The rigid analytical regulator and K_2 of Drinfeld modular curves

We evaluate a rigid analytical analogue of the Beilinson-Bloch-Deligne regulator on certain explicit elements in the K_2 of Drinfeld modular curves, constructed from analogues of modular units, and relate its value to special values of L-series using the Rankin-Selberg method.Comment: 38 pages, to appear in Publ. Res. Inst. Math. Sc

On the kernel and the image of the rigid analytic regulator in positive characteristic

We will formulate and prove a certain reciprocity law relating certain residues of the differential symbol dlog^2 from the K_2 of a Mumford curve to the rigid analytic regulator constructed by the author in a previous paper. We will use this result to deduce some consequences on the kernel and image of the rigid analytic regulator analogous to some old conjectures of Beilinson and Bloch on the complex analytic regulator. We also relate our construction to the symbol defined by Contou-Carrere and to Kato's residue homomorphism, and we show that Weil's reciprocity law directly implies the reciprocity law of Anderson and Romo.Comment: 28 pages, to appear in Publ. Res. Inst. Math. Sc

The pp-adic monodromy group of abelian varieties over global function fields of characteristic pp

We prove an analogue of the Tate isogeny conjecture and the semi-simplicity conjecture for overconvergent crystalline Dieudonn\'e modules of abelian varieties defined over global function fields of characteristic $p$. As a corollary we deduce that monodromy groups of such overconvergent crystalline Dieudonn\'e modules are reductive, and after a finite base change of coefficients their connected components are the same as the connected components of monodromy groups of Galois representations on the corresponding $l$-adic Tate modules, for $l$ different from $p$. We also show such a result for general compatible systems incorporating overconvergent $F$-isocrystals, conditional on a result of Abe.Comment: 56 pages, comments welcome

The Brauer-Manin obstruction to the local-global principle for the embedding problem

We study an analogue of the Brauer-Manin obstruction to the local-global principle for embedding problems over global fields. We will prove the analogues of several fundamental structural results. In particular we show that the (algebraic) Brauer-Manin obstruction is the only one to weak approximation when the embedding problem has abelian kernel. As a part of our investigations we also give a new, elegant description of the Tate duality pairing and prove a new theorem on the cup product.Comment: Referenced upgraded. 37 page

On the nilpotent section conjecture for finite group actions on curves

We give a new, geometric proof of the section conjecture for fixed points of finite group actions on projective curves of positive genus defined over the field of complex numbers, as well as its natural nilpotent analogue. As a part of our investigations we give an explicit description of the abelianised section map for groups of prime order in this setting. We also show a version of the 2-nilpotent section conjecture.Comment: Mathematika, to appear. 16 page

Applied arithmetic geometry

Paper presented at the 5th Strathmore International Mathematics Conference (SIMC 2019), 12 - 16 August 2019, Strathmore University, Nairobi, Kenya.The aim of this lecture series is to introduce some methods of arithmetic geometry which are applied in cryptographic research. Cryptography, including more sophisticated versions such as elliptic curve cryptography, allows for efficient protocols for information security, and is widely used in the banking sector including mobile money transfers, an industry in which Africa is a world leader. The methods presented can be used by African research groups to tackle a range of problems arising in technological challenges relevant to the African development context. Arithmetic geometry is a rather modern, highly prestigious and very developed area of pure mathematics, developed originally for studying Diophantine equations. I has very efficient methods to count points on algebraic varieties over finite fields which is closely related to the original motivating problem of finding rational points on varieties over number fields, a geometric reformulation of Diophantine equations. The former problem is very important in cryptography and related areas of secure communication, network building and hash functions. The lecture series will cover the necessary background on cryptography and point counting, and will introduce such tools as p-adic numbers, differential forms and Monsky-Washnitzer cohomology, from the ground up