Related papers: Markoff Surfaces and Strong Approximation: 1

We study finite dimensional representations of the projective modular group. Various explicit dimension formulas are given.

Based on the result on derived categories on K3 surfaces due to Mukai and Orlov and the result concerning almost-prime numbers due to Iwaniec, we remark the following fact: For any given positive integer N, there are N (mutually…

We obtain a lower bound for a number of primes in tuples. As applications, we obtain a lower bound for the Romanoff type representation functions.

We discuss the concepts of fine and coarse moduli spaces in the context of finite dimensional algebras over algebraically closed fields. In particular, our formulation of a moduli problem and its potential strong or weak solution is adapted…

In this paper, we call strongly modular those reducible semi-simple odd mod $l$ Galois representations for which the conclusion of the strongest form of Serre's original modularity conjecture holds. Under the assumption that the Serre…

We show that the "profinite direct sum" is a good notion of infinite direct sums for profinite modules having properties similar to direct sums of abstract modules. For example, the profinite direct sum of projective modules is projective,…

We discuss how Dokken's methods of approximate implicitization can be applied to triangular B\'ezier surfaces in both the original and weak forms. The matrices $\mathbf{D}$ and $\mathbf{M}$ that are fundamental to the respective forms of…

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between…

These notes grew out of our learning and applying the methods of Fock and Goncharov concerning moduli spaces of real projective structures on surfaces with ideal triangulations. We give a self-contained treatment of Fock and Goncharov's…

We introduce the notion of strong test module and show that a large number of such modules appear in the tight closure theory of complete domains: the test ideal (this has already been known), the parameter test module, and the module of…

We prove the existence of a finite set of moves sufficient to relate any two representations of the same 3-manifold as a 4-fold simple branched covering of S^3. We also prove a stabilization result: after adding a fifth trivial sheet two…

We consider the transcendental motive of three K3 surfaces $X$ conjectured to have complex multiplication (CM). Under this assumption, we match these to explicit algebraic Hecke quasi-characters $\psi_X$, and CM abelian threefolds $A$. This…

For a non-isotrivial family of surfaces of general type over a complex projective curve, we give upper bounds for the degree of the direct images of powers of the relative dualizing sheaf. They imply that, fixing the curve and the possible…

We continue our investigations of bilinear sums with modular square roots and the large sieve for square moduli in our recent article "On bilinear sums with modular square roots and applications II", arXiv:2603.00768. In the present…

We establish a result on the large sieve with square moduli. These bounds impro ve recent results by S. Baier(math.NT/0512228) and L. Zhao(math.NT/0508125).

We establish a sharp reciprocity inequality for modulus in compact metric spaces $X$ with finite Hausdorff measure. In particular, when $X$ is also homeomorphic to a planar rectangle, our result answers a question of K. Rajala and M.…

The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on…

We prove an analogue of a theorem of A. Pollington and S. Velani ('05), furnishing an upper bound on the Hausdorff dimension of certain subsets of the set of very well intrinsically approximable points on a quadratic hypersurface. The proof…

Following McShane, we employ the stable norm on the homology of the modular torus to investigate the Markov ordering on the set of relatively prime integer pairs $(q,p)$ with $q\ge p\ge0$. Our main theorem is a characterization of slopes…

In this paper we construct certain moduli spaces, which we call moduli spaces of (principal) $F$-bundles, and study their basic properties. These spaces are associated to triples consisting of a smooth projective geometrically connected…