Hyperbolic lattice-point counting and modular symbols

Risager, Morten S.

doi:10.5802/jtnb.698

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“…We have obtained different but related normal distribution results for modular symbols in [36][37][38]40]. One difference between these papers and the current one is in the ordering and normalization of the values of γ, α .…”

Section: Remark 112mentioning

confidence: 98%

Arithmetic statistics of modular symbols

Risager¹

2018

Invent. math.

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Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.

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Section: Remark 112mentioning

confidence: 98%