View source for Mordell Conjecture

= News = 

Sarah gave me the notes for her talk:

[http://mordell.fhoermann.org/fgpschemes-v2.pdf fgpschemes-v2.pdf]

= Introduction =

In this seminar we would like to understand Faltings' proof of the Mordell conjecture:

; Theorem (Mordell Conjecture)
: Let <math>K</math> be a number field and <math>C</math> a non-singular projective curve of genus <math>\ge 2</math>, defined over <math>K</math>. Then <math>C(K)</math> is finite.

Afterwards different proofs have been found, notably 
Vojta's (see [<ref name="BG"/>, Chapter 11] or [<ref name="HS"/>, Part E]), which use basically only Arakelov theory.

The main references for Falting's proof (our seminar) are [<ref name="CS"/>] and [<ref name="FW"/>].
I recommend that anyone attending the seminar reads the introduction of Henri Darmon in [<ref name="ClayI"/>, Darmon, 1-2].

Faltings' proof is based on the following strategy of constructing a sequence of maps to a
set which is known to be finite, and then proving that these maps are all finite-to-one.
The finite set of primes <math>S</math> of <math>K</math> appearing, depends only on <math>C</math> (basically the primes of good reduction of <math>C</math>).

<math> 
\begin{array}{rl}
C(K) \xrightarrow{R_1} &  
\begin{Bmatrix}
\text{Isomorphism classes of curves of genus } g' \text{ defined over } K  \\
\text{ of good reduction outside } S 
\end{Bmatrix} \\
& \\
\xrightarrow{R_2} &  
\begin{Bmatrix}
\text{Isomorphism classes of Abelian varieties of dimension } g' \text{ defined over } K  \\
\text{ of good reduction outside } S 
\end{Bmatrix} \\
& \\
\xrightarrow{R_3} &  
\begin{Bmatrix}
\text{Isogeny classes of Abelian varieties of dimension } g' \text{ defined over } K  \\
\text{ of good reduction outside } S 
\end{Bmatrix} \\
& \\
\xrightarrow{R_4} &  
\begin{Bmatrix}
\text{semi-simple }l\text{-adic representations of dimension } 2g' \text{ of } \text{Gal}(\overline{K}|K)  \\
\text{unramified outside } S 
\end{Bmatrix}
\end{array}
</math>

The map <math>R_1</math>, which makes this approach possible, is due to a clever construction of Parshin.
The finiteness of this map relies heavily on the geometric fact that a curve of genus <math>g\ge 2</math> has only finitely many 
automorphisms!

The map <math>R_2</math> is given by associating to a curve <math>C</math> its Jacobian <math>J(C)</math>.
The finiteness of the map is given by the classical geometric theorem of Torelli.

The map <math>R_3</math> is obvious. 
Its finiteness is one of the 2 cornerstones of Faltings proof (Theorem F below).

The map <math>R_4</math> is given by associating to an isogeny class of Abelian varieties the l-adic representation on its Tate module. 
The finiteness of the map is a consequence of the famous '''Tate conjecture'''. Its proof is the second cornerstone of the proof (Theorem D below). 

The proof of finiteness of the last set, again, is very classical. It relies on the theorems of Cebotarev and Hermite respectively.

Remark: The finiteness of the second set above was the content of the '''Shafarevich conjecture'''.

