Related papers: A relation between standard conjectures and their …

A cyclic proof system gives us another way of representing inductive definitions and efficient proof search. In 2011 Brotherston and Simpson conjectured the equivalence between the provability of the classical cyclic proof system and that…

In algebraic geometry there is the notion of a height pairing of algebraic cycles, which lies at the confluence of arithmetic, Hodge theory and topology. After explaining a motivating example situation, we introduce new directions in this…

For families of smooth complex projective varieties we show that normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. In particular, the zero loci of those…

In this paper we are concerned with a long-standing conjecture of Huneke and Wiegand. We introduce a new class of ideals and prove thateach ideal from such class satisfies the conclusion of the conjecture in question. We also study the…

We formulate a "correct" version of the Quillen conjecture on linear group homology for certain arithmetic rings and provide evidence for the new conjecture. In this way we predict that the linear group homology has a direct summand looking…

We introduce `canonical' classes in the Selmer groups of certain Galois representations with a conjugate-symplectic symmetry. They are images of special cycles in unitary Shimura varieties, and defined uniquely up to a scalar. The…

We extend to the topological setting the classical constructions of the Abel-Jacobi mapping on homologically trivial algebraic cycles and the height pairing between two such cycles. We further interpret the height pairing between…

The Colmez conjecture, proposed by Colmez, is a conjecture expressing the Faltings height of a CM abelian variety in terms of some linear combination of logarithmic derivatives of Artin L-functions. The aim of this paper to prove an…

The cyclicity and Koblitz conjectures ask about the distribution of primes of cyclic and prime-order reduction, respectively, for elliptic curves over $\mathbb{Q}$. In 1976, Serre gave a conditional proof of the cyclicity conjecture, but…

In this paper, we prove the following conjecture proposed by Gould, Hirohata and Keller [Discrete Math. submitted]: Let $G$ be a graph of sufficiently large order. If $\sigma_t(G) \geq 2kt - t + 1$ for any two integers $k \geq 2$ and $t…

Using morphic cohomology, we produce a sequence of conjectures, called morphic conjectures, which terminates at the Grothendieck standard conjecture A. A refinement of Hodge structures is given, and with the assumption of morphic…

We propose analogs of the classical Generalized Riemann Hypothesis and the Generalized Simplicity Conjecture for the characteristic p L-series associated to function fields over a finite field. These analogs are based on the use of absolute…

Heegner cycles are higher weight analogues of Heegner points. Their arithmetic intersection numbers also appear as Fourier coefficients of modular forms and often belong to abelian extensions of imaginary-quadratic fields. Rotger and Seveso…

The concept of the cyclic averages are introduced for a regular polygon $P_n$ and a Platonic solid $T_n$. It is shown that cyclic averages of equal powers are the same for various $P_n(T_n)$, but their number is characteristic of…

The middle levels conjecture asserts that there is a Hamiltonian cycle in the middle two levels of $2k+1$-dimensional hypercube. The conjecture is known to be true for $k \leq 17$ [I.Shields, B.J.Shields and C.D.Savage, Disc. Math., 309,…

We first propose a generalization of the image conjecture [Z3] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent…

We develop a theory of abstract arithmetic Chow rings where the role of the fibers at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. This theory allows the construction of many variants of the…

We prove some new cases of the Grothendieck-Serre conjecture for classical groups. This is based on a new construction of the Gersten-Witt complex for Witt groups of Azumaya algebras with involution on regular semilocal rings, with explicit…

Two new sufficient conditions for generalized cycles (including Hamilton and dominating cycles as special cases) in an arbitrary k-connected graph (k=1,2,...) are derived, which prove the truth of Bondy's (1980) famous conjecture for some…

The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of ``elementary transformations" which are Nielsen transformations augmented by arbitrary…