Related papers: The Hilbert-Polya strategy and height pairings

In 1996 as part of the development of arithmetic intersection theory and Arakelov theory, Zhang defined a "GIT height function" for semi-stable algebraic cycles in projective space. In the same work, Zhang conjectured that this height…

We establish a connection between the ratios conjecture for the Riemann zeta-function and a conjecture concerning correlations of convolutions of M\"{o}bius and divisor functions. Specifically, we prove that the ratios conjecture and an…

Associated to classical semi-simple groups and their maximal parabolics are genuine zeta functions. Naturally related to Riemann's zeta and governed by symmetries, including that of Weyl, these zetas are expected to satisfy the Riemann…

In this paper we give some evidence for the Tate (and Hodge) conjecture(s) for a class of Hilbert modular fourfolds X, whose connected components arise as arithmetic quotients of the fourfold product of the upper half plane by congruence…

We prove a conjecture of Shklyarov concerning the relationship between K. Saito's higher residue pairing and a certain pairing on the periodic cyclic homology of matrix factorization categories. Along the way, we give new proofs of a result…

We attach a mixed Hodge structure and associate two versions of heights to a pair of Bloch higher cycles. Both these heights generalize the biextension height attached to a pair of classical algebraic cycles homologous to zero. We also…

This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelov-theoretic version of motivic cohomology is the conjecture on special values of $L$-functions and zeta…

In this talk I give an introduction and present some recent progress towards understanding the cohomology rings of character varieties of Riemann surfaces, such as the proof of the $P=W$ conjecture and the computation of the…

The purpose of this work is to generalize, in the context of 1-motives, the $p$-adic height pairings constructed by B. Mazur and J. Tate on abelian varieties. Following their approach, we define a global pairing between the rational points…

We discuss a formal derivation of an integral expression for the Li coefficients associated with the Riemann xi-function which, in particular, indicates that their positivity criterion is obeyed, whereby entailing the criticality of the…

In this paper, we derive a more precise version of the Strong Pair Correlation Conjecture on the zeros of the Riemann zeta function under Riemann Hypothesis and Twin Prime Conjecture.

We define the Archimedean height pairing for fiberwise cohomologically trivial differential forms on a one-parameter degeneration of Riemann surfaces, and we study its asymptotic behavior. The proof relies on recent work by Dai--Yoshikawa…

Making use of topological periodic cyclic homology, we extend Grothendieck's standard conjectures of type C and D (with respect to crystalline cohomology theory) from smooth projective schemes to smooth proper dg categories in the sense of…

We discuss about the conjectural cohomological theory of dynamical zeta functions in the case of general Anosov flows. Our aim is to provide a functional-analytic framework that enables us to justify the basic part of the theory rigorously.…

We conjecture an expression for the dimensions of the Khovanov-Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the…

We introduce motivic zeta functions for matroids. These zeta functions are defined as sums over the lattice points of Bergman fans, and in the realizable case, they coincide with the motivic Igusa zeta functions of hyperplane arrangements.…

We show that the ample degree of a stable theory with trivial forking is preserved when we consider the corresponding theory of belles paires, if it exists. This result also applies to the theory of $H$-structures of a trivial theory of…

The Hilbert-Polya conjecture states that the imaginary parts of the zeros of the Riemann zeta function are eigenvalues of a quantum hamiltonian. If so, conjectures by Katz and Sarnak put this hamiltonian in Altland and Zirnbauer's…

The statement of the Riemann hypothesis makes sense for all global fields, not just the rational numbers. For function fields, it has a natural restatement in terms of the associated curve. Weil's work on the Riemann hypothesis for curves…

For elliptic curves over rationals, there are a well-known conjecture of Sato-Tate and a new computational guided murmuration phenomenon, for which the abelian Hasse-Weil zeta functions are used. In this paper, we show that both the…