= Schedule =

{| border="1" style="text-align:left;"
| 26.10.
|[[#Diophantine geometry in dimension 1|Diophantine geometry in dimension 1]]
|Matthias Wendt
|-
| 2.11.
|[[#Complex Abelian varieties|Complex Abelian varieties]]
|Magnus Engenhorst
|-
| 9.11.
|[[#Jacobians and the Torelli theorem over|Jacobians and the Torelli theorem over C]]
|Helene Sigloch
|-
| 16.11.
|[[#Algebraic theory of Abelian varieties|Algebraic theory of Abelian varieties I]]
|Stephen Enright-Ward
|-
| 23.11.
|[[#Algebraic theory of Abelian varieties|Algebraic theory of Abelian varieties II]]
|Clemens Jörder
|-
| 30.11.
|[[#Jacobians and Parshin's construction|Jacobians and Parshin's construction]]
|Daniel Greb
|-
| 7.12.
|[[#Classical Weil heights|Classical Weil heights]]
|Patrick Graf
|-
| 14.12.
|[[#The Mordell-Weil theorem|The Mordell-Weil theorem]]
|Maximilian Schmidtke
|-
| 21.12.
|[[#Proof of Shafarevich's conjecture|Proof of Shafarevich's conjecture]]
|Annette Huber-Klawitter
|-
| 11.1.
|[[#Arakelov theory on arithmetic surfaces|Arakelov theory on arithmetic surfaces]]
|Peter Wieland
|-
| 18.1.
|[[#Néron models|Néron models]]
|Wolfgang Soergel
|-
| 25.1.
|[[#The Moduli space of Abelian varieties and its compactifications|The Moduli space of Abelian varieties and its compactifications]]
|Stefan Kebekus, Alex Küronya
|-
| 1.2.
|[[#Integral models of the moduli space and Faltings heights|Integral models of the moduli space and Faltings heights]]
|Sebastian Goette, Matthias Wendt
|-
| 8.2.
|[[#Finite flat group schemes and p-divisible groups|Finite flat group schemes and <math>p</math>-divisible groups]]
|Sarah Kitchen, Wolfgang Soergel
|-
| 15.2.
|[[#Proof of Tate's conjecture|Proof of Tate's conjecture]]
|Fritz Hörmann
|}

= Description of Talks =

== Diophantine geometry in dimension 1 ==

'''First part: Generalities on Diophantine Geometry of Dimension 1'''

Present the Introduction to [<ref name="ClayI"/>, Darmon 1.] up to the top of page 11.
Explain what the ring <math>R</math> of <math>S</math>-integers is. Explain <math>R</math>-valued points of
a (quasi-)projective variety.

'''Second part: Overview of Falting's proof'''

Give an overview of the 4 maps in the introduction above (cf. also [<ref name="ClayI"/>, Darmon 2.8]).
Explain roughly why maps <math>R_1</math> and <math>R_4</math> are finite-to-one.
Present the following theorems schematically and explain roughly how they are used to prove finite-to-1-ness of the maps <math>R_2</math> and <math>R_3</math>... Do not yet explain explicitly what the Faltings height is. 

;Theorem A
: For integers <math>g, d, n</math> and positive constant <math>C</math>, there are only finitely many isomorphism classes of tripels <math>(A, \phi, \xi)</math> consisting of an Abelian variety <math>A</math>, a polarization <math>\phi</math> of degree <math>d</math>, and a level-<math>n</math>-structure <math>\xi</math>, defined over <math>K</math>, with <math>h(A)<C</math>.

The proof will be given in talk on the Moduli space of Abelian Varieties.

;Theorem A'
: For an integer <math>g</math>, there are only finitely many isomorphism classes of Abelian varieties <math>A</math> defined over <math>K</math> with <math>h(A) < C</math>

Follows from Theorem A by Zarhin's trick and the fact <math>h(A) = h({}^t A)</math>.

;Theorem B
: Let <math>A</math> be a semi-stable Abelian variety defined over <math>K</math>. There is a finite set of primes <math>S</math> such that for any isogeny <math>A \rightarrow A'</math> of degree coprime to <math>S</math> we have <math> h(A)=h(A') </math>

;Theorem C
: Let <math>A</math> be a semi-stable Abelian variety defined over <math>K</math>. Let <math>W \subset T_l(A)</math> be a saturated Galois stable sublattice and let <math>W_n = W/l^nW</math>. Then <math>h(A/W_n)</math> is independent of <math>n</math> for large <math>n</math>.

The proofs of Theorem B and C are very similar and use deep facts about p-divisible groups and
from p-adic Hodge theory.

;Theorem D (Tate conjecture)
: 
# The representation of <math>\text{Gal}(\overline{K}|K)</math> on <math>V_l(A)</math> is semi-simple. 
# <math>\text{End}(A) \otimes \Q_l = \text{End}_{\Q_l}(V_l(A))^{\text{Gal}(\overline{K}|K)}. </math>

Proof: A' + C + elementary arguments imply D for <math>A</math> semi-simple.
Then it holds for all Abelian varieties by the argument [<ref name="FW"/>IV (2.3)]

;Theorem E
: Let <math>S</math> be a finite set of primes and <math>A</math> an Abelian variety defined over <math>K</math>.  There are finitely many <math>K</math>-Isogenies <math>A_1 \rightarrow A, \cdots, A_n \rightarrow A</math> such that any <math>K</math>-isogeny <math>A' \rightarrow A</math> factors through a prime-to-<math>S</math> isogeny <math>A' \rightarrow A_i \rightarrow A</math>.

