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We describe recent work connecting combinatorics and tropical/non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of Chabauty--Coleman lies at the heart of this connection, and we emphasize the clarification that tropical geometry affords throughout the theory of $p$-adic integration, especially to the comparison of analytic continuations of $p$-adic integrals and to the analysis of zeros of integrals on domains admitting monodromy.
We provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the curve.
In 1922, Mordell conjectured the striking statement that for a polynomial equation $f(x,y)=0$, if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was proved by Falti
Let $X$ be a curve of genus $ggeq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic geometry, known
We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manins conjecture for a cubic surface whose singularity type is A_5+A_1.
Let $k$ be a number field, let $X$ be a Kummer variety over $k$, and let $delta$ be an odd integer. In the spirit of a result by Yongqi Liang, we relate the arithmetic of rational points over finite extensions of $k$ to that of zero-cycles over $k$ f