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Higher Hida and Coleman theories on the modular curve. (English) Zbl 1507.11038

In his seminal work in the 80s, H. Hida [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 2, 231–273 (1986; Zbl 0607.10022)] interpolated the ordinary part of the 0-th degree cohomology of the modular sheaves over the classical modular curves. This was the first example of what is now called a \(p\)-adic family of modular forms: the interpolation is over the weight space, meaning that the variable that lives in \(p\)-adic families is the weight. This construction has been generalized by R. F. Coleman [Invent. Math. 127, No. 3, 417–479 (1997; Zbl 0918.11026)], who introduced the notion of overconvergent modular forms (i.e., forms defined over a neighborhood of the ordinary locus) and by several other authors to various Shimura varieties in different contexts.
The origin of this idea comes from Galois representations, which can be easily deformed. Indeed, one of the main strategy to prove that a given representation is modular (and has hence good properties) is to deform it in a \(p\)-adic family, study the corresponding family of modular forms (whose existence if often easier to prove) and then invoke a classicality result to get a classical modular form. In the paper at hand, the authors consider also the interpolation of the 1-th degree of the cohomology. They construct two projective modules over the Iwasawa algebra (i.e., over the weight space) that interpolate the cohomology and that are in perfect duality (interpolating Serre duality). They also define an action of the unramified Hecke algebra and of a compact operator at \(p\), showing a “small slopes forms are classical” criterion. In particular, they are able to construct two coherent sheaves over the eigencurve that interpolate finite slope eigenforms. The work has already been generalized by the authors and others to the context of more general Shimura variety, but in the case of modular curves the construction is more transparent and the results are more precise.

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
11F85 \(p\)-adic theory, local fields
14G22 Rigid analytic geometry
14G35 Modular and Shimura varieties
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