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Basic Notions (6 hours)
Natalia Garcia-Fritz
Program
In this course, we present the prerequisites for the continuation of the program. We suppose that the students have some notions about elliptic
curves, Galois theory and algebraic number theory.
1. A few reminders about elliptic curves : Torsion points of elliptic curves over number fields Weil pairing Representation associated to Galois action on p-torsion points
Statement of Serre's theorem Examples
2. Reduction of elliptic curves Good reduction : ordinary or supersingular Bad reduction : multiplicative or additive, Tate curve
3. Classification of subgroups of GL2(Fp) Statement and ideas of the proof (using Weber)
| Action of inertia (12 hours)
Francesco Campagna and Marusia Rebolledo
Program
A key step in the proof of Serre?s theorem is based on the way the inertia group acts on the torsion subgroup. After introducing all the relevant notions, we give the main statements which are useful in the proof.
1. Reminder of (local) Galois theory, inertia, wild inertia and tame inertia groups
2. Characters of inertia group
3. Formal groups : définition of a formal group of one parameter, hight, action of inertia group on the kernel of multiplication by p and characters
4. Application 1 : representation defined by an elliptic curve with good reduction (ordinary case and supersingular case)
5. Application 2 : representation defined by an elliptic curve with bad multiplicative reduction
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CM Theory, Hecke characters, class field theory
(10 hours)
Francesco Pappalardi and Peter Stevenhagen
Program
The proof of Serre's theorem proceeds by showing that an elliptic curve for which the image of the Galois representation on its p-torsion points is not the full group GL2(Fp) for infinitely many p is actually a CM-curve. This course treats the arithmetic of CM-curves, and highlights their role in the broader setting of class field theory.
1. Historic introduction Reciprocity laws, Kronecker-Weber theorem, Frobenius symbols, Artin reciprocity for abelian extensions
2. Class field theory Ideal theoretic and idelic formulation of the theory, computational aspects.
3. Complex multiplication Endomorphisms of complex elliptic curves, j-invariant, singular moduli, Hilbert class field
4. Kronecker's Jugendtraum Division polynomials, ray class fields from complex multiplication, Shimura reciprocity
5. L- functions of CM elliptic cruves Dirichlet L-series, Grössencharacter associated to a CM-elliptic curve, Hecke L-series, L-function of a CM-elliptic curve There will be sets of exercises for all topics mentioned above.
Literature:
[1] H. Cohen and P. Stevenhagen, Computational Class Field Theory, in: Algorithmic Number Theory, Lattices, Number Fields, Curves and cryptography, pp. 497-534, edited by J.P. Buhler and P. Stevenhagen, Cambridge University Press
[2] J.H. Silverman, Advanced Topics in the Theory of Elliptic Curves, Chapter II, pp. 95-187
| Diophantine Approximation and Diophantine Equations (12 hours)
Amalia Pizarro and Michel Waldschmidt
Program
The end of the proof of Serre's theorem on the image of the Galois representations attached to elliptic curves rests on the following result: Let K be a number field, Δ a nonzero element of K, S a finite set of places of K including the archimedean places and OS the ring of S-integers in K. Then there are only finitely many U, V in OS satisfying U3 - 27V2=Δ. This auxiliary result was proved by C.L. Siegel when S is the set of archimedean places, and by K. Mahler and S. Lang in the general case. It is the main object of the present course on Diophantine approximation and Diophantine equations.We start by reducing the proof to the finiteness of the S-unit equation u+v=1 with u, v units in the ring OS . Next, there are mainly two methods for solving this so--called S-unit equation: an ineffective one, which rests on the Thue-Siegel-Roth Theorem in Diophantine approximation, and an effective one, which is based on the transcendence method of Gel'fond and Baker. The topics covered in this course will include:
1. The S- unit equation S-integers, S-units in a number field. Statements of finiteness results on the solutions of an S-unit equation: upper bounds for the number of solutions, upper bounds for the solutions. Application of the S-unit equation to S-integral points on curves, in particular on elliptic curves.
2. Diophantine approximation. The theorems of Thue-Siegel-Roth and Ridout (no proof); they lead to upper bounds for the number of solutions of the S-unit equation. Siegel's Theorem on integral points on elliptic curves. Example: Mordell curve y2=x^3+k.
3. Effective methods. Lower bounds for linear forms (no proof); they lead to upper bounds for the solutions of the S- unit equation. Linear forms in classical logarithms of algebraic numbers. Archimedean and p-adic theory. Linear forms in elliptic logarithms
References
Baker, A. (1990). Transcendental number theory. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition.
Bugeaud, Y. (2018). Linear forms in logarithms and applications, volume 28 of IRMA Lect. Math. Theor. Phys. Zurich: European Mathematical Society (EMS).
Evertse, J.-H. and Gyory, K. (2015). Unit equations in Diophantine number theory, volume 146 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge.
Lang, S. (1978). Elliptic curves: Diophantine analysis, volume 231 of Grundlehren Math. Wiss. Springer
Schmidt, W. M. (1991). Diophantine approximations and Diophantine equations, volume 1467 of Lecture Notes in Mathematics. Springer-Verlag, Berlin.
Serre, J.-P. (1997). Lectures on the Mordell-Weil theorem. Aspects of Mathematics. Friedr. Vieweg and Sohn, Braunschweig, third edition. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, With a foreword by Brown and Serre.
