Related papers: Effectivity in Mochizuki's work on the $abc$-conje…

In this paper we give a detailed proof of a result we announced a year ago. This result is an effective version of the theorem of Mazur-Kamienny-Merel concerning uniform bounds for rational torsion points on elliptic curves over number…

Based on many experts' former work in the Jacobian conjecture and an essential analysis of intrinsic topology of linear maps, I completely prove the Jacobian conjecture by demonstrating the injectivity of real Keller map of any…

Condensed mathematics, developed by Clausen and Scholze over the last few years, is a new way of studying the interplay between algebra and geometry. It replaces the concept of a topological space by a more sophisticated but better-behaved…

Let $f \colon X \to B$ be a complex elliptic surface and let $\DD \subset X$ be an integral divisor dominating $B$. It is well-known that the Parshin-Arakelov theorem implies the Mordell conjecture over complex function fields by a…

There are three aims of this note. The first one is to report some advances around the dynamical Mordell-Lang (=DML) conjecture. Second, we generalize some known results. For example, the Dynamical Mordell-lang conjecture was known for…

Following a line of research initiated in \cite{BBNN}, I describe a general framework for turning reduction concepts of relative computability into diagrams forming an analogy with the Cicho\'n diagram for cardinal characteristics of the…

We obtain an effective version of Matsusaka's theorem for arbitrary smooth algebraic surfaces in positive characteristic, which provides an effective bound on the multiple which makes an ample line bundle D very ample. The proof for…

At a prime of ordinary reduction, the Iwasawa ``main conjecture'' for elliptic curves relates a Selmer group to a $p$-adic $L$-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the…

It is proved that, if $K$ is a complete discrete valuation field of mixed characteristic $(0,p)$ with residue field satisfying a mild condition, then any abelian variety over $K$ with potentially good reduction has finite…

This paper develops further the theory of the automorphic group of non-constant entire functions. This theory essentially started with two remarkable papers of Tatsujir\^o Shimizu that were published in 1931. There are three results in this…

Suppose a sequence $M_j$ of Alexandrov spaces collapses to a space $X$ with only weak singularities. Yamaguchi constructed a map $f_j:M_j\to X$ called an almost Lipschitz submersion for large $j$. We prove that if $M_j$ has a uniform…

We give a novel and effective criterion for algebraicity of rational normal analytic surfaces constructed from resolving the singularity of an irreducible curve-germ on $CP^2$ and contracting the strict transform of a given line and all but…

We prove the following special case of Mazur's conjecture on the topology of rational points. Let $E$ be an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$. For a class of elliptic pencils which are quadratic twists of $E$ by…

We prove that the $abc$-Conjecture implies upper bounds on Zsigmondy sets that are uniform over families of unicritical polynomials over number fields. As an application, we use the $abc$-Conjecture to prove that there exist uniform bounds…

We express the kernel of Griffiths' Abel-Jacobi map by using the inductive limit of Deligne cohomology in the generalized sense (i.e. the absolute Hodge cohomology of A. Beilinson). This generalizes a result of L. Barbieri-Viale and V.…

In this paper, we generalize the existence result in [14] and prove convergence theorems of the iterative scheme in [12, 16] for monotone generalized alpa-nonexpansive mappings in uniformly convex partially ordered hyperbolic metric spaces.…

We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the L\"uck Approximation Theorem…

This work is motivated by two central questions in the birational geometry of moduli spaces of curves -- Fulton's conjecture and the effective cone of $\bar M_g$. We study the algebro-geometric aspect of Teichmuller curves parameterizing…

We show that an earlier conjecture of the author, on diophantine approximation of rational points on varieties, implies the ``abc conjecture'' of Masser and Oesterl'e. In fact, a weak form of the former conjecture is sufficient, involving…

We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has…