Abstract
This survey paper is focused on a connection between the geometry of \(\mathop{\text{GL}}\nolimits _{d}\) and the arithmetic of \(\mathop{\text{GL}}\nolimits _{d-1}\) over global fields, for integers d ≥ 2. For d = 2 over \(\mathbb{Q}\), there is an explicit conjecture of the third author relating the geometry of modular curves and the arithmetic of cyclotomic fields, and it is proven in many instances by the work of the first two authors. The paper is divided into three parts: in the first, we explain the conjecture of the third author and the main result of the first two authors on it. In the second, we explain an analogous conjecture and result for d = 2 over \(\mathbb{F}_{q}(t)\). In the third, we pose questions for general d over the rationals, imaginary quadratic fields, and global function fields.
The original version of this chapter was revised. An erratum to this chapter can be found at DOI 10.1007/978-3-642-55245-8_17
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-642-55245-8_17
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Notes
- 1.
This is still not quite the right object unless we invert 2. In Sect. 4, we take the point of view that the right object is the relative homology of the quotient of the space X 1(Np r) by the action of complex conjugation.
- 2.
Actually, g α, β is a root of a unit, but the difficulties this causes are resolvable by passing to the projective limit and descending, so we ignore this for simplicity of presentation. We will be very careless about denominators in several places, omitting them where they occur for simplicity of the discussion that follows.
- 3.
There is one potentially confusing aspect: the action of \(\varLambda \hookrightarrow \mathfrak{h}\) on \(\mathfrak{S} \subset \varLambda [\![q]\!]\) is not given by multiplication of the coefficients of q-expansions by the element of Λ. It is instead this multiplication after first applying the inversion map λ ↦ λ ∗ on Λ that takes group elements to their inverses.
- 4.
It should actually be possible to allow either or both of p = 3 and \(p\mid \varphi (N)\) in what follows.
- 5.
The reader may wish to ignore the involutions in order to focus on the idea of the argument.
- 6.
Another, more usual, way to approach injectivity is to use \(I_{\theta } +\xi _{\theta }\mathfrak{h}_{\theta }\) in place of I θ until one recovers the equality of these ideals through a proof of the main conjecture, as in 2.5.7 below.
- 7.
In the projective limit, this gives another way of defining the isomorphism \(\mathfrak{h}_{\theta }/I_{\theta }\stackrel{\sim }{\rightarrow }\varLambda _{\theta }/(\xi _{\theta })\).
- 8.
To make sense of this, note that the tensor product in the sum is taken over \(\mathbb{Z}_{p}[\theta ]\).
- 9.
Actually, we know that the kernel coincides with (ξ θ ), but this weaker statement is enough.
- 10.
Note that q appears in this sentence as the order of the residue field of \(\mathcal{O}_{\infty }\).
- 11.
Actually, g α, β as we have described it is not well-defined until we take its q 2 − 1 power. The assumption that \(p \nmid (q^{2} - 1)\) is used to avoid this issue when we work with étale cohomology.
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Acknowledgements
The work of the first two authors (resp., third author) was supported in part by the National Science Foundation under Grant Nos. DMS-1001729 (resp., DMS-0901526).
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Fukaya, T., Kato, K., Sharifi, R. (2014). Modular Symbols in Iwasawa Theory. In: Bouganis, T., Venjakob, O. (eds) Iwasawa Theory 2012. Contributions in Mathematical and Computational Sciences, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55245-8_5
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