We prove a pro-p Hom-form of the birational anabelian conjecture for function fields over sub-p-adic fields. Our starting point is the corresponding Theorem of Mochizuki in the case of transcendence degree 1.

Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d,r such that (diag(d)−A)r=0, where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag(d)−A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients 2n−1n−1, and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.

This paper studies groups of maximal size acting harmonically on a finite graph. Our main result states that these maximal graph groups are exactly the finite quotients of the modular group Γ=〈x,y∣x2=y3=1〉 of size at least 6. This characterization may be viewed as a discrete analogue of the description of Hurwitz groups as finite quotients of the (2,3,7)-triangle group in the context of holomorphic group actions on Riemann surfaces. In fact, as an immediate consequence of our result, every Hurwitz group is a maximal graph group, and the final section of the paper establishes a direct connection between maximal graphs and Hurwitz surfaces via the theory of combinatorial maps.

This paper studies the analogue of Hurwitz groups and surfaces in the context of harmonic group actions on finite graphs. Our main result states that maximal graph groups are exactly the finite quotients of the modular group $\Gamma= $ of size at least 6. As an immediate consequence, every Hurwitz group is a maximal graph group, and the final section of the paper establishes a direct connection between maximal graphs and Hurwitz surfaces via a construction due to Brooks and Makover.

This paper develops a harmonic Galois theory for finite graphs, thereby classifying harmonic branched G-covers of a fixed base X in terms of homomorphisms from a suitable fundamental group of X together with G-inertia structures on X. As applications, we show that finite embedding problems for graphs have proper solutions and prove a Grunwald Wang type result stating that an arbitrary collection of fibers may be realized by a global cover.

This paper develops graph analogues of the genus bounds for the maximal size of an automorphism group of a compact Riemann surface of genus $g\ge 2$. Inspired by the work of M. Baker and S. Norine on harmonic morphisms between finite graphs, we motivate and define the notion of a harmonic group action. Denoting by M(g) the maximal size of such a harmonic group action on a graph of genus $g\ge 2$, we prove that $4(g-1)\le M(g)\le 6(g-1)$, and these bounds are sharp in the sense that both are attained for infinitely many values of g. Moreover, we show that the values $4(g-1)$ and $6(g-1)$ are the only values taken by the function $M(g)$.

This paper investigates Galois branched covers of the open p-adic disc and their reductions to characteristic p. Using the field of norms functor of Fontaine and Wintenberger, we show that the special fiber of a Galois cover is determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. As applications, we derive a criterion for good reduction in the abelian case, and give an arithmetic reformulation of the local Oort Conjecture concerning the liftability of cyclic covers of germs of curves.

We prove a pro-$p$ Hom-form of the birational anabelian conjecture for function fields over sub-$p$-adic fields. Our starting point is the Theorem of Mochizuki in the case of transcendence degree 1.

We prove a Hom-form of the pro-p birational anabelian conjecture for function fields over sub-p-adic fields. Our starting point is the corresponding Theorem of Mochizuki in the case of transcendence degree 1.

ARITHMETIC AND GEOMETRY OF THE OPEN P -ADIC DISC Scott Corry Florian Pop, Advisor Motivated by the local lifting problem for Galois covers of curves, this thesis investigates Galois branched covers of the open p-adic disc. Our main result is that the special fiber of an abelian cover is completely determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. This determination of the special fiber in terms of characteristic zero data is accomplished via the field of norms functor of Fontaine and Wintenberger. As a consequence of our result, we derive a characteristic zero reformulation of the abelian local lifting problem, and as an application we give a new proof of the p-cyclic case of the Oort Conjecture, which states that cyclic covers should always lift.

This paper is a partial summary of the survey paper [1]. In particular, we are interested in telling the following story: given a lattice polytope, P , one would like to find an efficient way of counting the lattice points contained in P . One of the nicest ways to accomplish this is to use algebraic geometry in a clever and beautiful way. Namely, from P one can construct a toric variety, XP , together with a line bundle, LP , on XP . Then it turns out that the Euler characteristic χ(XP , LP ) is equal to the number of lattice points contained in P . Moreover, the Hirzebruch-Riemann-Roch theorem tells us how to calculate this Euler characteristic in terms of the Todd class of the toric variety XP . This yields an efficient method for counting the lattice points in P , because there is a polynomial time algorithm that computes the Todd class of XP given the polytope P .