05950557

j

05950557 Ghitza, Alexandru Distinguishing Hecke eigenforms. Int. J. Number Theory 7, No. 5, 1247-1253 (2011). 2011 World Scientific, Singapore EN modular forms Hecke eigenforms Fourier coefficients Zbl 0805.11040 Zbl 0910.11018

doi:10.1142/S1793042111004708

Let $f=\sum_{n\geq 0}a_n(f)e(n\tau)$ be a modular form of weight $k$ for the congruence subgroup $\Gamma_0(N)$. It is an interesting question how many Fourier coefficients $a_n(f)$ (of the Fourier expansion at $\infty$) are sufficient to determine the modular form $f$. There are several papers which studied this problem. In [{\it D. Goldfeld} and {\it J. Hoffstein}, ``On the number of Fourier coefficients that determine a modular form", Contemp. Math. 143, 385--393 (1993; Zbl 0805.11040)], the problem is solved for the special case of newforms by means of the theory of $L$-functions. In [``Congruences between modular forms", Lond. Math. Soc. Lect. Note Ser. 247, 309--320 (1997; Zbl 0910.11018)], {\it M. Ram Murty} provided some generalizations of the results in [Goldfeld and Hoffstein, loc. cit.]. He was able to prove the following theorem without using the theory of $L$-functions in a very elegant way. Theorem 1. Let $f$ and $g$ be two holomorphic Heck newforms of distinct weights $k_1$ and $k_2$ on $\Gamma_0(N_1)$ and $\Gamma_0(N_2)$ respectively. Then, there is an $n < 4(\log(N))^2$ with $N=\text{lcm}(N_1,N_2)$ so that $$a_n(f) \not= a_n(g).$$ The cited theorem above is the starting point of the paper under review. The author states that one disadvantage of the result above is that it holds (only) for relatively large $N$, see page 1248. Based on Theorem 1, he proves Theorem 2. Let $f$ and $g$ be cuspidal eigenforms of weights $k_1\not=k_2$ on the group $\Gamma_0(N)$. Then there exists $$n \leq 4(\log(N) + 1)^2 \text { such that } a_n(f) \not= a_n(g).\tag$*$ $$ In contrast to the upper bound of Theorem 1, the inequality of ($*$) holds for all level $N\geq 1$. In a further theorem the author provides some sharper bounds for $n$ than that given in ($*$), see Theorem 5 of the paper. Two of these bounds are obtained under the assumption of the Riemann hypothesis and Cram\'ers hypothesis respectively. The paper ends with some numerical experiments for the case of level $N=1$. Oliver Stein (Bonn)