Mordell Conjecture

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Sarah gave me the notes for her talk:

Introduction

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

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

Afterwards different proofs have been found, notably Vojta's (see [[1], Chapter 11] or [[2], Part E]), which use basically only Arakelov theory.

The main references for Falting's proof (our seminar) are [[3]] and [[4]]. I recommend that anyone attending the seminar reads the introduction of Henri Darmon in [[5], 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 S of K appearing, depends only on C (basically the primes of good reduction of C).

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

The map R1, 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 $g\ge 2$ has only finitely many automorphisms!

The map R2 is given by associating to a curve C its Jacobian J(C). The finiteness of the map is given by the classical geometric theorem of Torelli.

The map R3 is obvious. Its finiteness is one of the 2 cornerstones of Faltings proof (Theorem F below).

The map R4 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

 26.10. Diophantine geometry in dimension 1 Matthias Wendt 2.11. Complex Abelian varieties Magnus Engenhorst 9.11. Jacobians and the Torelli theorem over C Helene Sigloch 16.11. Algebraic theory of Abelian varieties I Stephen Enright-Ward 23.11. Algebraic theory of Abelian varieties II Clemens Jörder 30.11. Jacobians and Parshin's construction Daniel Greb 7.12. Classical Weil heights Patrick Graf 14.12. The Mordell-Weil theorem Maximilian Schmidtke 21.12. Proof of Shafarevich's conjecture Annette Huber-Klawitter 11.1. Arakelov theory on arithmetic surfaces Peter Wieland 18.1. Néron models Wolfgang Soergel 25.1. The Moduli space of Abelian varieties and its compactifications Stefan Kebekus, Alex Küronya 1.2. Integral models of the moduli space and Faltings heights Sebastian Goette, Matthias Wendt 8.2. Finite flat group schemes and p-divisible groups Sarah Kitchen, Wolfgang Soergel 15.2. 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 [[5], Darmon 1.] up to the top of page 11. Explain what the ring R of S-integers is. Explain R-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 [[5], Darmon 2.8]). Explain roughly why maps R1 and R4 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 R2 and R3... Do not yet explain explicitly what the Faltings height is.

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

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

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

Follows from Theorem A by Zarhin's trick and the fact h(A) = h(tA).

Theorem B
Let A be a semi-stable Abelian variety defined over K. There is a finite set of primes S such that for any isogeny $A \rightarrow A'$ of degree coprime to S we have h(A) = h(A')
Theorem C
Let A be a semi-stable Abelian variety defined over K. Let $W \subset T_l(A)$ be a saturated Galois stable sublattice and let Wn = W / lnW. Then h(A / Wn) is independent of n for large n.

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)
1. The representation of $\text{Gal}(\overline{K}|K)$ on Vl(A) is semi-simple.
2. $\text{End}(A) \otimes \Q_l = \text{End}_{\Q_l}(V_l(A))^{\text{Gal}(\overline{K}|K)}.$

Proof: A' + C + elementary arguments imply D for A semi-simple. Then it holds for all Abelian varieties by the argument [[4]IV (2.3)]

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

Proof: Follows from D by (rather) elementary arguments

Theorem F
Finiteness of the map R3.

Proof: A' + B + E.

That it is legitimate to use Theorem B despite A not being of semi-stable reduction is justified in [[4], V, 3. on p. 169].

Complex Abelian varieties

The main reference is [[3], Chapter IV]. Cf. also [[6], [7]]. 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 End(A)
• Neron-Severi group and the dual Abelian variety
• Polarizations
• Moduli of principally polarized Abelian varieties over $\mathbb{C}$

Jacobians and the Torelli theorem over $\mathbb{C}$

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

• Definition of Jacobian
• The canonical map from C to its Jacobian
• Statement of Torelli's theorem
• Sketch of the proof of Torelli's theorem (conclude that the map R2 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 [[3], 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 [[8], Chapter V, VI], [[3], 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 R1 above.

The main reference is [[5], Darmon, Theorem 2.4] or [[4], V, 4.3, 4.4]. Prove also that the map R1 is finite-to-one (this uses that a geometric gurve of genus ≥ 2 has only finitely many automorphisms).

cf. also [[9], X, 2.]