Proof: Follows from D by (rather) elementary arguments

;Theorem F: Finiteness of the map <math>R_3</math>.

Proof: A' + B + E.

That it is legitimate to use Theorem B despite <math>A</math> not being of semi-stable reduction is justified in 
[<ref name="FW"/>, V, 3. on p. 169].

== Complex Abelian varieties ==

The main reference is [<ref name="CS"/>, Chapter IV]. Cf. also [<ref name="BL"/>, <ref name="GH"/>]. Please coordinate with the speaker of the next talk. 

* Complex tori
* Isogenies of complex tori
* Criterion for a complex tori to be an Abelian variety (with an idea of the proof)
* Poincare's reducibility theorem and the semi-simplicity of <math>\text{End}(A)</math>
* Neron-Severi group and the dual Abelian variety
* Polarizations
* Moduli of principally polarized Abelian varieties over <math>\mathbb{C}</math>

== Jacobians and the Torelli theorem over <math>\mathbb{C}</math> ==

This is a continuation of the previous talk. Please coordinate with the speaker.

* Definition of Jacobian
* The canonical map from <math>C</math> to its Jacobian
* Statement of Torelli's theorem
* Sketch of the proof of Torelli's theorem (conclude that the map <math>R_2</math> is finite-to-one)

== Algebraic theory of Abelian varieties ==

'''Two talks'''

Give an overview of the algebraic theory of Abelian varieties as e.g. in [<ref name="CS"/>, Chapter V] in particular including...
* The seesaw principle and the theorems of the square and cube
* The dual Abelian variety
* Polarizations from an algebraic point of view
* The Tate module and its structure
* Weil-Pairing
* Rosati involution
* Discuss the Galois representation on the Tate module and the Zeta-function of the Abelian variety (include a sketch of proof of the Riemann hypothesis), especially discuss the determinant of the Tate module - this is needed in the talk on "Tate's conjecture".

== Jacobians and Parshin's construction ==

'''First part: Complements on Jacobians'''

The main reference is [<ref name="Serre2"/>, Chapter V, VI], [<ref name="CS"/>, Chapter VII]

* Mention the algebraic construction of the Jacobian
* Generalized Jacobians
* Explain how Jacobians (resp. generalized Jacobians) can be used to construct unramified (resp. controlledly ramified) coverings of curves.

'''Second part: Parshin's construction'''

This is about the construction of the map <math>R_1</math> above. 

The main reference is [<ref name="ClayI"/>, Darmon, Theorem 2.4] or [<ref name="FW"/>, V, 4.3, 4.4].
Prove also that the map <math>R_1</math> is finite-to-one (this uses that a geometric gurve of genus &ge; 2 has only
finitely many automorphisms).

cf. also [<ref name="Szpiro"/>, X, 2.]

== Classical Weil heights ==

The main reference is [<ref name="CS"/>, Chapter VI]. Do not present the extension to logarithmic singularities. It will be
subject of the talk on Faltings heights.

* Recall fundamental facts about rings of integers in number fields: finite generation, factorization of primes, ramification, valuations
* Heights on projective space and on projective varieties,
* Metrized line bundles on <math>Spec(R)</math>, 
* Metrized line bundles on varieties,
* Present as many proofs as possible, in particular, we have to understand 3.4  of [loc.cit.].

Other references for this talk are [<ref name="BG"/>, Chapter 1, 2], [<ref name="HS"/>, Part B].

== The Mordell-Weil theorem ==

The Mordell-Weil theorem is not needed in the proof of Mordell's conjecture. However, it constitutes
a nice application of the elementary theory of heights and decent. 
 
