This blog is designed to be a reflection of my inner mathematical monologue (ramblings). It will host discussions of: current research and open problems that are interesting to me, talks and seminars that I have given or attended, and various expository pieces (mostly related to number theory, p-adic geometry, knot theory, and low-dimensional topology).
Here I will review Tate’s uniformization of Elliptic curves over and over –adic fields. Historically, Tate’s work served as motivation for Mumford’s uniformization of –adic hyperbolic curves, which I plan on discussing in a subsequent post. If is an elliptic curve defined over , the full Uniformization Theorem of Riemann surfaces implies that , where is a lattice inside . Without loss of generality assume . By factoring the quotient, note that . Consider the analytic isomorphism given by the exponential map . Define and . Observe that . Moreover observe that induces an analytic isomorphism
given by . In fact, we can give an explicit analytic description of the resulting isomorphism using q-expansions: set , , , , and .
Theorem 1 If and , then there exists a complex elliptic curve and a complex–analytic isomorphism
Proof: Let denote the curve cut out by the Weierstrass equation
and define the map in the Theorem by
where indicates a fixed origin point on . A full proof of the Theorem can be found in Silverman’s second book on elliptic curves.
The ability to compute cohomological invariants of curves is one of the main applications of classical uniformization theories. So when one studies the –adic cohomology of –adic abelian varieties, a natural question that arises is if there exists an analogue of Theorem~1. Let be a complete non–archimedian field of residue characteristic with ring of integers and maximal ideal . If one tries to generalize the usual description of elliptic curves as the quotient of by some lattice, issues quickly arise. First of all, 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 is mixed characteristic, does not contain non-trivial discrete subgroups. And when is of characteristic , 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 contains non-trivial lattices: for any with , is a discrete subgroup in . So one might hope that the uniformization in Theorem~1 can be generalized to the –adic setting. This is indeed the case.
Theorem 2 (Tate) For each with , there exists an elliptic curve over such that is an isomorphism of rigid–analytic spaces. In particular, is an isomorphism for each algebraic extension of , which is –equivariant whenever is Galois over .
Proof: In fact, one defines using the same formula as in Theorem~1. The point being that and define –adically convergent series. The analytic structure of was originally formulated in terms of rigid–analytic spaces. Recall that . Then the rigid–analytification of is given by the base space together with the Grothendieck topology of admissible covers , where , denotes the maximal spectrum, and is the Tate algebra in two variables. Note that acts on the discontinuously via , so the quotient is a well defined rigid–analytic space covered by the images of and . One then shows that is isomorphic to . The full proof can be found in Tate’s article A review of non-Archimedean elliptic functions.
The situation for higher–dimensional abelian varieties is similar. Consider and note there exists a natural group homomorphism given by . We say that is a lattice in if is a lattice in . It can be shown that then has the structure of a rigid–analytic space that is also an abelian variety.
I do not intend to prove any results in this post, but for the sake of exposition as I work towards understaning Mochizuki’s –adic uniformization theory, I would like to review the uniformization of complex–analytic hyperbolic curves. There are three main ways to realize such curves as the quotient of a topological space by the action of some group that acts discontinuously. We will focus primarily on the first two uniformizations described below.
Let (resp. ) denote the upper–half plane (resp. hyperbolic 3–space), and denote (resp. ) by (resp. ) . Let be a –punctured genus curve over such that (i.e has negative Euler characteristic and admits a hyperbolic structure). A hyperbolic structure on is, by definition, a choice of an atlas on with transition maps . We take as fact that such a choice induces an isometry , where is the universal cover of .
Fuchsian–Koebe Uniformization: The group acts transitively on via fractional linear transformations:
In fact, . It follows that the identification induces a representation
which is sometimes called the canonical representation. Set . The resulting isometry
is called the Fuchsian–Koebe uniformization of .
Schottky Uniformization: Observe that gives a natural –representation after composition with the inclusion . Moreover, we may compactify by ordering and filling its set of punctures so that they are viewed as marked points. Let denote the compactification of . Topologically speaking, –pointed and –punctured curves carry the same information and define the same point in the relevant moduli stack. The distinction between and becomes necessary when discussing uniformization. Now recall from the post “Cusps on Bianchi Orbifolds I”, the group acts on by Poincaré extending the fractional linear transformation on the sphere at to all of . Formulaically,
Similar to the case in two dimensions, one can show that .
