L函数卷积族对群对称性的影响
The effect of convolving families of L-functions on the underlying group symmetries
摘要：
L-函数，使得当N，M -> oo时，L函数在F_N中的低零点的统计行为（1-能级密度）（分别为，G_M）与G_1中矩阵的特征值在1附近的性质（分别为，G_2），其中每个G_i是一个经典紧群（酉、辛或正交）。假设L函数F_N x G_M的卷积族是自守的，我们研究了它们的单能级密度. (We也研究了f × G_M形式的卷积族。在关于族的自然假设下（在许多情况下成立），我们可以将等于0的对称常数c_L与L函数的每个族L相关联（分别地，1或-1）如果相应的低位零统计量与酉统计量一致（分别，辛或正交）群。我们的主要结果是：c_{F x G} = c_G * c_G：卷积族的对称型是两族对称型的乘积。对于卷积族f x G_M也有类似的陈述。我们提供的例子建立从狄利克雷L函数和全纯模形式和他们的对称权力。一个有趣的特殊情况是卷积两族具有秩的椭圆曲线。在这种情况下，根据对称常数的乘法性的一般原理，卷积的对称群与秩无关（但在取极限N，M -> oo作为低阶项之前，秩仍然存在）。
英文摘要：
L-functions for GL_n(A_Q) and GL_m(A_Q), respectively, such that, as N,M --> oo, the statistical behavior (1-level density) of the low-lying zeros of L-functions in F_N (resp., G_M) agrees with that of the eigenvalues near 1 of matrices in G_1 (resp., G_2) as the size of the matrices tend to infinity, where each G_i is one of the classical compact groups (unitary, symplectic or orthogonal). Assuming that the convolved families of L-functions F_N x G_M are automorphic, we study their 1-level density. (We also study convolved families of the form f x G_M for a fixed f.) Under natural assumptions on the families (which hold in many cases) we can associate to each family L of L-functions a symmetry constant c_L equal to 0 (resp., 1 or -1) if the corresponding low-lying zero statistics agree with those of the unitary (resp., symplectic or orthogonal) group. Our main result is that c_{F x G} = c_G * c_G: the symmetry type of the convolved family is the product of the symmetry types of the two families. A similar statement holds for the convolved families f x G_M. We provide examples built from Dirichlet L-functions and holomorphic modular forms and their symmetric powers. An interesting special case is to convolve two families of elliptic curves with rank. In this case the symmetry group of the convolution is independent of the ranks, in accordance with the general principle of multiplicativity of the symmetry constants (but the ranks persist, before taking the limit N,M --> oo, as lower-order terms).
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