* Heights on Abelian Varieties, Neron-Tate height [<ref name="Serre"/>, 3.1-3.4] (present as many proofs as possible)
* Recall Minkowski's theorem
* Hermite's finiteness theorem [<ref name="Serre"/>, 4.1] (this, in fact, will be needed later!)
* The Chevalley-Weil theorem [<ref name="Serre"/>, 4.2]
* The weak Mordell-Weil Theorem [<ref name="Serre"/>, 4.3]
* The Mordell-Weil theorem using the Descent-Lemma [<ref name="CS"/>, Chapter IV, 5.1, 5.2]

== Proof of Shafarevich's conjecture ==

[<ref name="Szpiro"/>, IX]

* Recall Hermite's theorem from the talk on "Mordell-Weil"
* Cebotarev, Finiteness of the set of semi-simple l-adic representations with certain ramification properties
* Conclude the proof of Shafarevich's conjecture and hence Mordell's conjecture ''using'' that the maps <math>R_2</math> and <math>R_3</math> are finite-to-one (which will be proven in the last 4 talks)
* if time is left - Siegel's theorem [<ref name="FW"/>, V, 5.]

== Arakelov theory on arithmetic surfaces ==

This talk is not strictly needed for the sequel. Its purpose is to understand heights as
arithmetic intersection numbers (in the special case of arithmetic surfaces).

Main references are [<ref name="CS"/>, Chapter XII], [<ref name="Soule"/>] and [<ref name="Lang"/>] (if you like Serge Lang's books).

Emphasize the analogy with intersection products, degree, etc. on geometric surfaces.

According to taste you may present also something about the arithmetic Riemann-Roch for surfaces or
higher dimensional Arakelov Chow groups [<ref name="SABK"/>].

== Néron models ==

Main references [<ref name="CS"/>, Chapter VIII], [<ref name="Neron"/>] and [<ref name="Groth"/>]

* Define group schemes and Abelian schemes
* Define semi-Abelian schemes and discuss basic properties [<ref name="FC"/>, Chapter 1 ?]
* Definition of a Néron model by its universal property
* Understand the connection between semi-Abelian extensions and the (connected) Néron model
* State the existence of Néron models for Abelian schemes (without proof)
* Explain semi-stability and Grothendieck's semi-stability theorem [<ref name="Groth"/>, 3.]
* Explain the filtration of the Tate module induced by semi-Abelian reduction and Grothendieck's orthogonality theorem [<ref name="Groth"/>, 5.] - this will be needed in "Proof of Tate's conjecture"; coordinate with the speaker

== The Moduli space of Abelian varieties and its compactifications ==

Coordinate with the speaker of the next talk. Concentrate on properties over <math>\mathbb{C}</math> in this talk.

* Generalities on moduli problems [<ref name="FW"/>, I.2]
* Moduli schemes of Abelian schemes with polarization and level-structure
* Construction of the minimal compactification over <math>\mathbb{C}</math>
* The toroidal compactification over <math>\mathbb{C}</math>
* Discuss explicitly the degeneration of the universal Abelian scheme along the boundary
* Connections to semi-stable degeneration of curves...

References: [<ref name="FW"/>, Chapter I], [<ref name="FC"/>, <ref name="GIT"/>, <ref name="AMRT"/>, <ref name="Namikawa"/>]

cf. also [<ref name="Alexeev"/>]

== Integral models of the moduli space and Faltings heights ==

This is a direct sequel to the previous talk.

* Cite properties of integral models of the moduli space from [<ref name="FC"/>] (without proof!), cf. also [<ref name="Deligne"/>, (a), (b), (c) on p.32]
* Distance Functions and logarithmic Singularities from [<ref name="CS"/>, chapter VI]
* Finiteness result: 8.2 of [<ref name="CS"/>, chapter VI]
* The metric on the moduli space of Abelian varieties has logarithmic singularities along the boundary [<ref name="FW"/>, Chapter I]
* Definition of Faltings height and comparison with the height on the moduli space in the sense of the talk on heights (in the logarithmic extension just discussed), [<ref name="FW"/>, II] - please keep in mind that an integral model of the compactification now exists and this can be simplified a lot! Theorem 3.1 in [loc. cit.] is '''not''' needed anymore! cf. also [<ref name="Deligne"/>, p. 32] the seven lines proof of Proposition 1.12 using (a), (b), (c) of [loc. cit.].
* Conclude Theorem A

* Finally: Theorem A implies Theorem A' (Zarhin's trick + <math>h(A)=h({}^t A)</math>), [<ref name="FW"/>, IV, Proposition 3.7, Lemma 3.8] - Prop. 3.7. maybe without proof at this point ...

