Refined dimensions of cusp forms, and equidistribution and bias of signs

Abstract

We refine known dimension formulas for spaces of cusp forms of squarefree level, determining the dimension of subspaces generated by newforms both with prescribed global root numbers and with prescribed local signs of Atkin-Lehner operators. This yields precise results on the distribution of signs of global functional equations and sign patterns of Atkin-Lehner eigenvalues, refining and generalizing earlier results of Iwaniec, Luo and Sarnak. In particular, we exhibit a strict bias towards root number +1 and a phenomenon that sign patterns are biased in the weight but perfectly equidistributed in the level. Another consequence is lower bounds on the number of Galois orbits.Date: November 9, 2018.

Low-lying zeros of L-functions for Quaternion Algebras

“…It can be observed that the determination of the precise type of symmetry, among the three orthogonal ones, depends on the proportion of automorphic forms having sign −1 in the functional equation. Such information is not available in the level of generality of this paper; however, with mild assumption on the conductor, Martin [21] shows (despite an interesting bias) that the limiting proportion is 1/2, pleading for an orthogonal type of symmetry for A(G). The family restricted to positive (resp.…”

Computing $\protect \mathcal{L}$-invariants via the Greenberg–Stevens formula

“…In the case of N squarefree, building on previous works of Yamauchi, Skoruppa and Zagier, Martin gave a formula [18,Equation (1.6)] for the trace of the Atkin-Lehner operator W p on S k (Γ 0 (pN )). In the simple case where N = q > 3 is a prime and p > 3, we have…”

“…In this step, we actually only compute the dimension d ε of such eigenspaces. As shown in Section 4, this computation can be done via Corollary 4.2, Equation (1.6) in [18] and by recursion.…”

“…For this purpose, we explain how to efficiently compute higher derivatives of the characteristic series and how the L-invariants can be read off from these derivatives. In order for this approach to work and not increase too much the precision required to perform the computations, one first needs to decompose the space of p-newforms by their Atkin-Lehner eigenvalue at p. Since the needed precision grows with the dimension of these subspaces, we give explicit dimension formulas, analogous to the ones presented in [18].…”

On Bhargava’s Heuristics for GL2(𝔽p)-Number Fields and the Number of Elliptic Curves of Bounded Conductor

“…Remark 4.2. Applying the trace formula to compute the trace of products w i 1 · · · w i k acting on S 2 (N, Q) new , Martin [12] shows that d χ,Q := dim Q S 2 (N, Q) new,χ all roughly equal d/2 s . This does not directly imply the above assumption, since d χ,Q does not generally equal d χ .…”