Related papers: On higher regulators of Picard modular surfaces

We construct some integral elements in the motivic cohomology of the Hesse cubic curves and express their regulators in terms of generalized hypergeometric functions and Kamp\'e de F\'eriet hypergeometric functions. By using these…

We survey over some recent applications of motivic homotopy theory in the definition and the study of $p$-adic cohomology theories. In particular, we revisit the proof of the $p$-adic weight-monodromy conjecture for smooth projective…

We prove Zagier's conjecture on the value at s=4 of the Dedekind zeta-function of a number field F. For any field F, we define a map from the appropriate pieces of algebraic K-theory of F to the cohomology of the weight 4 polylogarithmic…

Let $\mathcal{L}=(L,[\cdot\,,\cdot],\delta)$ be an algebraic Lie algebroid over a smooth projective curve $X$ of genus $g\geq 2$ such that $L$ is a line bundle whose degree is less than $2-2g$. Let $r$ and $d$ be coprime numbers. We prove…

This work builds on earlier work of the first three authors where a notion of congruence modules in higher codimension is introduced. The main new results are a criterion for detecting regularity of local rings in terms of congruence…

We compute Benois $\mathscr{L}$-invariants of weight $1$ cuspforms and of their adjoint representations and show how this extends Gross' $p$-adic regulator to Artin motives which are not critical in the sense of Deligne. Benois'…

Using the $L^2$ norm of the Higgs field as a Morse function, we study the moduli spaces of $U(p,q)$-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on $(p,q)$. A key…

We investigate Cousin (bi-)complexes in the setting of motives. Over essentially smooth local schemes, the columns of the Cousin bicomplex with coefficients in any stable motivic homotopy type are shown to be acyclic. On the other hand, we…

We explain how some results of M. Nori (on motives) and F. Ivorra (on perverse motives) can be used to define "motivic" versions of Lyubeznik numbers, a set of numerical invariants for local rings. We also discuss on additional structures…

We present a general and effective algebraic framework for enumerating finite-length quotients of a torsion-free sheaf of arbitrary rank (the Quot zeta function) and finite-length coherent sheaves (the Coh zeta function) over reduced…

We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with…

Multiple polylogarithms are equipped with rich algebraic structures including the motivic coaction and the single-valued map which both found fruitful applications in high-energy physics. In recent work arXiv:2312.00697, the current authors…

In this paper, we construct and study a Serre-type spectral sequence for motivic cohomology associated to a map of bisimplicial schemes with motivically cellular fiber. Then, we show how to apply it in order to approach the computation of…

We determine the class of the Hilbert scheme of points on a surface in the Grothendieck group of varieties. As a corollary we obtain its class in the Grothendieck group of motives. We give some applications to moduli spaces of sheaves on…

We construct motivic cohomology cycles in the group $H^3_{\mathcal M}(Z,{\mathbb Q}(2))$ where $Z$ is a K3 surface obtained as a double cover of a del Pezzo surface $X$ branched at a curve in $|-2K_X|$. The construction uses (-1) curves on…

Thanks to Hrushovski-Loeser's work on motivic Milnor fibers, we give a model-theoretic proof for the motivic Thom-Sebastiani theorem in the case of regular functions. Moreover, slightly extending of Hrushovski-Loeser's construction adjusted…

We construct the Weil restriction map for l-adic cohomology and, more generally, for mixed Weil cohomology theories. We study its compatibility with the motivic cycle class map and show that these constructions admit a natural…

We have recently proved a homological stability theorem for moduli spaces of r-Spin Riemann surfaces, which in particular implies a Madsen--Weiss theorem for these moduli spaces. This allows us to effectively study their stable cohomology,…

We study the Picard groups of moduli spaces in positive characteristics and we give a "$p$-adic" proof that the Picard group of moduli of vector bundles of fixed determinant is isomorphic to the group of integers. Along the way we prove…

In this article we study the subgroup of the Picard group of Voevodsky's category of geometric motives generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann, but we also provide an alternative…