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        Questions tagged [formal-groups]

        The tag has no usage guidance.

        2
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        63 views

        Exercise on formal group laws over an algebraically closed field

        There is an exercise in Weinstein's notes on Lubin--Tate theory, namely show that there is a unique (up to isomorphism) one-dimensional formal group law of given finite height $h$ over an ...
        2
        votes
        1answer
        337 views

        Formal group law and Koenigs function conjecture?

        Let $f(x,y)$ be a symmetric real function and a formal group law $$G(x + y) = f(G(x),G(y)). \tag{1}$$ Consider the equation $$ h(2x) = f(h(x),h(x)) = A(h(x)). \tag{2}$$ This equation has many ...
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        91 views

        Computation of the Newton polygon of formal modules under unramfied condition

        When reading a very short paper "A generalization of the Chowla-Selberg formula" which is avaliable at http://www.math.titech.ac.jp/~taguchi/bib/cs.pdf by Yukiyoshi Nakkajima and Yuichiro Taguchi, ...
        9
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        1answer
        348 views

        Nontrivial p-divisible groups over $\mathbb Z$ for general prime $p$

        In Tate's famous paper about $p$-divisible groups, for a prime number $p$ he asks whether there exists a $p$-divisible group $G$ over $\mathbb Z$ such that $G$ is not a direct sum of $\mu_{p^\infty}$ ...
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        81 views

        Geometric intuition and computation for Cartier theory

        I am learning Cartier theory of commutative formal groups by the book of Zink. It is a powerful tool but I don't understand it's motivation. The Cartier module of a formal group $G$ over a $\mathbb{Z}...
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        3answers
        1k views

        Characterizing positivity of formal group laws

        The formal group law associated with a generating function $f(x) = x + \sum_{n=2}^\infty a_n \frac{x^n}{n!}$ is $$f(f^{-1}(x) + f^{-1}(y)).$$ In my thesis, I found a large number of examples of ...
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        96 views

        Formal Group Laws in a lined topos

        I am aware of the following: in the context of synthetic differential geometry (SDG) one obtains a Lie algebra by exponentiating a microlinear group by a standard infinitesimal object and taking the ...
        8
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        1answer
        271 views

        What is the essential image of $AbVar$ in $p-div$?

        Given an abelian variety $A$ over a base scheme $\text{Spec } \mathcal{O}_{K_p}$, we define the functor $P$ as taking $A \mapsto \text{colim}_n A[p^n]$, its associated $p$-divisible group. What is the ...
        8
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        1answer
        223 views

        (Pre)orientation vs. formal completion

        Let $\mathbb G$ be an abelian vatiety over an $\mathbb E_\infty$-ring $A$. That is to say, it consists of an abelian group object in the $\infty$-category of relative schemes $\mathbb G\to \...
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        3answers
        197 views

        Morphisms of formal group laws $\,F_a \rightarrow F_m\,$ and $\,F_m\to F_m$

        While studying cohomology theories on the stable homotopy setting, I have come up with the following basic question: Consider the additive formal group law, $F_a$, and the multiplicative formal group ...
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        1answer
        462 views

        deformation theory in positive characteristic

        The idea "Formal deformation theory in characteristic zero is controlled by a differential graded Lie algebra (dgla)" goes back to Goldman-Millson, Deligne, Drinfeld among others; see Lurie's ICM talk....
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        208 views

        Rational cohomology of formal multiplicative group

        Let $\hat{\mathbb G}$ be a formal group over a field $k$, and let $V$ be a finite dimensional algebraic representation of $\hat{\mathbb G}$ (meaning we have fixed a homomorphism of algebraic groups $\...
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        349 views

        Geometry underlying a comparison of Dieudonné theories

        Maybe these hypotheses aren't necessary, but for me $\mathbb G$ will be a smooth formal group of dimension 1 and finite height over a perfect field $k$. There are several presentations of the ...
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        2answers
        654 views

        How to compute the formal group law of a Shimura variety (using its invariant differentials)?

        I have a 3 dimensional abelian variety whose formal group law breaks into a formal summand where one of the pieces is one-dimensional. I am desperately wondering how to compute the $p$-series of ...
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        0answers
        179 views

        Definition of logarithm for universal vector extension

        Let $S$ be a topological $\mathbb{Z}_p$-algebra and $R\to S$ a surjection (where $R$ has $p$ nilpotent) with topologically nilpotent kernel which has a PD structure. We know that if $G/R$ is a $p$-...

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