A simple method to extract zeros of certain Eisenstein series of small level
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A simple method to extract zeros of certain Eisenstein series of small level ARADHITA CHATTOPADHYAYA Department of Centre for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India Present Address: Department of Mathematics, Trinity College Dublin, Ireland Email: [email protected]; [email protected] MS received 11 March 2019; revised 15 June 2019; accepted 22 June 2019 Abstract. This paper provides a simple method to extract the zeros of some weight two Eisenstein series of level N where N = 2, 3, 5 and 7. The method is based on the observation that these Eisenstein series are integral over the graded algebra of modular forms on S L(2, Z ) and their zeros are ‘controlled’ by those of E 4 and E 6 in the fundamental domain of 0 (N ). Keywords.
Zeros of Eisenstein series; zeros of modular forms of 0 (N ).
2000 Mathematics Subject Classification.
11F11, 11F03.
1. Introduction In the work of Rankin and Swinnerton-Dyer [16], the location of zeros of all Eisenstein series E k (of weight k ≥ 4, even) of full modular group S L 2 (Z) had been determined. In the fundamental domain this was found to be always on the arc |τ | = 1, with 2π/3 ≥ arg(τ ) ≥ π/2. The method has been generalized to Fricke groups in recent works of [17] and to the subgroups of S L 2 (Z) in [13,14]. The zeros of weight two Eisenstein series E 2 (q) were studied by [11,19]. In this work, we shall find the zeros of Eisenstein series E˜ N (in general, E˜ N is defined as negative of what it is defined here in (1.1) which are holomorphic modular forms of weight 2 of 0 (N ) defined as 1 (N E 2 (N τ ) − E 2 (τ )), (1.1) E˜ N (τ ) := N −1 where E 2 (τ ) is the quasimodular Eisenstein series defined by ∞ E 2 (τ ) := 1 − 24 σ1 (n)q n , q = e2πiτ , (1.2) n=1
with σ1 (n) being the sum over all the divisors of n. The method which we present is quite different from that of [16], however can only be applied for N = 2, 3, 5, 7. Our main observation is that E˜ N is integral over the graded algebra Mk (S L 2 (Z)) = C[E 4 , E 6 ] M(S L 2 (Z)) = k≥0
© Indian Academy of Sciences 0123456789().: V,-vol
6
Page 2 of 9
Proc. Indian Acad. Sci. (Math. Sci.)
(2020) 130:6
Table 1. Zeros of E˜ N (τ ) for N = 2, 3, 5, 7. N
2
3
τ
1 − i+1
−
1 e2πi/3 +2
5
7
1 , − i+2
1 , e2πi/3 +3 1 − 2πi/3 e +5
1 − i+3
−
with E 4 , E 6 being the Eisenstein series of weight 4 and 6 defined as E 4 (τ ) := 1 + 240 E 6 (τ ) := 1 − 504
∞ n=1 ∞
σ3 (n)q n ,
(1.3)
σ5 (n)q n ,
(1.4)
n=1
where q = e2πiτ , τ ∈ H. Let us denote the fundamental domain for 0 (N ) by FN . For N = 2, 3, 5, 7, the zeros of E˜ N are ‘controlled’ (2) by those of E 4 and E 6 in FN . We now state the main result of our paper which would be proved in section 4. Theorem 1. All the zeros of E˜ N (τ ) in the fundamental domain of 0 (N ) lie as in Table 1. Before we move into the proof of the theorem, we shall first provide a brief description where these modular forms occur in the context of string theory in physics.
2. Physics motivation Partition functions defined in str
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