Suppose that is a free, finitely generated Kleinian group that consists only of loxodromic elements: see Cusps on Bianchi Orbifolds I for the classification of Möbius transformations. A Theorem of Maskit in A Characterization of Schottky Groups implies that acts discontinuously on some nonempty connected domain bounded by Jordan curves whose interiors are pairwise disjoint and such that for each there exists with the properties and . Any such is known as a Schottky group and any domain in that is the maximal connected locus of discontinuity for the action of some Schottky group is called a Schottky domain. Set and let denote the full locus of discontinuity for the action of . The condition that all elements are loxodromic is equivalent to containing no unipotent, i.e parabolic, elements. Passing to the quotient, one obtains homeomorphisms
where denotes the Handlebody obtained by attaching handles to a solid torus, and denotes the boundary surface of genus . Intuitively, the boundary components of are identified in pairs via the action of , and the base solid torus of manifests in the Poincaré extension of the –action to . Note that is compact, ergo closed. The hyperbolic metric on induces a hyperbolic structure on , so can be promoted to an isometry of Riemann surfaces. The description of as the quotient is called the Schottky uniformization of .
It turns out that every closed Riemann surface of genus admits a Schottky uniformization; however, the closedness condition excludes punctured surfaces. There is a way to handle surfaces with an even number of punctures, using so–called extended Schottky groups, but we will not discuss these here. Instead, we consider marked surfaces. This will allow us to relate the Fuchsian–Koebe and Schottky uniformizations on the level of covering maps and fundamental groups.
Suppose is free on generators . A marked Schottky group is a Schottky group equipped with an ordered choice of free generators. Let denote the moduli space of marked Schottky groups of genus . consists of classes, under Möbius transformation, of ordered –tuples that generate a Schottky group. Let denote the holomorphic fibration over whose fiber over a point is the configuration spaces of –points on . can be thought of as the space of equivalence classes of tuples , where the indicate marked points on . Note that the define a –orbit of points, whose representatives are also denoted , in . Fix , set , and let . Then for some Fuchsian goup . Let denote the union of with the cusps of and recall that the –invariant defines a bijective Hauptmodul
for some points . Without loss of generality assume that for each . Note that lifts to a bijection
that sends the –orbit of the to the –orbit of . In particular, after restricting the codomain, lifting the domain to , and then composing with , one obtains covering maps
Suppose is generated by a standard meridian–longitude basis , and let denote the smallest normal subgroup of containing . One can show that is the topological covering space corresponding to the subgroup . From and the Fuchsian uniformization it follows that
Bers (Simultaneous) Uniformization: We conclude this discussion by mentioning the Bers simultaneous uniformization Theorem, which looks similar to the Schottky uniformization, but applies to a compact Riemann surface that admits two hyperbolic structures and . We call a Kleinian group quasi–Fuchsian of type I if its limit set is equal to a –invariant Jordan curve.
Theorem 1: If a compact Riemann surface admits hyperbolic structures and , then there exists a quasi–Fuchsian group of type I such that
where is the invariant Jordan curve of and indicates equipped with the hyperbolic structure .
The main observation towards proving such a result is that divides into two open discs and , each of which is conformally equivalent to . So it suffices to find Fuchsian groups and such that .
I need to clarify a minor mistake (now corrected) that was made out of haste in my previous post. Thankfully this leads nicely into a discussion that I was planning on writing about anyways. Recall the following result from part I (Lemma 8):
Lemma 1: If is an orientable hyperbolic 3–orbifold of finite volume, then has finitely many ends (i.e. cusp neighborhoods) and each end is isometric to , where is some quotient of the 2–torus . Moreover, the stabilizer subgroup of each cusp , generated by a pair of parabolics.
Before the correction, I had written that the cusp neighborhoods are isometric to . As the proof of the above Lemma implies, this is true if and only if the covering group does not contain torsion elements that fix the point at . The existence of such elements introduces interesting delicacies that are relevant to some of the problems I have been working on. In general, if there are elliptic elements that fix a cusp at , then the cusp is called rigid. This means that it cannot be deformed; and in particular, Dehn surgery cannot be performed (unless the cusp is ). We will explain and investigate these last statements in a later post when we discuss arithmetic knot complements (Cusps on Bianchi Orbifolds III).
Assume from here-on-out that is a Bianchi group. Then the number of cusps on is equal to the class number of , so there is always at least one cusp. In what follows we will show that the cusp cross sections of are tori unless . Moreover, unless , all cusp cross sections are non–rigid.
Lemma 2: If has finite order, then is an elliptic Möbius transformation.
