ETD

• Chabauty's method
• Galois representation
• Jacobian
• modular curve
• twist

This work was inspired by a paper of B. Poonen, E. Schaefer and M. Stoll, where the authors solve the Diophantine equation x^2 + y^3 = z^7.
Chapter 1 describes Chabauty’s method, which can be used to estimate the number of rational points on a curve C. We prove two theorems:
Theorem 1 (Chabauty) Let C/Q be a smooth projective curve of genus g. Let r be the rank of the Jacobian of C, and suppose that g > r. Then C(Q) is finite.
Theorem 2 (Chabauty-Coleman) Let C/Q be a smooth projective curve of genus g, let r be the rank of the Jacobian of C and suppose g > r. Let p > 2g be a prime of good reduction for C, and let C be the (smooth) curve over F_p obtained by reducing C modulo p. Then
|C(Q)| ≤ |C(F_p)| + 2g − 2.
While Chabauty’s method can be very useful for the problem of determining the rational points on a curve, it requires knowing the rank of the Jacobian of the curve, which is notoriously difficult to calculate. Chapter 2 is devoted to the problem of computing the rank of the Jacobian of a curve C defined over a number field K. More specifically, we describe a strategy to calculate the rank in the case where C is the smooth projective model of the affine curve with equation y^2 = f(x), where f ∈ K[x] is a separable, monic polynomial of odd degree. We then use this strategy to compute the rank of the Jacobian of C in the case where K = Q and f(x) = x(x−1)(x−2)(x−5)(x−6).
In Chapter 3 we first establish a relationship between the rational points on twists of curves, then we construct the modular curve X(7) as a moduli space of elliptic curves with a level-7 structure, and we conclude by studying how twisting X(7) affects the level-7 structure. Finally, in Chapter 4, we give a brief overview of the strategy used by Poonen, Schaefer and Stoll to solve the equation x^2 +y^3 = z^7.