## Betti and Hodge numbers of configuration spaces of a punctured elliptic curve from its zeta functions

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- by Gilyoung Cheong and Yifeng Huang PDF
- Trans. Amer. Math. Soc.
**375**(2022), 6363-6383 Request permission

## Abstract:

Given an elliptic curve $E$ defined over $\mathbb {C}$, let $E^{\times }$ be an open subset of $E$ obtained by removing a point. In this paper, we show that the $i$-th Betti number of the unordered configuration space $\mathrm {Conf}^{n}(E^{\times })$ of $n$ points on $E^{\times }$ appears as a coefficient of an explicit rational function in two variables. We also compute its Hodge numbers as coefficients of another explicit rational function in four variables. Our result is interesting because these rational functions resemble the generating function of the $\mathbb {F}_{q}$-point counts of $\mathrm {Conf}^{n}(E^{\times })$, which can be obtained from the zeta function of $E$ over any fixed finite field $\mathbb {F}_{q}$. We show that the mixed Hodge structure of the $i$-th singular cohomology group $H^{i}(\mathrm {Conf}^{n}(E^{\times }))$ with complex coefficients is pure of weight $w(i)$, an explicit integer we provide in this paper. This purity statement implies our main result about the Betti numbers and the Hodge numbers. Our proof uses Totaro’s spectral sequence computation that describes the weight filtration of the mixed Hodge structure on $H^{i}(\mathrm {Conf}^{n}(E^{\times }))$.## References

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## Additional Information

**Gilyoung Cheong**- Affiliation: Department of Mathematics, University of California–Irvine, 340 Rowland Hall, Irvine, California 92697-3875
- MR Author ID: 1096770
- Email: gilyounc@uci.edu
**Yifeng Huang**- Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109-1043
- MR Author ID: 1414559
- ORCID: 0000-0003-2210-5287
- Email: huangyf@umich.edu
- Received by editor(s): September 21, 2021
- Received by editor(s) in revised form: February 2, 2022
- Published electronically: June 30, 2022
- Additional Notes: The second author was supported by Research Training Grant (RTG): Number Theory and Representation Theory at the University of Michigan while completing this work.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 6363-6383 - MSC (2020): Primary 14F45
- DOI: https://doi.org/10.1090/tran/8668
- MathSciNet review: 4474895