We give algorithms to compute Coleman integrals on superelliptic curves over unramified extensions of the p-adics, and apply these via Chabauty methods to determine the set of rational points on such curves. We also determine the solution to an explicit instance of the Shafarevich conjecture by finding all elliptic curves with good reduction outside of the first 6 primes, subject to a heuristic. We use a combination of non-abelian Chabauty and the Mordell--Weil sieve to determine the rational points on several quotient modular curves, and therefore classify pairs of elliptic curves over the rationals with 67-, 73-, and 107-isogenies. We give methods to explicitly compute Coleman integrals on modular curves using a canonical lift of Frobenius and canonical local coordinates in each residue disk, and discuss the problem of computing the Weil pairing in finite rings.

abstract: Diophantine arithmetic is one of the oldest branches of mathematics, the search for integer or rational solutions of algebraic equations. Pythagorean triangles are an early instance. Diophantus of Alexandria wrote the first related treatise in the fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat. The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of elliptic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Diophantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations $y^2=x^6+k$, $k=-39,\,-47$, the two previously unsolved cases for $|k|<50$, are solved using algebraic number theory and the ‘elliptic Chabauty’ method. The thesis also studies the genus three quartic curves $F(x^2,y^2,z^2)=0$ where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals. The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form $n=(x+y+z+w)(1/x+1/y+1/z+1/w).$ Further, an example, the first such known, of a quartic surface $x^4+7y^4=14z^4+18w^4$ is given with remarkable properties: it is everywhere locally solvable, yet has no non-zero rational point, despite having a point in (non-trivial) odd-degree extension fields of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

Let X be an Atkin-Lehner quotient of the modular curve X_0(N) whose Jacobian J_f is a simple quotient of J_0(N)^{new} over Q. We give analytic methods for determining the rational points of X using quadratic Chabauty by explicitly computing two p-adic Gross--Zagier formulas for the newform f of level N and weight 2 associated with J_f when f has analytic rank 1. Combining results of Gross-Zagier and Waldspurger, one knows that for certain imaginary quadratic fields K, there exists a Heegner divisor in J_0(N)(K) whose image is finite index in J_f(Q) under the action of Hecke. We give an algorithm to compute the special value of the anticyclotomic p-adic L-function of f constructed by Bertolini, Darmon, and Prasanna, assuming some hypotheses on the prime p and on K. This value is proportional to the logarithm of the Heegner divisor on J_f with respect to the differential form f dq/q. We also compute the p-adic height of the Heegner divisor on J_f using a p-adic Gross-Zagier formula of Perrin-Riou. Additionally, we give algorithms for the geometric quadratic Chabauty method of Edixhoven and Lido. Our algorithms describe how to translate their algebro-geometric method into calculations involving Coleman-Gross heights, logarithms, and divisor arithmetic. We achieve this by leveraging a map from the Poincaré biextension to the trivial biextension.

Dans cette thèse, nous nous intéressons à des compactifications géométriques variées. Nous décrivons l'espace des sous-groupes fermés du groupe RxZ. Nous étudions la compactification de Chabauty des espaces symétriques de type non compact. Nous définissons et étudions la compactification de Chabauty de l'espace des plats maximaux des espaces symétriques de SL3(R) et de SL4(R). Nous étudions les limites géométriques de plats maximaux de l'espace symétrique ou de l'immeuble de Bruhat-Tits associé à SL3 sur un corps local. Nous définissons et étudions une compactification à la Thurston des espaces de classes d'isométrie de réseaux marqués. Nous définissons une compactification à la Thurston de l'espace de Torelli d'une surface et nous décrivons la stratification naturelle d'une partie de son bord. / In our thesis, we focus on various geometric compactifications. We describe the space of closed subgroups of RxZ. We study the Chabauty compactification of symmetric spaces of non-compact type. We define and study the Chabauty compactification of the space of maximal flats of the symmetric spaces of SL3(R) and SL4(R). We study the geometric limits of maximal flats in the symmetric space or in the Bruhat-Tits building associated to SL3 over a local field. We define and study a Thurston-like compactification of spaces of isometry classes of marked lattices. We define a Thurston-like compactification of the Torelli space of a surface and we describe the natural stratification of a subset of the boundary.

Dans cette thèse, nous nous intéressons à des compactifications géométriques variées. Nous décrivons l'espace des sous-groupes fermés du groupe RxZ. Nous étudions la compactification de Chabauty des espaces symétriques de type non compact. Nous définissons et étudions la compactification de Chabauty de l'espace des plats maximaux des espaces symétriques de SL3(R) et de SL4(R). Nous étudions les limites géométriques de plats maximaux de l'espace symétrique ou de l'immeuble de Bruhat-Tits associé à SL3 sur un corps local. Nous définissons et étudions une compactification à la Thurston des espaces de classes d'isométrie de réseaux marqués. Nous définissons une compactification à la Thurston de l'espace de Torelli d'une surface et nous décrivons la stratification naturelle d'une partie de son bord.

No description available.

Ce mémoire présente des résultats dans trois directions. En géométrie riemannienne, on montre une généralisation de l'inégalité de Günther sur le volume, et en dimension 4 une inégalité isopérimétrique pour les variétés à courbure majorée. En géométrie des espaces de Wasserstein, issus du transport optimal, on montre des résultats plongement et de non-plongement, on calcule des groupes d'isométries, et on étudie la dynamique de l'action sur les mesures des applications dilatantes du cercle. En topologie de Chabauty, on montre que l'espace des sous-groupes fermés de $R^n$ est simplement connexe.