The distribution of values of the Poincaré pairing for hyperbolic Riemann surfaces

Risager, Morten S.

doi:10.1515/crll.2005.2005.579.159

Cited by 12 publications

(12 citation statements)

References 20 publications

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“…Proof. This follows from Lemma 2.1 in [16], for example, where in the present case the relevant modular symbol is formed using the cohomology class ω i .…”

Section: Counting Closed Geodesics On Riemann Surfacesmentioning

confidence: 89%

Equidistribution of geodesics on homology classes and analogues for free groups

Risager¹

2008

Forum Mathematicum

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We investigate how often geodesics have homology in a fixed set of the homology lattice of a compact Riemann surface. We prove that closed geodesics are equidistributed on any sets with asymptotic density with respect to a specific norm. We explain the analogues for free groups, conjugacy classes and discrete logarithms, in particular, we investigate the density of conjugacy classes with relatively prime discrete logarithms.

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“…Proof. This follows from Lemma 2.1 in [16], for example, where in the present case the relevant modular symbol is formed using the cohomology class ω i .…”

Section: Counting Closed Geodesics On Riemann Surfacesmentioning

confidence: 89%

Equidistribution of geodesics on homology classes and analogues for free groups

Risager¹

2008

Forum Mathematicum

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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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“…Apart from the use of this identity our proof is elementary. Our method is very powerful and has been used by the authors in a series of papers [6,7,9,8] to prove distribution results of additive homomorphisms in various different contexts. In fact we were motivated by our previous results to investigate the normal distribution on the level of the group without considering its action as isometries on hyperbolic space.…”

Section: Theorem 11mentioning

confidence: 99%