Nonvanishing of HeckeL-Functions Associated to Cusp Forms inside the Critical Strip
Nonvanishing of HeckeL-Functions Associated to Cusp Forms inside the Critical Strip
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DOI：
10.1006/jnth.1997.2178
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发表时间：
1997-12
影响因子：
0.7
通讯作者：
W. Kohnen
中科院分区：
文献类型：
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作者：
W. Kohnen
设f是满模群SL 2（Z）上的整数权为k的非零尖点Hecke特征形，用L*（f，s）（s# C）表示相应的Hecke L-函数及其自然1-因子.众所周知，L*（f，s）的零点只能出现在临界带（k&1）<$2 < Re（s）<（k+ 1）<$2内，并且根据广义黎曼假设，它们都应该位于线Re（s）= k <$2上。虽然目前黎曼假设的证明似乎遥不可及，但证明L-函数的平均非零结果是相当容易的。事实上，在本文中，我们将证明，给定一个真实的数t0和一个正的真实的数=，对于所有足够大的k，f在（适当归一化的）权重k的Hecke本征形在线段Im（s）= t0，（k +1）< Re（s）<（k +1）< Re（s）<（k+ 1）<2上不为零。为了证明，我们考虑L*（f，s）的Petersson标积的尖点形式对偶，这些函数是[3]中研究的周期函数的推广。我们计算它们的傅里叶展开，并以适当的方式估计第一傅里叶系数。结果就会出来。
Let f be a non-zero cuspidal Hecke eigenform of integral weight k on the full modular group SL2 (Z) and denote by L*(f, s)(s# C) the associated Hecke L-function completed with its natural 1-factor. As is well-known, zeroes of L*(f, s) can occur only inside the critical strip (k&1) Ā2< Re (s)<(k+ 1) Ā2, and according to the generalized Riemann hypothesis they should all lie on the line Re (s)= kĀ2. While at present a proof of the Riemann hypothesis seems to be out of reach, it turns out to be rather easy to show nonvanishing results for L-functions on the average. In fact, in the present paper we shall prove that, given a real number t0 and a positive real number=, for all k large enough the sum of the functions L*(f, s) with f running over a basis of (properly normalized) Hecke eigenforms of weight k does not vanish on the line segments Im (s)= t0,(k&1) Ā2< Re (s)<(kĀ2) &=,(kĀ2)+=< Re (s)<(k+ 1) Ā2. For the proof we consider the cusp forms dual wrt the Petersson scalar product of the values L*(f, s); these functions are generalizations of the period functions studied in [3]. We compute their Fourier expansion and estimate the first Fourier coefficient in an appropriate way. The result then will turn out.