Riemann hypothesis for period polynomials of modular forms
Riemann hypothesis for period polynomials of modular forms
复制标题
模形式周期多项式的黎曼假设
DOI：
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发表时间：
2016
影响因子：
11.1
通讯作者：
K. Soundararajan
中科院分区：
文献类型：
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作者：
S. Jin;Wenjun Ma;K. Ono;K. Soundararajan
模L-函数的临界值是算术几何和数论中的重要对象。这些数字被预测为编码深算术信息的伯奇和Swinnerton-Dyer猜想和布洛赫-加藤猜想。这里我们考虑这些值的生成函数，即所谓的周期多项式。这些多项式的Riemann假设是断言这些多项式的零点位于由标准函数方程产生的对称圆上。这个假设的真实性对临界L值的大小有很强的约束。这一论断在这里得到了证明。对于偶数权k≥4的周期多项式rf（z），新形式f∈Sk（Γ0（N））是L（f，s）临界值的母函数.它有一个将rf（z）与rf（− 1 Nz）联系起来的函数方程。我们证明了这些多项式的黎曼假设：rf（z）的零点位于圆上|z| =1/N。我们证明，这些零是equidistributed当k或N是大的。
Significance Critical values of modular L-functions are objects of central importance in arithmetic geometry and number theory. These numbers are predicted to encode deep arithmetic information by the Birch and Swinnerton-Dyer conjecture and the Bloch–Kato conjecture. Here we consider the generating functions for these values, the so-called period polynomials. The Riemann hypothesis for these polynomials is the assertion that the zeros of these polynomials are located on the circle of symmetry that arises from the standard functional equations. The truth of this hypothesis places strong constraints on the size of the critical L-values. This assertion is proved here. The period polynomial rf(z) for an even weight k≥4 newform f∈Sk(Γ0(N)) is the generating function for the critical values of L(f,s). It has a functional equation relating rf(z) to rf(−1Nz). We prove the Riemann hypothesis for these polynomials: that the zeros of rf(z) lie on the circle |z|=1/N. We prove that these zeros are equidistributed when either k or N is large.