Mordell Conjecture
From Mathematics
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− | + | = 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 ≥ 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.'' | ||
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