Let K be a complete subfield of Cp. Consider a rigid space X over K with good reduction and a differential of the second kind w over X. Coleman theory of p-adic integration tells us how to give a meaning to an expression of the form:
integral from P to Q of w,        P,Q in X(K)-(w)_poles
The work of Coleman relies on using the Dwork principle of continuation along Frobenius to overcome the topological problems coming from the ultrametric nature of K.
Between 2006 and 2011, K.S. Kedlaya and J. Balakrishnan have constructed algorithms to compute explicitly Coleman’s integrals on hyperelliptic curves and, together with R. Bradshaw, they have implemented these methods in SAGE.
In this thesis, I study the theory of Coleman both from the theoretical and the algorithmic point of view and I provide the results of some explicit computations.
After a review of some fundamental ideas in rigid geometry, I present the theory of Coleman as it appears in his original articles. The second part of this work is devoted to the computational approach: I describe the ideas of Kedlaya and Balakrishnan and I produce
some concrete examples. Finally, the last Chapter deals with one application of Coleman’s integrals: I study the method of Chabauty and Coleman and I show how it can be used to effectively detect rational points on curves.