Calcul effectif de points spéciaux

Antonin Riffaut 1, 2
2 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : Starting for André’s Theorem in 1998, which is the first non-trivial contribution to the celebrated André-Oort conjecture on the special subvarieties of Shimura varieties, the main purpose of this thesis is to study Diophantine properties of singular moduli, by characterizing CM-points (x; y) satisfying a given type of equation of the form F(x; y) = 0, for an irreducible polynomial F(X; Y ) with complex coefficients. More specifically, we treat two different equations involving powers of singular moduli. On the one hand, we show that two distinct singular moduli x; y such that the numbers 1, xm and yn are linearly dependent over Q, for some positive integers m; n, must be of degree at most 2. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in C2 defined over Q. On the other hand, we show that, with “obvious” exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to hyperbolas xy = A, where A 2 Qx. The methods we develop lie mainly on the properties of ring class fields generated by singular moduli, on estimations of the j-function and on estimations of linear forms in logarithms. We also determine fields generated by sums and products of two singular moduli x and y : we show that the field Q(x; y) is generated by the sum x + y, unless x and y are conjugate over Q, in which case x + y generate a subfield of degree at most 2 ; the same holds for the product xy. Our proofs are assisted by the PARI/GP package, which we use to proceed to verifications in particular explicit cases.
Document type :
Theses
Complete list of metadatas

Cited literature [40 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-01931307
Contributor : Abes Star <>
Submitted on : Thursday, November 22, 2018 - 4:24:06 PM
Last modification on : Wednesday, January 23, 2019 - 1:20:11 AM
Long-term archiving on: Saturday, February 23, 2019 - 3:35:19 PM

File

RIFFAUT_ANTONIN_2018.pdf
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-01931307, version 1

Collections

Citation

Antonin Riffaut. Calcul effectif de points spéciaux. Mathématiques générales [math.GM]. Université de Bordeaux, 2018. Français. ⟨NNT : 2018BORD0100⟩. ⟨tel-01931307⟩

Share

Metrics

Record views

153

Files downloads

99