Mordell Conjecture

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Sarah gave me the notes for her talk:
  
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== Getting started ==
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[http://mordell.fhoermann.org/fgpschemes-v2.pdf fgpschemes-v2.pdf]
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= Introduction =
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 +
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>

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