Serre, J.-P. (1998). Abelian l-adic representations and elliptic curves, volume 7 of Res. Notes Math. Wellesley, MA:
Shorey, T. N. and Tijdeman, R. (1986). Exponential Diophantine equations, volume 87 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge.
Shorey, T. N., Van der Poorten, A. J., Tijdeman, R., and Schinzel, A. (1977). Applications of the Gel'fond- Baker method to Diophantine equations. pages 59-77.
Siegel, C. L. (2014). On some applications of Diophantine approximations, volume 2 of Quaderni/Monographs. Edizioni della Normale, Pisa. A translation of Carl Ludwig Siegels Uber einige Anwendungen diophantischer Approximationen by Clemens Fuchs, With a commentary and the article Integral points on curves: Siegel's theorem after Siegel's proof by Fuchs and Umberto Zannier, Edited by Zannier.
] Silverman, J. H. (2009). The arithmetic of elliptic curves, volume 106 of Grad. Texts Math. New York, NY: Springer, 2nd ed. edition.
Sprind?zuk, V. G. (1993). Classical Diophantine equations, volume 1559 of Lecture Notes in Mathematics. Springer-Verlag, Berlin. Translated from the 1982 Russian original, Translation edited by Ross Talent and Alf van der Poorten, With a foreword by van der Poorten.
Zannier, U. (2009). Lecture notes on Diophantine analysis, volume 8 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) Edizioni della Normale, Pisa. With an appendix by Francesco Amoroso.
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Compatible systems of l-adic representations on elliptic curves and algebraic tori (12 hours)
Hector Pasten and Valerio Talamanca
Program
This course will develop from scratch the theory of (compatible) l- adic representation which is one of the most crucial technical tool in Serre's proof.
1. Definition and some examples of l-adic representations. Frobenius elements and rational l-adic representation. Rational l-adic representations and their compatibility. Compatible and strictly compatible system of l-adic representations.
2. Preliminaries on algebraic groups: the notion of restriction of scalars. Construction of the groups T and S, via the idele class group and restriction of scalars. The canonical l-adic representation with values in S and its main
properties. The canonical l-adic representation with values in S form a strictly compatible system of l-adic representation with values in S.
3. Linear representations of S: rationality and properties of the images of Frobenius elements. l-adic representation associated to a linear representation of S: definition and main properties. The rational case: properties of the compatible system of l-adic representations associated to a linear representation of S.
4. Representations of the Galois group of the maximal abelian extension of K defined by a character of S and conversely how to define a character of S from a system of characters mod l. Application: proof that certain system of l-adic representations are abelian.
5. Locally algebraic representation: definition in the p-adic case. Locally algebraic representations are semi-simple if restricted to the inertia subgroup. Alternative characterisation of locally algebraic representation. Locally algebraic representation and field extensions. The global case definition and main properties. The modulus of a locally algebraic representation: proof of existence. The l-adic representation associated to a linear representation of S_m is a locally algebraic representation and of modulo m. Main theorem: every abelian rational locally algebraic representation of modulus m is the l-adic representation associated to a linear representation of S.
6. Construction of l-adic representation associated to an elliptic curves. Proof that is abelian, semi-simple, rational and locally algebraic.
References
J.P. Serre Abelian l-adic representations and elliptic curves. Advance book Classic. Addison-Wesley
J.P. Serre Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Inventiones mathematicae volume 15; pp. 259 - 331 (1971)
| Division polynomials and explicit computation of the images of representations with GP/PARI (2 hours)
Francesco Pappalardi
Program
This is a programming session devoted to explicit computations of the images of representations with GP/PARI
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Proof of Serre's theorem (6 hours)
René Schoof
Program
Give the proof (building on result proven in the other courses) of the following
Theorem. Let E be an elliptic curve over a number field F and suppose that for an infinite set of primes p the image of Gal(Fbar/F) acting on the p-torsion points of E is strictly smaller than GL2(Fp). Then E has CM
I. By extending F we reduce to the case that E has either good or split multiplicative reduction. Moreover, the primes p in S are unramified in F and have the property that det(im Gal(Fbar/F)) = Fp*
II. Using the list of finite subgroups of PGL2(Fp) we may assume that for infinitely many p, the image of Galois is in a Borel subgroup,the Normalizer(split/non-split Cartan) or is exceptional ( image in PGL2(Fp) is either A4, S4, A5)
III. Eliminate exceptional case by studying the inertia subgroup at p. This involves the formal group.
IV. By Safarevic, the isogeny class of E contains only finitely many isomorphism classes of curves. This is proved using Siegel?s theorem on integral points on elliptic cuvrves. It follows that in the Borel case $ has CM by a pigeon hole argument going back to Deuring.
V. Making an unramified quadratic extension of the base field, we may assume that for infinitely many primes p, the image of Gal(Fbar/F) acting on E[p] is contained in a non-split Cartan subgroup. More precisely, it is a cyclic subgroup of order p2 - 1.
VI. Show that there is a Hecke character whose mod p representations are for infinitely many p isomorphic to E[p]. Deduce that for all primes p the Qp representation coming from the elliptic curve, is coming from a Hecke character. Then show that there is another infinite set of primes for which the image of Gal(Fbar.F) acting on E[p] is contained in a split Cartan subgroup and hence in a Borel subgroup. So E has CM by the previous step.
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