# Motivating Lubin-Tate theory

The Lubin-Tate theory gives an amazingly clean and streamlined way of constructing the subfield (usually denoted) $F_\pi\subset F^\mathrm{ab}$ for a local field $F$ fixed by the Artin map associated to the prime element $\pi$ (i.e. such that $F^{\mathrm{ab}}=F_\pi\cdot F^{\mathrm{un}}$ with the usual notations). The idea to consider 1-dim. formal groups over the ring of integers $\mathcal{O}_F$ is a deus ex machina for me, and I wonder if anyone can explain Lubin-Tate's motivation to consider such a thing?

Related, on page 50 of J.S. Milne's online notes on the class field theory, he offers the speculation that the motivation comes from complex multiplication of elliptic curves and how one might try to get an analogue of the theory for local fields. But this requires again that it is somehow natural to consider formal groups as an analogue which I think still needs a motivation.

What is the motivation to consider formal groups a la Lubin-Tate theory? Is there a way to motivate their construction?

• Maybe @lubin can comment... Oct 13, 2015 at 15:50
• The prehistory explains a lot, but there is still a jump, that, perhaps, can only be explained by Tate's great originality.
– anon
Oct 13, 2015 at 16:09
• "Tate's great originality"? Is this why this is called the "LUBIN-Tate" theory? Geez... Oct 13, 2015 at 20:23
• It’s true that it was Lubin who found the formal groups that do everything for you, but it was Tate who understood all the implications of their existence, and put everything together in the paper you’re referring to. Oct 16, 2015 at 0:59

Sorry I didn’t see this earlier. My memory is vague, and probably colored by subsequent events and results, but here’s how I recall things happening.

Since I had read and enjoyed Lazard’s paper on one-dimensional formal group (laws), which dealt with the case of a base field of characteristic $p$, I decided to look at formal groups over $p$-adic rings. For whatever reason, I wanted to know about the endomorphism rings of these things, and gradually recognized the similarity between, on the one hand, the case of elliptic curves and their supersingular reduction mod $p$, when that phenomenon did occur, and, on the other hand, formal groups over $p$-adic integer rings of higher height than $1$.

I had taken, or sat in on, Tate’s first course on Arithmetic on Elliptic Curves, and was primed for all of this. In addition, I was aware of Weierstrass Preparation, and the power it gave to anyone who wielded it. And in the attempt to prove a certain result for my thesis, I had thought of looking at the torsion points on a formal group, and I suppose it was clear to me that they formed a module over the endomorphism ring. Please note that it was not my idea at all to use them as a representation module for the Galois group.

But Tate was looking over my shoulder at all times, and no doubt he saw all sorts of things that I was not considering. At the time of submission of my thesis, I did not have a construction of formal groups of height $h$ with endomorphism ring $\mathfrak o$ equal to the integers of a local field $k$ of degree $h$ over $\Bbb Q_p$. Only for the unramified case, and I used extremely tiresome degree-by-degree methods based on the techniques of Lazard. Some while after my thesis, I was on a bus from Brunswick to Boston, and found not only that I could construct formal groups in all cases that had this maximal endomorphism structure, but that one of them could take the polynomial form $\pi x+x^q$. Tate told me that when he saw this, Everything Fell Into Place. The result was the wonderful and beautiful first Lemma in our paper, for which I can claim absolutely no responsibility. My recollection, always undependable, is that the rest of the paper came together fairly rapidly. Remember that Tate was already a master of all aspects of Class Field Theory. But if the endomorphism ring of your formal group is $\mathfrak o$ and the Tate module of the formal group is a rank-one module over this endomorphism ring, can the isomorphism between the Galois group of $k(F[p^\infty]])$ over $k$ and the subgroup $\mathfrak o^*\subset k^*$ fail to make you think of the reciprocity map?

• One other thing, for people’s amusement. Much earlier in my grad-school career, I went in to Tate and asked if he could recommend a good problem in Class Field Theory. He said, and here I’m sure my memory is reliable, “There’s nothing I’d like better than to know a good problem in Class Field Theory.” Oct 16, 2015 at 1:23
• great inspiring and motivating information and facts
– MAS
Oct 9, 2019 at 1:38

Abelian extensions of $\mathbb{Q}$ can be described using torsion points in the multiplicative group. If $K$ is a quadratic imaginary field, and $E$ is an elliptic curve where $\mathcal{O}_K$ acts by CM, then abelian extensions of $K$ can be described using torsion points of $E$. Shimura proved similar results about CM number fields and higher dimensional abelian varieties. That definitely suggests trying to build class field theory based on algebraic groups. Unfortunately, no one can get past the CM case in the global theory.

Hasse and Hilbert already showed that it was helpful to consider local fields in formulating Class Field Theory.

With all of this as prehistory, it doesn't strike me as that strange to consider formalizing the local study of algebraic groups and apply it to class field theory.

• It's important to note that if $p$ is inert in $K$, so $K_p$ is a local field, then all the $p$-power-torsion points reduce to $0$ mod $p$ and thus are points of the formal group of the elliptic curve. As you get abelian extensions of $K_p$ by considering the $p$-power-torsion points, one might try to use general formal groups after failing to generalize the theory using general abelian varieties. Oct 13, 2015 at 15:40
• Lubin's thesis was in the 1960s. Local Kronecker-Weber was certainly proved before that. For one thing it is not very hard to deduce it from the global result; it could also be in Hasse(?). More to the point, I wanted to point out Shafarevich's paper (A new proof of the Kronecker-Weber theorem) of 1951, which proves $p$-adic K-W and then notes: "Once we have proved the $p$-adic analog, K-W is obtained automatically by applying Minkowski's theorem [that $\mathbb{Q}$ has no unramified extensions]." This is the proof that Narkiewicz reproduces in his book on algebraic numbers (excluding CFT). Oct 13, 2015 at 17:31
• @VesselinDimitrov Thanks! So much for my skim then. Oct 13, 2015 at 17:46
• @VesselinDimitrov For the record, it was my error, not Wikipedia's. Wikipedia's article on Kronecker-Weber wrote that "Lubin and Tate (1965, 1966) proved the local Kronecker-Weber theorem", but if I had read more carefully, I would have seen that Wikipedia meant the case of a general local field, not just $\mathbb{Q}_p$ Oct 13, 2015 at 23:19
• Right. So before Lubin-Tate, one had Kronecker-Weber for $\mathbb{Q}_p$ (from $\mathbb{G}_m$) and its quadratic extensions (from CM elliptic curves), and then just the abstract local class-field correspondence. But one did not know or expect that a completely explicit generation of all the abelian extensions was possible for all the local fields. By the way, for $\mathbb{Q}_p$, I'd be curious to know where local K.-W. was first stated/considered explicitly. Shafarevich, commenting on his new proof, writes in his 1951 paper: "It turns out that the K.-W. theorem is essentially a $p$-adic fact." Oct 14, 2015 at 0:45