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Formal Groups Reading Notes
Formal Groups and Spectra
Formal Groups
Basics
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Motivation: why should one care about [[Formal group|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?
- What is a formal group law?
- Why is the group law called “formal”?
- How “should” one think of a formal group?
- What is a formal exponential and formal logarithm?
- What are examples of formal groups? What is a morphism? When are they isomorphic?
- What are the additive and multiplicative formal groups?
- How are the additive and multiplicative formal groups related?
- What is the height of a formal group?
- How can one extend formal groups over R to R-algebras?
- What is the formal scheme interpretation of formal groups?
- How does iterated K-theory show up in the theory of formal groups?
Other Connections to Elliptic Curves
- What is the invariant differential of an elliptic curve?
- How does one extract the formal group of an elliptic curve using differential forms?
- How does the formal group of a curve relate to torsion and the Mordell-Weil theorem?
- How do formal groups arise from elliptic curves?
- What can a formal group tell you about your curve?
Lazard Ring
- What is the Lazard ring?
- What does it mean to be a “universal” coefficient ring?
- How exactly is the Lazard ring universal?
- What is a complex oriented cohomology theory?
- How is the Lazard ring directly related to cobordism classification?
- What is the moduli stack of formal groups?
Stable Homotopy
tmf
- How is [tmf\] related to modular forms?
- How is tmf defined in terms of the moduli stack of elliptic curves?
K-Theories
- How is the algebraic K-theory of a commutative ring defined/constructed in terms of monoidal categories?
