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Najib Idrissi’s math page
We will start by reviewing Adams’s computation of the image of $J$. Using motivations from modular forms, we construct a family of Dirichlet $J$-spectra for each Dirichlet character. When conductor of the character is an odd prime $p$, the $p$-completion of the Dirichlet $J$-spectra splits as a wedge sum of $K(1)$-local invertible spectra. These summands are elements of finite orders in the $K(1)$-local Picard group.
We will then introduce a spectral sequence to compute homotopy groups of the Dirichlet $J$-spectra. The $1$-line in this spectral sequence is closely related to congruences of certain Eisenstein series. This explains appearance of special values of Dirichlet $L$-functions in the homotopy groups of these Dirichlet $J$-spectra. Finally, we establish a Brown-Comenetz duality for the Dirichlet $J$-spectra that resembles the functional equations of the corresponding Dirichlet $L$-functions. In this sense, the Di
To date we have developed a largely 2-categorical theory of adjunctions, monadicity, cartesian fibrations, modules, limits and colimits, (pointwise) Kan extensions, final and initial functors, Yoneda's lemma, presheaf categories and many other important categorical structures besides. All of this work follows a mild re-working of a well trodden path of 2-being, which not only provides for a foundational redevelopment of the category theory of quasi-categories but is also couched in terms that is both model
Home | Sanath Devalapurkar’s home page
Najib Idrissi’s math page