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

Abstract

Let {FN } and {GM } be families of primitive automorphic L-functions for GLn(A Q ) and GLm(A Q ), 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).

From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond

“…While the exact definition of family is still a work in progress, roughly it is a collection of L-functions coming from a common process. Examples include Dirichlet characters, elliptic curves, cuspidal newforms, symmetric powers of GL(2) L-functions, Maass forms on GL(3), and certain families of GL(4) and GL(6) L-functions; see for example [AAILMZ,AM,DM1,DM2,FiM,FI,Gao,Gü,HM,HR,ILS,KaSa2,LM,Mil1,MilPe,OS1,OS2,RR,Ro,Rub,Ya,Yo2]. This correspondence between zeros and eigenvalues allows us, at least conjecturally, to assign a definite symmetry type to each family of L-functions (see [DM2,ShTe] for more on identifying the symmetry type of a family).…”

“…There we average the Satake parameters over a family of L-function, and in the limit as the conductors tend to infinity only the first and second moments contribute to the main term (at least under the assumption of the Ramanujan conjectures for the sizes of these parameters). The first moment controls the rank at the central point, and the second moment determines the symmetry type (see [DM2,ShTe]). For example, families of elliptic curves with very different arithmetic (complex multiplication or not, or different torsion structures) have the same limiting behavior but have different rates of convergence to that limiting behavior.…”

Determining Optimal Test Functions for Bounding the Average Rank in Families of 𝐿-Functions

“…There is now an extensive body of work supporting this for numerous families and various levels of support, including Dirichlet characters, elliptic curves, cuspidal newforms, symmetric powers of GLp2q Lfunctions, and certain families of GLp4q and GLp6q L-functions; see for example [DM1, DM2, ER-GR, FiM, FI, Gao, Gü, HM, HR, ILS, KaSa2, LM, Mil1, MilPe, OS1, OS2, RR, Ro, Rub, Ya, Yo2]. This correspondence between zeros and eigenvalues allows us, at least conjecturally, to assign a definite symmetry type to each family of L-functions (see [DM2,ShTe] for more on identifying the symmetry type of a family).…”

The Classical Theory of Zeta and L-Functions

“…Instead, the j-th zero of L(s, E ⊗χ), E an elliptic curve, follows the O prediction (two cases, according to the sign ± in the functional equation, corresponding to the two connected components O ± of O). Note however that there are intermediate cases and the prediction of the correct behavior is not quite automatic, see the paper [25] by Duñez and Miller.…”