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).
Tate Uniformization
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.
Uniformization of Complex Hyperbolic Curves
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
and
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
.
Cusps on Bianchi Orbifolds II
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
.
Cusps on Bianchi Orbifolds
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
:
Definition 2:
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
and
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.
Concerning p-adic Floer Theory
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.
Temporarily absent
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.