Abstract
In this paper we introduce the concept of integral Frobenius to formulate an integral analogue of the classical compatibility condition linking the collection of rational Tate modules V λ (A) arising from abelian varieties over number fields with real multiplication. Our main result gives a recipe for constructing an integral Frobenius when the real multiplication field has class number one. By exploiting algorithms already existing in the literature, we investigate this construction for three modular abelian surfaces over Q.
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Notes
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- 2.
More generally, this freeness holds if \(S_{\mathfrak {p}}\otimes \mathbf {Z}_\ell \) is a Gorenstein ring (see [11, Remark, p. 502]).
- 3.
These ideals are always defined in our computations as we never found a prime p for which \(a_p^2-4p=0\).
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Acknowledgements
We wish to thank Gebhard Böckle for all the useful discussions we had on this project. We thank Gaetan Bisson for an intense email correspondence where he helped us understanding better his work. The first author thanks Jordi Guàrdia and Josep Gonzàlez for the warm hospitality he received during his visit to Universitat Politècnica de Catalunya in July 2015. This work was supported by the DFG Priority Program SPP 1489 and the Luxembourg FNR.
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Centeleghe, T.G., Theisen, C. (2017). Integral Frobenius for Abelian Varieties with Real Multiplication. In: Böckle, G., Decker, W., Malle, G. (eds) Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-70566-8_6
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DOI: https://doi.org/10.1007/978-3-319-70566-8_6
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