Mazur–Rubin's central limit conjecture for modular symbols

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Let f∈S2(Γ0(q))f\in S_2(\Gamma_0(q)), let cc tend to infinity through positive integers satisfying (c,q)=d(c,q)=d, where dd divides qq, and let aa range over (Z/cZ)∗(\mathbb Z/c\mathbb Z)^*. Let CfC_f and Df,dD_{f,d} be the variance slope and variance shift from the preceding variance conjecture. Mazur–Rubin's central limit conjecture. The limiting distribution of the data

⟨a/c⟩(Cflog⁡c+Df,d)1/2\frac{\langle a/c\rangle}{(C_f\log c+D_{f,d})^{1/2}}

is the standard normal distribution. This is the conjectured normal distribution for modular symbols at fixed level gcd class; the supplied text gives no resolution status.

References

Primary source

Petru Constantinescu, “Distribution of modular symbols in H^3”, arXiv:2005.07629 (2020).

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