An awful problem about integers in base four
In previous articles, we showed that for number fields in a certain large class, there are at most elliptic points on a Shimura curve of Γ₀(p)-type for every sufficiently large prime number p. In this article, we obtain an effective bound for such p.
We give an effective procedure to find minimal bases for ideals of the ring of polynomials over the integers.
Let be a hyperelliptic curve of genus over a number field with good reduction outside a finite set of places of . We prove that has a Weierstrass model over the ring of integers of with height effectively bounded only in terms of , and . In particular, we obtain that for any given number field , finite set of places of and integer one can in principle determine the set of -isomorphism classes of hyperelliptic curves over of genus with good reduction outside .
We prove some new effective results of André-Oort type. In particular, we state certain uniform improvements of the main result in [L. Kühne, Ann. of Math. 176 (2012), 651-671]. We also show that the equation X + Y = 1 has no solution in singular moduli. As a by-product, we indicate a simple trick rendering André's proof of the André-Oort conjecture effective. A significantly new aspect is the usage of both the Siegel-Tatuzawa theorem and the weak effective lower bound on the class number of an...