== Finite flat group schemes and <math>p</math>-divisible groups ==

Main references are [<ref name="CS"/>, Chapter III] and [<ref name="FW"/>, Chapter III]

'''First part: Finite flat group schemes'''

* Explain the definitions of (commutative) finite flat group schemes with as many examples as possible
* Differentials of finite flat group schemes
* etale and connected group schemes and the exact sequence
* etale group schemes and Galois representations
* Duality for finite flat group schemes

'''Second part: p-divisible groups'''

* Definition of p-divisible groups
* The p-divisible group associated with an Abelian scheme
* Height and dimension
* Serre's equivalence of divisible formal groups and connected p-divisible groups
* Duality for p-divisible groups
* Tate module of a p-divisible group

== Proof of Tate's conjecture ==

''' First part: Some <math>p</math>-adic Hodge Theory '''

Just state needed results.
Main references [<ref name="Tate"/>] and [<ref name="Hodge"/>] 

* Recall local Galois representations
* Explain the notion of Hodge-Tate (maybe cite Faltings' general theorem which is analogous to the analytic Hodge decomposition)
* Unramified implies Hodge-Tate
* Explicit determination of 1-dimensional HT-representations
* The Galois representation associated with a p-divisible group is Hodge-Tate [<ref name="Tate"/>, p.180]
* Determine weights and multiplicities of the Galois representation associated with a p-divisible group
* The goal is to understand the formula [<ref name="FW"/>, IV, 3.6]:

<math>h \frac{[K:\mathbb{Q}]}{2} = \sum_{\nu|l} [K_{\nu}:\mathbb{Q}_l] d_{\nu}</math>

''' Second part: Proofs of Theorems B-F (including Tate's conjecture) '''

* Explain how heights vary in isogeny classes
* The Isogeny formula [<ref name="FW"/>, IV, 3.1] 

* Present different formulation of Tate's conjecture. Prove that A and C imply Tate's conjecture (Theorem F) [<ref name="FW"/>, IV, 2]

* Sketch of the proof of Theorem C [<ref name="FW"/>, IV, 2.7] in [loc. cit., pp. 130--140] (maybe give first a sketch of the proof for l of good reduction!) (3.6) should have been presented in the talk about p-adic Hodge theory! 

* Omitted: The proof of Theorem B [<ref name="FW"/>, V, 3.5] in [loc.cit., section V, 3.4]

* Sketch of the proof of theorem E [<ref name="FW"/>, V, 3.2-3.4] (note that this uses Tate's conjecture in an essential way, too!)

Alternative: Theorem B is [<ref name="Deligne"/>, 2.4] and Theorem C is [<ref name="Deligne"/>, 2.6] and 
the use of Theorem E for proving F is substituted by the argument [<ref name="Deligne"/>, proof of Corollary 2.8]. 
According to taste, you
may also present this but will have to explain finiteness of class numbers for reductive groups. 
A third, nice possibility to circumvent Theorem E seems [<ref name="ClayI"/>, Darmon, 2.13 (2)], see proof [loc.cit., proof of theorem 2.11], I couldn't find
an explicit reference for this however, which doesn't boil down to the strategies above.

cf. also:<br>
Faltings' original article [<ref name="CS"/> Chapter II, 4-5]<br>
[<ref name="Szpiro"/>, VIII 3]

