منابع مشابه
On the Asymptotic Distribution of Zeros of Modular Forms
1.1. Our purpose in this note is to study the limiting distribution of zeros of modular forms. We review some definitions: A modular form of weight k for SL2(Z) is a holomorphic function on the upper half-plane H, transforming as f( cz+d ) = (cz + d)f(z), for all ( a b c d ) ∈ SL2(Z) (this forces k to be even), and “holomorphic at the cusp” (see § 2.1). A form is cuspidal if it vanishes at the ...
متن کاملInterlacing of Zeros of Weakly Holomorphic Modular Forms
We prove that the zeros of a family of extremal modular forms interlace, settling a question of Nozaki. Additionally, we show that the zeros of almost all forms in a basis for the space of weakly holomorphic modular forms of weight k for SL2(Z) interlace on most of the lower boundary of the fundamental domain.
متن کاملExceptional zeros and L-invariants of Bianchi modular forms
Let f be a Bianchi modular form, that is, an automorphic form for GL2 over an imaginary quadratic field F . In this paper, we prove an exceptional zero conjecture in the case where f is new at a prime above p. More precisely, for each prime p of F above p we prove the existence of an L-invariant Lp, depending only on p and f , such that when the p-adic L-function of f has an exceptional zero at...
متن کاملA Generalization of a Theorem of Rankin and Swinnerton-dyer on Zeros of Modular Forms
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...
متن کاملOn the Zeros and Coefficients of Certain Weakly Holomorphic Modular Forms
For this paper we assume familiarity with the basics of the theory of modular forms as may be found, for instance, in Serre’s classic introduction [12]. A weakly holomorphic modular form of weight k ∈ 2Z for Γ = PSL2(Z) is a holomorphic function f on the upper half-plane that satisfies f( cτ+d ) = (cτ + d)f(τ) for all ( a b c d ) ∈ Γ and that has a q-expansion of the form f(τ) = ∑ n≥n0 a(n)q , ...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2012
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2012.04.013