Formal Groups Reading Notes

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Formal Groups and Spectra

Formal Groups

Basics

  • Motivation: why should one care about formal groups?

    • Idea: substitute for Lie algebras in positive characteristic
    • Formal group and lie algebra carry the same info in char 0
    • Functor: LieGroups (or AlgGrp) → LieAlgebras factors through FormalGroups

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  • Motivation: if a formal group were for sale, why might you buy it?

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  • What is a formal group law?

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  • Why is the group law called “formal”?

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  • How “should” one think of a formal group?

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  • What is a formal exponential and formal logarithm?

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  • What are examples of formal groups? What is a morphism? When are they isomorphic?

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  • What are the additive and multiplicative formal groups?

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  • How are the additive and multiplicative formal groups related?

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  • What is the height of a formal group?

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  • How can one extend formal groups over R to R-algebras?

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  • What is the formal scheme interpretation of formal groups?

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  • How does iterated K-theory show up in the theory of formal groups?

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Other Connections to Elliptic Curves

  • What is the invariant differential of an elliptic curve?

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  • How does one extract the formal group of an elliptic curve using differential forms?

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  • How does the formal group of a curve relate to torsion and the Mordell-Weil theorem?

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  • What can a formal group tell you about your curve?

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Lazard Ring

  • What is the Lazard ring?

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  • What does it mean to be a “universal” coefficient ring?

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  • How exactly is the Lazard ring universal?

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  • How are formal groups related to spectra?

  • What is MU?

  • How are formal groups related to Motivation?

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  • How is the Lazard ring directly related to cobordism classification?

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  • What is the moduli stack of formal groups?

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Stable Homotopy

tmf

  • How is [tmf\] related to modular forms?

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  • How is tmf defined in terms of the moduli stack of elliptic curves?

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K-Theories

  • How is the algebraic K-theory of a commutative ring defined/constructed in terms of monoidal categories?

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