= References =
<references>
<ref name="FW">Faltings, G.; Wüstholz, G.; et al. ''Rational points''. Second edition. Papers from the seminar held at the Max-Planck-Institut für Mathematik, Bonn/Wuppertal, 1983/1984. Aspects of Mathematics, E6. Friedr. Vieweg & Sohn, Braunschweig, 1986. vi+268 pp.</ref>
<ref name="FC">Faltings, G.; Chai, C.-L.; ''Degeneration of abelian varieties''. With an appendix by David Mumford. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 22. Springer-Verlag, Berlin, 1990. xii+316 pp. </ref>
<ref name="HS">Hindry, M.; Silverman J. H.; ''Diophantine Geometry''. An introduction. Graduate Texts in Mathematics, 201. Springer-Verlag, New York, 2000. xiv+558 pp.</ref> 
<ref name="BG">Bombieri, E.; Gubler W.; ''Heights in Diophantine Geometry''. New Mathematical Monographs, 4. Cambridge University Press, Cambridge, 2006. xvi+652 pp.</ref>
<ref name="ClayI">Darmon, H.; Ellwood, D. A.; Hassett, B.; Tschinkel, Y. (eds.); ''Arithmetic Geometry''. Clay Mathematics Proceedings. Volume 8. Darmon's article is available online: [http://www.math.mcgill.ca/darmon/pub/Articles/Expository/12.Clay/paper.pdf http://www.math.mcgill.ca/darmon/pub/Articles/Expository/12.Clay/paper.pdf]</ref>
<ref name="Serre">Serre, J.-P.; ''Lectures on the Mordell-Weil theorem''.
Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. Aspects of Mathematics, E15. Friedr. Vieweg & Sohn, Braunschweig, 1989. x+218 pp.</ref>
<ref name="Serre2">Serre, J.-P.; 
''Algebraic groups and class fields.''
Translated from the French. Graduate Texts in Mathematics, 117. Springer-Verlag, New York, 1988. x+207 pp. ISBN: 0-387-96648-X
</ref>
<ref name="CS">Cornell, G.; Silverman, J. H. (eds.); ''Arithmetic Geometry''. Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30–August 10, 1984. Springer-Verlag, New York, 1986. xvi+353 pp. </ref>
<ref name="Deligne">Deligne, P. 
[http://archive.numdam.org/article/SB_1983-1984__26__25_0.pdf ''Preuve des conjectures de Tate et de Shafarevitch (d'après G. Faltings)'']. Seminar Bourbaki, Vol. 1983/84.</ref>
<ref name="Szpiro">Szpiro, L. (ed.); ''Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell.''
Papers from the seminar held at the École Normale Supérieure, Paris, 1983–84. Astérisque No. 127 (1985). Société Mathématique de France, Paris, 1985. pp. i–vi and 1–287. </ref>
<ref name="Tate">Tate, J. T.; [http://fhoermann.org/Tate%20-%20p-Divisible%20Groups.pdf ''p?divisible groups'']. 1967 Proc. Conf. Local Fields (Driebergen, 1966) pp. 158–183 Springer, Berlin</ref>
<ref name="Hodge">Brinon, O.; Conrad, B.; ''CMI Summer School Notes on p-adic Hodge Theory.'' Available online at
[http://math.stanford.edu/~conrad/ http://math.stanford.edu/~conrad/]</ref>
<ref name="Neron">Bosch, S.; Lutkebohmert, W.; Raynaud, M.; ''Néron Models'', Springer-Verlag, 1980.</ref>
<ref name="BL">Birkenhake, C.; Lange, H.; ''Complex Abelian Varieties''. Grundlehren der mathematischen
Wissenschaften 302, Springer 1992.</ref>
<ref name="Soule">Soulé, Ch.;
''Géométrie d'Arakelov des surfaces arithmétiques''. Séminaire Bourbaki, Vol. 1988/89.
Astérisque No. 177-178 (1989), Exp. No. 713, 327–343.</ref>
<ref name="SABK">Soulé, Ch.;
''Lectures on Arakelov geometry''. With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer. Cambridge Studies in Advanced Mathematics, 33. Cambridge University Press, Cambridge, 1992. viii+177 pp.</ref> 
<ref name="Lang">Lang, S.
''Introduction to Arakelov theory''. Springer-Verlag, New York, 1988. x+187 pp.</ref>
<ref name="Groth">Grothendieck, A.; ''Modèles de Néron et monodromie'', in Groupes de Monodromie en géometrie algébrique, SGA 7 I.</ref>
<ref name="GH">Griffiths, Ph.; Harris, J.
''Principles of algebraic geometry.''
Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978. xii+813 pp. </ref>
<ref name="AMRT">Ash, A.; Mumford, D.; Rapoport, M.; Tai, Y.; ''Smooth compactification of locally symmetric varieties.'' Math. Sci. Press, Brookline, Mass., 1975. Lie Groups: History,
Frontiers and Applications, Vol. IV.
</ref>
<ref name="Namikawa">Namikawa, Y.;
''Toroidal compactification of Siegel spaces.''
Lecture Notes in Mathematics, 812. Springer, Berlin, 1980. viii+162 pp. ISBN: 3-540-10021-0 </ref>
<ref name="GIT">Mumford, D.; Fogarty, J.; Kirwan, F.;
''Geometric invariant theory.''
Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. ISBN: 3-540-56963-4</ref>
<ref name="Alexeev">Alexeev, V.;
''Complete moduli in the presence of semiabelian group action.''
Ann. of Math. (2) 155 (2002), no. 3, 611–708.</ref> 
</references>