Tate Uniformization

Here I will review Tate’s uniformization of Elliptic curves over {{\mathbb C}} and over {p}–adic fields. Historically, Tate’s work served as motivation for Mumford’s uniformization of {p}–adic hyperbolic curves, which I plan on discussing in a subsequent post. If {E} is an elliptic curve defined over {{\mathbb C}}, the full Uniformization Theorem of Riemann surfaces implies that {E({\mathbb C}) \cong {\mathbb C}/({\mathbb Z}\oplus {\mathbb Z}\tau)}, where {{\mathbb Z}\oplus {\mathbb Z}\tau} is a lattice inside {{\mathbb C}}. Without loss of generality assume {\mathrm{Im}(\tau) > 0}. By factoring the quotient, note that {E({\mathbb C})\cong \left({\mathbb C}/{\mathbb Z}\right)/{\mathbb Z}\tau}. Consider the analytic isomorphism {\mathrm{exp}: {\mathbb C}/{\mathbb Z}\rightarrow {\mathbb C}^{\times}} given by the exponential map {z\mapsto e^{2\pi i z}}. Define {q = e^{2\pi i \tau}} and {q^{{\mathbb Z}} = \{q^n : n\in {\mathbb Z}\}}. Observe that {|q| < 1}. Moreover observe that {\mathrm{exp}} induces an analytic isomorphism
\displaystyle \left({\mathbb C}/{\mathbb Z}\right)/{\mathbb Z}\tau \xrightarrow{\sim} {\mathbb C}^{\times}/q^{{\mathbb Z}}

given by {z \mapsto e^{2\pi i z}}. In fact, we can give an explicit analytic description of the resulting isomorphism {E({\mathbb C})\cong {\mathbb C}^{\times}/q^{{\mathbb Z}}} using q-expansions: set {s_k = \sum_{n\geq 1} \frac{n^k q^n}{1-q^n}}, {a_4 = -5s_3(q)}, {a_6(q) = \frac{a_4(q) - 7s_5(q)}{12}}, {x_q(t) = -2s_1(q) + \sum_{n\in {\mathbb Z}} \frac{q^nt}{(1-q^nt)^2}}, and {y_q(t) = s_1(q) + \sum_{n\in {\mathbb Z}} \frac{(q^nt)^2}{(1-q^nt)^3}}.

Theorem 1 If {q \in {\mathbb C}^{\times}} and {|q|<1}, then there exists a complex elliptic curve {E_q} and a complex–analytic isomorphism
\displaystyle {\mathbb C}^{\times}/q^{{\mathbb Z}} \xrightarrow{\sim} E_q({\mathbb C})

Proof: Let {E_q} denote the curve cut out by the Weierstrass equation

\displaystyle E_q: y^2+xy = x^3 + a_4(q)x + a_6(q)

and define the map in the Theorem by
\displaystyle t \mapsto \begin{cases} (x_q(t),y_q(t)) & e^{2\pi i z} \not\in q^{{\mathbb Z}} \\ P & e^{2\pi i z}\in q^{{\mathbb Z}} \end{cases}

where {P} indicates a fixed origin point on {E_q}. A full proof of the Theorem can be found in Silverman’s second book on elliptic curves. \Box

The ability to compute cohomological invariants of curves is one of the main applications of classical uniformization theories. So when one studies the {l}–adic cohomology of {p}–adic abelian varieties, a natural question that arises is if there exists an analogue of Theorem~1. Let {K} be a complete non–archimedian field of residue characteristic {p} with ring of integers {{\cal O}_K} and maximal ideal {m}. If one tries to generalize the usual description of elliptic curves as the quotient of {{\mathbb C}} by some lattice, issues quickly arise. First of all, {K} is totally disconnected with respect to its natural topology. So the notion of analyticity becomes more complicated. This issue is resolved by passing to the rigid–analytic, or more generally, the adic category. Additionally, when {K} is mixed characteristic, {K} does not contain non-trivial discrete subgroups. And when {K} is of characteristic {(p,p)}, then such subgroups may exist, but their quotients do not define Abelian varieties. The latter phenomena leads to the study of Drinfeld modules, which is an incredible subject in its own right (and I may dedicate a future blog post to it). Nevertheless, Tate’s insight was that {K^{\times}} contains non-trivial lattices: for any {q\in K^{\times}} with {|q|< 1}, {q^{{\mathbb Z}}} is a discrete subgroup in {K^{\times}}. So one might hope that the uniformization in Theorem~1 can be generalized to the {p}–adic setting. This is indeed the case.

Theorem 2 (Tate) For each {q \in K^{\times}} with {|q| < 1}, there exists an elliptic curve {E_q} over {K} such that {\mathbb{G}_m^{an}/q^{{\mathbb Z}} \xrightarrow{\sim} E_q^{an}} is an isomorphism of rigid–analytic spaces. In particular, {L^{\times}/q^{{\mathbb Z}} \cong E_q(L)} is an isomorphism for each algebraic extension {L} of {K}, which is {\mathrm{Gal}(L/K)}–equivariant whenever {L} is Galois over {K}.

Proof: In fact, one defines {E_q} using the same formula as in Theorem~1. The point being that {a_4} and {a_6} define {p}–adically convergent series. The analytic structure of {E_q} was originally formulated in terms of rigid–analytic spaces. Recall that {\mathbb{G}_{m} = \text{Spec }K[T,U]/(TU - 1) = \text{Spec }K[T,T^{-1}]}. Then the rigid–analytification of {\mathbb{G}_{m}} is given by the base space {\mathbb{G}_{m}^{an} = \text{Spm }K/(TU - 1) = \text{Spm }K} together with the Grothendieck topology of admissible covers {\{U_n\}_{n\geq 0}}, where {U_n = \{z\in K^{\times}: |q|^{\frac{n+1}{2}}< |z|< |q|^{\frac{n}{2}}\} = \text{Spm }K<T,T^{-1},X,Y>/(TY - q^{\frac{n+1}{2}}, T - |q|^{\frac{n}{2}}X)}, {\text{Spm}} denotes the maximal spectrum, and {K<T,U>} is the Tate algebra in two variables. Note that {q} acts on the {U_n} discontinuously via {U_n\mapsto U_{n+2}}, so the quotient {\mathbb{G}_m^{an}/q^{{\mathbb Z}}} is a well defined rigid–analytic space covered by the images of {U_0} and {U_1}. One then shows that {\mathbb{G}_m^{an}/q^{{\mathbb Z}}} is isomorphic to {E_q^{an}}. The full proof can be found in Tate’s article A review of non-Archimedean elliptic functions. \Box

The situation for higher–dimensional abelian varieties is similar. Consider {(\mathbb{G}_m^{an})^g = \text{Spm }K<T_1,T_1^{-1},\dots ,T_g,T_g^{-1}>} and note there exists a natural group homomorphism {{\cal L}: (\mathbb{G}_m^{an})^g \rightarrow {\mathbb R}^g} given by {{\cal L}(z) = (-\log|z_1|,\dots,-\log|z_g|)}. We say that {\Lambda} is a lattice in {(\mathbb{G}_m^{an})^g} if {{\cal L}(\Lambda)} is a lattice in {{\mathbb R}^g}. It can be shown that {(\mathbb{G}_m^{an})^g/\Lambda} then has the structure of a rigid–analytic space that is also an abelian variety.

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