Integral points on moduli schemes of elliptic curves

Abstract

We combine the method of Faltings (Arakelov, Paršin, Szpiro) with the Shimura–Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems, such as for example S‐unit and Mordell equations, this gives an effective method which does not rely on Diophantine approximation or transcendence techniques.

The effective Shafarevich conjecture for abelian varieties of -type

“…Further, we point out that the above isogeny estimates are independent of and . As already mentioned, this is absolutely crucial for Theorem 6.6 and for certain Diophantine applications such as, for example, [40,41]. On calculating the constant of Lemma 3.1 (i) explicitly, we see that the bound in Corollary 6.5 (ii) is better in terms of and than Lemma 3.1 (i).…”

“…This fully explicit Diophantine inequality establishes Conjecture ( ) for abelian schemes of product GL 2 -type and it proves new cases of the effective Shafarevich conjecture for curves (see Corollary 6.4). In addition, Theorem A leads to Corollary 6.5, giving new isogeny estimates for abelian schemes over of product GL 2 -type: Our isogeny estimates are uniform in the sense that they only depend on and , which is crucial for Theorem B and for the Diophantine applications in [40]. In Theorem A it is important that we do not make any semistable assumption or assume that is simple, because these assumptions would be too restrictive for many Diophantine applications of interest.…”

“…Now, our proof of Conjecture ( ) for GL 2 , ( ) with ≥ 2 opens the way for the effective study of some classes of Diophantine equations that appear to be beyond the reach of the known effective methods. For instance, Theorems A and B are the main tools of the joint project with Kret [40,41]: Therein we combine the results of the present article with suitable Paršin constructions in order to prove explicit upper bounds for the height and the number of -integral points on Hilbert modular varieties. To this end, if is not too large, say ≤ 100, then our explicit height bounds in Theorem A are in fact already sufficiently strong for practical computations when combined with efficient sieves (see [42]) that in practice can deal with huge initial bounds.…”

“…For example, if is a curve over , then Conjecture ( ) would also allow controlling the finite places of where the reductions of and Pic 0 ( ) are different; see the discussion at the end of Section 6 for more details. Furthermore, it is shown in [40] that already special cases of Conjecture ( ), such as, for example, Theorem 6.2, have direct applications to the effective study of Diophantine equations. On the other hand, one needs to prove Conjecture ( ) * in quite general situations to get effective Diophantine applications.…”

“…If → ∞, then ℎ ( ) ≤ The bound in (ii) and the estimate (6.9), which we obtained in a more precise form in (6.10), both asymptotically improve the bound in Theorem 5.1 (ii). On the other hand, Theorem 5.1 (ii) makes no additional semistable or real multiplication assumptions, and this is crucial for many Diophantine applications (e.g., [40]). In view of this, it would be interesting to remove in Proposition 6.9 the semistable and real multiplication assumptions.…”