Proof: Suppose for some positive integer . Observe that is not conjugate to a matrix of the form for some complex number since otherwise would be parabolic, hence have infinite order. So has two distinct fixed points on , therefore is conjugate to a marix of the form . It is an elementary group–theoretic fact that conjugates of torsion elements are also torsion, which implies that . It follows that is a root of unity, so is real and is less than . Since the square trace map is invariant under conjugation. and the trace is invariant up to multiplication by , we conclude that is elliptic.
The Dirichlet Unit Theorem implies that the group of units in any imaginary quadratic field has rank . Hence any unit in is a root of unity. It turns out that the existence of such elements is rare, as the following Lemma shows.
Lemma 3: contains a non-real root of unity if and only if .
Proof: The “if” direction is clear given the element and in . Conversely, suppose contains an –root of unity . Then contains the subfield , which is Galois of degree . Recall the fact that implies . In particular, one has for all primes dividing . It follows that (otherwise contains a subfield of strictly larger degree), and so . Now note if , then . We conclude that . When , then ; and when , . Any imaginary quadratic field containing (resp. or ) must also contain (resp. ), hence be equal to (resp. ) by comparing degrees.
Lemma 4: contains a non–identity element fixing if and only if .
Proof: Let and suppose is an element fixing . Observe that we can identify the point with in for any . Consider . It is easy to see that if and only if , so . Then since , hence is a unit. By Dirichlet’s unit Theorem, is necessarily a root of unity; and since by hypothesis, Lemma~3 implies . For future reference we note that these elements are and , respectively, where .
Theorem 5: The cusp cross sections of the Bianchi orbifold are tori unless . When (resp. ), the cusp cross section is a pillowcase (resp. ).
Proof: When , Lemma~4 shows that there are no torsion elements in that fix the cusp cross sections, hence each such cross section is a torus by Lemma~1. When , the unique tosion element fixing the cusp at is , which acts as on . In particular the action of on induces a degree two quotient map . So is a pillowcase, which has orbifold structure . This last statement can be seen directly by noting that stabilizes the group of –roots of unity in , and the successive pairwise dihedral angles of the corresponding lines in have cone angle . If , then fixes , where . Note that has order and that it permutes the third roots of unity in . Passing to we see that permutes the lines , which clearly have successive pairwise cone–angle . It follows that the cusp cross section in is topologically with three singular points of order , i.e .
Setup: Let be an imaginary quadratic field with discriminant and class number , and denote its ring of integers by . Set and consider the Bianchi orbifold , where denotes hyperbolic –space. The goal of this note is to prove the following Theorem:
Theorem 1: The cusp set of is in bijection with , viewed as a subset of . Moreover, the number of ends of is equal to .
First recall that acts on via the fractional linear transformation . Geometrically, each such is the product of an even number of inversions within circles and lines in . Suppose acts by inversions in circles and lines . The action of can be extended to
as follows. First note that we can identify with . Then, given any circle (resp. line ) in , observe that there exists a unique hemisphere (resp. plane ) in that is simultaneously orthogonal to and intersects at (resp. ). The Poincaré extension of to is obtained by applying the corresponding inversions in and . We now recall the classification of elements in :
is elliptic if and .
is parabolic if .
is loxodromic otherwise.
It is an easy exercise to show that is parabolic if and only if it has a unique fixed point on , in which case is conjugate to the standard translation . acts transitively on points in , so the stabilizer of any point is conjugate to that of , which can easily be worked out to be . Hence we recover as the symmetric space . It is well-known that , which in turn is diffeomorphic to the 3–sphere . Similarly, acts transitively on , the sphere at infinity, so all point stabilizers (in ) are conjugate to the subgroup of upper–triangular matrices
We are particularly interested in point stabilizers inside discrete subgroups of .
Definition 3: A Kleinian group (resp. Bianchi group) is a discrete subgroup of (resp. for some ). A hyperbolic orbifold is called a Kleinian orbifold (resp. Bianchi orbifold) if is commensurable with a Kleinian group (resp. Bianchi group).
Definition 4: Let be any number field with exactly one complex place and ring of integers . Suppose is a quaternion algebra over that is ramified at all real places and let be a –embedding. Then a subgroup of is an arithmetic Kleinian group if it is commensurable with some , where is an –order in and is its elements of norm 1.
The following Lemma can be found in Shimura or Maclachlan–Reid's book, and for brevity we state it without proof.
Lemma 5: Bianchi groups are arithmetic Kleinian groups.
