## Residual equidistribution of modular symbols and cohomology classes for quotients of hyperbolic $n$-space

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- by Petru Constantinescu and Asbjørn Christian Nordentoft PDF
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## Abstract:

We provide a new and simple automorphic method using Eisenstein series to study the equidistribution of modular symbols modulo primes, which we apply to prove an average version of a conjecture of Mazur and Rubin. More precisely, we prove that modular symbols corresponding to a Hecke basis of weight 2 cusp forms are asymptotically jointly equidistributed mod $p$ while we allow restrictions on the location of the cusps. As an application, we obtain a residual equidistribution result for Dedekind sums. Furthermore, we calculate the variance of the distribution and show a surprising bias with connections to perturbation theory. Additionally, we prove the full conjecture in some particular cases using a connection to Eisenstein congruences. Finally, our methods generalise to equidistribution results for cohomology classes of finite volume quotients of $n$-dimensional hyperbolic space.## References

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## Additional Information

**Petru Constantinescu**- Affiliation: Department of Mathematics, University College London, 25 Gordon Street, London, WC1H 0AY, United Kingdom
- Address at time of publication: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
- ORCID: 0000-0002-2719-4592
- Email: petru.constantinescu.17@ucl.ac.uk
**Asbjørn Christian Nordentoft**- Affiliation: Mathematical Institute of the University of Bonn, Endenicher Allee 60, Bonn 53115, Germany
- Address at time of publication: LAGA, Institut Galilée, 99 avenue Jean Baptiste Clément, 93430 Villetaneuse, France
- MR Author ID: 1416007
- ORCID: 0000-0003-2998-7139
- Email: acnordentoft@outlook.com
- Received by editor(s): March 21, 2021
- Received by editor(s) in revised form: November 11, 2021, and January 10, 2022
- Published electronically: July 29, 2022
- Additional Notes: The first author was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London. The second author was supported by the Independent Research Fund Denmark DFF-7014-00060B
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 7001-7034 - MSC (2020): Primary 11F67; Secondary 11M36, 11E45, 11E88
- DOI: https://doi.org/10.1090/tran/8646