An explicit bound for integral points on modular curves
نویسندگان
چکیده
In this paper, we give a constant $C$ in \cite[Theorem 1.2]{sha2014bounding} by using an explicit Baker's inequality, hence have bound of the integral points on modular curves.
منابع مشابه
Explicit methods for rational points on curves
One of the “big problems” of number theory is to understand the set of rational points on a variety, or equivalently, the rational solutions to a system of polynomial equations. Despite thousands of years of research, we are very far from having a general method for solving all such problems. There is even some evidence that deciding the existence of a rational solution is an undecidable proble...
متن کاملIntegral Points on Hyperelliptic Curves
Let C : Y 2 = anX + · · · + a0 be a hyperelliptic curve with the ai rational integers, n ≥ 5, and the polynomial on the right irreducible. Let J be its Jacobian. We give a completely explicit upper bound for the integral points on the model C, provided we know at least one rational point on C and a Mordell–Weil basis for J(Q). We also explain a powerful refinement of the Mordell–Weil sieve whic...
متن کاملTorsion Points on Modular Curves
Let N ≥ 23 be a prime number. In this paper, we prove a conjecture of Coleman, Kaskel, and Ribet about the Q-valued points of the modular curve X0(N) which map to torsion points on J0(N) via the cuspidal embedding. We give some generalizations to other modular curves, and to noncuspidal embeddings of X0(N) into J0(N).
متن کاملComputing S-Integral Points on Elliptic Curves
1. Introduction. By a famous theorem of Siegel [S], the number of integral points on an elliptic curve E over an algebraic number field K is finite. A conjecture of Lang and Demyanenko (see [L3], p. 140) states that, for a quasiminimal model of E over K, this number is bounded by a constant depending only on the rank of E over K and on K (see also [HSi], [Zi4]). This conjecture was proved by Si...
متن کاملIntegral points on congruent number curves
We provide a precise description of the integer points on elliptic curves of the shape y2 = x3 − N2x, where N = 2apb for prime p. By way of example, if p ≡ ±3 (mod 8) and p > 29, we show that all such points necessarily have y = 0. Our proofs rely upon lower bounds for linear forms in logarithms, a variety of old and new results on quartic and other Diophantine equations, and a large amount of ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Communications in Mathematics
سال: 2022
ISSN: ['2336-1298', '1804-1388']
DOI: https://doi.org/10.46298/cm.9389