The effect of convolving families of L‐functions on the underlying group symmetries

Let {ℱN} and {𝒢M} be families of primitive automorphic L‐functions for GLn(𝔸ℚ) and GLm(𝔸ℚ), respectively, such that, as N, M → ∞, the statistical behavior (1‐level density) of the low‐lying zeros of L‐functions in ℱN and 𝒢M agrees with that of the eigenvalues near 1 of matrices in G1 and G2, respectively, as the size of the matrices tend to infinity, where each Gi is one of the classical compact groups (unitary U, symplectic Sp, or orthogonal O, SO(even), SO(odd)). Assuming that the convolved families of L‐functions ℱN × 𝒢M are automorphic, we study their 1‐level density. (We also study convolved families of the form f × 𝒢M 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 cℒ equal to 0, 1, or−1 if the corresponding low‐lying zero statistics agree with those of the unitary symplectic, or orthogonal group, respectively. Our main result is that cℱ×𝒢=cℱ·c𝒢: 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 × 𝒢M. 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 positive 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 → ∞, as lower‐order terms).