Classical Weil heights

The main reference is [[3], 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 Spec(R),
• 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 [[1], Chapter 1, 2], [[2], 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 [[10], 3.1-3.4] (present as many proofs as possible)
• Recall Minkowski's theorem
• Hermite's finiteness theorem [[10], 4.1] (this, in fact, will be needed later!)
• The Chevalley-Weil theorem [[10], 4.2]
• The weak Mordell-Weil Theorem [[10], 4.3]
• The Mordell-Weil theorem using the Descent-Lemma [[3], Chapter IV, 5.1, 5.2]

Proof of Shafarevich's conjecture

[[9], 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 R2 and R3 are finite-to-one (which will be proven in the last 4 talks)
• if time is left - Siegel's theorem [[4], 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 [[3], Chapter XII], [[11]] and [[12]] (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 [[13]].

Néron models

Main references [[3], Chapter VIII], [[14]] and [[15]]

• Define group schemes and Abelian schemes
• Define semi-Abelian schemes and discuss basic properties [[16], 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 [[15], 3.]
• Explain the filtration of the Tate module induced by semi-Abelian reduction and Grothendieck's orthogonality theorem [[15], 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 $\mathbb{C}$ in this talk.

• Generalities on moduli problems [[4], I.2]
• Moduli schemes of Abelian schemes with polarization and level-structure
• Construction of the minimal compactification over $\mathbb{C}$
• The toroidal compactification over $\mathbb{C}$
• Discuss explicitly the degeneration of the universal Abelian scheme along the boundary
• Connections to semi-stable degeneration of curves...

References: [[4], Chapter I], [[16], [17], [18], [19]]

cf. also [[20]]

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 [[16]] (without proof!), cf. also [[21], (a), (b), (c) on p.32]
• Distance Functions and logarithmic Singularities from [[3], chapter VI]
• Finiteness result: 8.2 of [[3], chapter VI]
• The metric on the moduli space of Abelian varieties has logarithmic singularities along the boundary [[4], 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), [[4], 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 [[21], 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 + h(A) = h(tA)), [[4], IV, Proposition 3.7, Lemma 3.8] - Prop. 3.7. maybe without proof at this point ...

Finite flat group schemes and p-divisible groups

Main references are [[3], Chapter III] and [[4], 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 p-adic Hodge Theory

Just state needed results. Main references [[22]] and [[23]]

• 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 [[22], p.180]
• Determine weights and multiplicities of the Galois representation associated with a p-divisible group
• The goal is to understand the formula [[4], IV, 3.6]:

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

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

• Explain how heights vary in isogeny classes
• The Isogeny formula [[4], IV, 3.1]
• Present different formulation of Tate's conjecture. Prove that A and C imply Tate's conjecture (Theorem F) [[4], IV, 2]
• Sketch of the proof of Theorem C [[4], 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 [[4], V, 3.5] in [loc.cit., section V, 3.4]
• Sketch of the proof of theorem E [[4], V, 3.2-3.4] (note that this uses Tate's conjecture in an essential way, too!)

Alternative: Theorem B is [[21], 2.4] and Theorem C is [[21], 2.6] and the use of Theorem E for proving F is substituted by the argument [[21], 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 [[5], 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:
Faltings' original article [[3] Chapter II, 4-5]
[[9], VIII 3]

References

1. 1.0 1.1 Bombieri, E.; Gubler W.; Heights in Diophantine Geometry. New Mathematical Monographs, 4. Cambridge University Press, Cambridge, 2006. xvi+652 pp.
2. 2.0 2.1 Hindry, M.; Silverman J. H.; Diophantine Geometry. An introduction. Graduate Texts in Mathematics, 201. Springer-Verlag, New York, 2000. xiv+558 pp.
3. 3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11 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.
4. 4.00 4.01 4.02 4.03 4.04 4.05 4.06 4.07 4.08 4.09 4.10 4.11 4.12 4.13 4.14 4.15 4.16 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.
5. 5.0 5.1 5.2 5.3 5.4 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
6. Birkenhake, C.; Lange, H.; Complex Abelian Varieties. Grundlehren der mathematischen Wissenschaften 302, Springer 1992.
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