An effective result of André–Oort type II

Let X ⊂ Y (1) n be a subvariety defined over a number field F and let (P 1 , . . . , Pn) ∈ X be a special point not contained in a positive-dimensional special subvariety of X. We show that the if a coordinate P i corresponds to an order not contained in a single exceptional Siegel-Tatuzawa imaginary quadratic field K * then the associated discriminant |∆(P i )| is bounded by an effective constant depending only on deg X and [F : Q]. We derive analogous effective results for the positive-dimensional maximal special subvarieties.From the main theorem we deduce various effective results of André-Oort type. In particular we define a genericity condition on the leading homogeneous part of a polynomial, and give a fully effective André-Oort statement for hypersurfaces defined by polynomials satisfying this condition.

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“…Proof of Corollary 1. The proof is similar to that of [17,Corollary 4]. We reduce to the case F X = Q as above.…”

Section: 2mentioning

confidence: 79%

Some effective estimates for André–Oort in Y(1)ⁿ

Binyamini

Kowalski

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…For all details, see, for instance, [6, §7 and §11] Starting from the ground-breaking article of André [2] equations involving singular moduli were studied by many authors, see [1,4,8] for a historical account and further references. In particular, Kühne [7] proved that equation x + y = 1 has no solutions in singular moduli x and y, and Bilu et al [5] proved the same for the equation xy = 1. These results where generalized in [1] and [4].…”

Section: Introductionmentioning

confidence: 95%

“…We have xy σ −1 = x σ −1 y with both x σ −1 and y not dominant. Arguing as in Subsection 5.1.1, we bound |xy σ −1 | from below with the righthand side of (7), and |x σ −1 y| from above by the right-hand side of (8), again arriving to a contradiction when |∆| ≥ 95. Thus, in any case we have |∆| ≤ 395, and for these small values of |∆| it can be verified that G = H using PARI.…”

Section: The Case ε = −1mentioning

confidence: 99%

Fields generated by sums and products of singular moduli

Faye

Riffaut

2018

Journal of Number Theory

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We show that the field Q(x, y), generated by two singular moduli x and y, is generated by their sum x + y, unless x and y are conjugate over Q, in which case x + y generates a subfield of degree at most 2. We obtain a similar result for the product of two singular moduli. CDC, the IRN GandA (CNRS) and the ALGANT Program. Our calculations were performed using the PARI/GP package [9]. The sources are available from the second author.

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“…Up to now, all non-trivial effectively solvable cases ( [1,2,3,11,12]) of the André-Oort conjecture have been restricted to the case of curves. In particular, the only known examples of algebraic subvarieties in Shimura varieties that are known to contain no special points are either curves or weakly special subvarieties.…”

Section: Introductionmentioning

confidence: 99%