# Dwork conjecture

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In mathematics, the **Dwork unit root zeta function**, named after [Bernard Dwork](/source/Bernard_Dwork), is the [L-function](/source/L-function) attached to the [p-adic](/source/P-adic) [Galois representation](/source/Galois_representation) arising from the p-adic [etale cohomology](/source/Etale_cohomology) of an [algebraic variety](/source/Algebraic_variety) defined over a [global function field](/source/Global_function_field) of [characteristic](/source/Characteristic_(algebra)) *p*. The **Dwork conjecture** (1973) states that his unit root zeta function is p-adic [meromorphic](/source/Meromorphic_function) everywhere.[1] This conjecture was proved by [Wan](/source/Daqing_Wan) (2000).[2][3][4]

## References.

1. Dwork, Bernard (1973), "Normalized period matrices II", *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **98** (1): 1–57, [doi:10.2307/1970905](https://doi.org/10.2307/1970905). [JSTOR 1970905](https://www.jstor.org/stable/1970905).

1. Wan, Daqing (1999), "Dwork's conjecture on unit root zeta functions", *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **150** (3): 867–927, [arXiv:math/9911270](https://arxiv.org/abs/math/9911270). [doi:10.2307/121058](https://doi.org/10.2307/121058). [JSTOR 121058](https://www.jstor.org/stable/121058).

1. Wan, Daqing (2000), "Higher rank case of Dwork's conjecture", *[Journal of the American Mathematical Society](/source/Journal_of_the_American_Mathematical_Society)*. **13** (4): 807–852, [doi:10.1090/S0894-0347-00-00339-8](https://doi.org/10.1090/S0894-0347-00-00339-8).

1. Wan, Daqing (2000), "Rank one case of Dwork's conjecture", *[Journal of the American Mathematical Society](/source/Journal_of_the_American_Mathematical_Society)*. **13** (4): 853–908, [doi:10.1090/S0894-0347-00-00340-4](https://doi.org/10.1090/S0894-0347-00-00340-4).

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