Definition~4 is the same as that in the theory of Shimura varieties. The discreteness condition implies that such act discontinuously on . In particular, the –stabilizer of any point in is finite and the stabilizer of a point on the sphere at infinity is conjugate to a discrete subgroup . By inspection one observes that can take on of the three forms:
Finite cyclic, a finite extension of generated by a parabolic or loxodromic element, or a finite extension of generated by a pair of parabolics.
The only delicate part about the above classification involves noting that any loxodromic element in is conjugate to a matrix of the form with . So if there were two loxodromic elements, they would be –translates of eachother, meaning that item (3) can only occur if the two summands are generated by parabolic elements.
Definition 6: A point is a cusp if its stabilizer subgroup contains a free abelian group of rank .
Note that we can always take to be one of the parabolic generators.
In order to gain traction on the cusp set of hyperbolic orbifolds , we need a “smallness" condition on . The theory of Tamagawa numbers shows that Kleinian groups of the form have covolume . Hence,
Lemma 7: Arithmetic Kleinian groups have finite covolume.
Lemma 8: If is an orientable hyperbolic 3–orbifold of finite volume, then has finitely many ends (i.e. cusp neighborhoods) and each end is isometric to , where is some quotient of the 2–torus . Moreover, the stabilizer subgroup of each cusp is of type above.
Proof: If there were infinitely many ends, then would not have finite volume. By the classification of discrete subgroups of and the definition of a cusp, the stabilizer of each end is generated by a pair of parabolics and . Without loss of generality we may assume that and , where . In particular, and represent independent translations in . Let denote the torsion subgroup of , and set . A consequence of the fact that is the free product of its torsion-free and torsion parts is we can factor the orbit space as
Since is torsion–free, is a manifold. In particular, the translations and , viewed as elements of , generate independent copies of in that intersect at a single point. In otherwords, they generate a copy of . By construction of the Poincaré extension of to , one sees that the stabilization locus of the in is isometric to . Yet again appealing to the Poincaré extension, it follows that the cusp neighborhoods in are isometric to .
Corollary 9: All Bianchi orbifolds have at least one cusp.
Proof: Let be a Bianchi group and consider the standard –basis for , where
Then and are independent parabolic elements in . By Lemma~8 their stabilization locus is an end of .
In order to prove Theorem~1, it suffices now to prove the following Lemma. Recall that each can be described as a fraction , written in lowest terms, with . Recall also that elements can be described as equivalence classes of points in modulo the equivalence relation if and only if there exists such that . Since is a rank –module, every fractional ideal can be generated by a pair of elements. Let denote the equivalence class of ideals (in the class group of ) with reprepresentative the ideal .
Lemma 10: Let and be points in . Then there exists such that if and only if .
Proof: First assume there exists such that . Let for some satisfying . Then . By definition there exists some nonzero such that and . Hence as ideals, which implies . Set and . It suffices now to show that . Indeed the containment is clear. Note that
So for all . It follows that , ergo . We conclude that in the class group of . Conversely, suppose . By definition there exists nonzero and in such that . Note this is equivalent to . It is a well-known fact that ideals in an imaginary quadratic number field embed as lattices in . We fix such embeddings of and , and denote their images by and , respectively. Since by hypothesis, there exists some linear transformation such that ; in particular, . Without loss of generality we can assume since and have coordinates in . As is a constant, one has . So on the level of one has
Proof of Theorem~1: Any Bianchi group has finite covolume by Lemmas~5 and~7. Lemma~8 then implies that has finitely many cusps. Let denote the cusp set of . From the previous exposition it is clear that every cusp in (parabolic element) gives rise to an element . Conversely, given , the parabolic element
fixes . This proves that is bijective with , which is the first claim of the Theorem. Next define the map by , where is the ideal class group of . As previously noted, every –ideal in can be generated by two elements; so if is any ideal class, there exists such that is a representative for . Hence , proving that is surjective. Using the previous paragraph and Lemma~10, one sees that descends to a bijection
We conclude that , completing the proof.
The first few sections of this article are designed to provide precursory evidence and motivation for “arithmetic” analogues of Classical Floer (co)homology theories, with applications to number theory. The work is partially based on Minhyong Kim’s recent paper (mentioned in my last post), and partially inspired by my recent trip to AWS 2018. Later sections in the document are either informal or incomplete.
Since creating this site I faced a rather busy academic term, submitted grad school apps, and have been otherwise preoccupied with a project. There are several posts that I would like to write up as soon as time admits, especially one concerning Minhyong Kim’s paper titled Arithmetic Gauge Theory that was recently posted to the arXiv. Until then, adieu.