Iterated Coleman integration for hyperelliptic curves

Signatures of paths transformed by polynomial maps

“…In stochastic analysis, the study of the signatures of paths arises in the theory of rough paths, where [FV10,FH14] are textbook references. Iterated integrals and the noncommutative series that encode them have also arisen in a variety of contexts in geometry and arithmetic, including the work of R. Hain in [Hai02], M. Kapranov in [Kap09], and J. Balakrishnan in [Bal13]. The results we derive in this paper have the potential for future applications in all of these contexts.…”

Coleman integration for even-degree models of hyperelliptic curves

“…With this observation in hand, we now state the generalization of our method to iterated integrals on hyperelliptic curves of even-degree model and refer the interested reader to [1] for the details in the odd-model case, which carry over directly to the even-model case.…”

“…The key theoretical input was Kedlaya's algorithm [12] to compute the action of Frobenius on Monsky-Washnitzer cohomology, formulated for odd-degree models of hyperelliptic curves. In a similar vein, we also gave algorithms to compute iterated Coleman integrals on hyperelliptic curves with odd-degree models [1] and used these algorithms to carry out the quadratic Chabauty method on hyperelliptic curves with odd-degree models [2].…”

“…Thus, in the current paper, building on Harrison's work, we extend the Coleman integration algorithms in [3] and [1] to even-degree models of hyperelliptic curves over unramified extensions of Q p and discuss what would be needed to carry out quadratic Chabauty for even-degree models.…”

Coleman–Gross height pairings and the p-adic sigma function

“…As a numerical check, one can use previous methods (as in [1,3]), to compute the invariant ratio correspond to the integral points W 0 1 D .4; 0/ and W 0 2 D .3; 0/, respectively, on E 0 . 0 ; Á 0 denote the pullbacks of !…”