerton-Dyer correspondence. We also prove a generalization of a criterion due to Ligozat for an eta-quotient to be a modular function.
متن کامل