Related papers: Meromorphic modular forms with rational cycle inte…
We give a new method for the evaluation of a class of integrals of rational symmetric functions in N pairs of variables {x_a, y_a}_{a=1,... N} arising in coupled matrix models, valid for a broad class of two-variable measures. The result is…
We consider a class of differential equations for multi-loop Feynman integrals which can be solved to all orders in dimensional regularisation in terms of iterated integrals of meromorphic modular forms. We show that the subgroup under…
We introduce a simple procedure to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. It gives rise to an intrinsic regularization of such singular integrals in terms of the underlying conformal geometry.…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
In this paper, we introduce new classes of functions that extend the known classes of functions of complex variable, such as entire functions, meromorphic functions, rational functions and polynomial functions and take values in the set of…
We present relations between cycles with rational coefficients modulo algebraic equivalence on the Jacobian of a curve. These relations depend on the linear systems the curve admits. They are obtained in the tautological ring, the smallest…
In this paper we prove the connectedness of isoperiodic moduli spaces of meromorphic differentials with at least three simple poles on homologically marked smooth curves whose periods are either not contained in a real line, or not…
We survey recent developments on rationality problems for algebraic varieties, with a particular emphasis on cycle-theoretic and combinatorial methods and their applications to hypersurfaces.
We study one variable meromorphic functions mapping a planar real algebraic set $A$ to another real algebraic set in the complex plane. By using the theory of Schwarz reflection functions, we show that for certain $A$, these meromorphic…
The main goal of this paper is to study properties of the iterated integrals of modular forms in the upper halfplane, eventually multiplied by $z^{s-1}$, along geodesics connecting two cusps. This setting generalizes simultaneously the…
In this paper, we are mainly interested in the two questions "which are the commutative rings on which every finitely presented modules is [Formula: see text]-periodic (respectively, [Formula: see text]-periodic)?". It is proved that these…
Consider a finite-dimensional algebra $A$ and any of its moduli spaces $\mathcal{M}(A,\mathbf{d})^{ss}_{\theta}$ of representations. We prove a decomposition theorem which relates any irreducible component of…
We study modular ortholattices in the variety generated by the finite dimensional ones from an equational and geometric point of view. We relate this to coordinatization results.
We prove a recursive formula for the exterior and symmetric powers of modules for a cyclic 2-group. This makes computation straightforward. Previously, a complete description was only known for cyclic groups of prime order.
We study natural D-modules on the moduli stack of elliptic curves over a field of characteristic zero. We use this to produce an algebro-geometric version of the algebra of higher depth mock modular forms, studied from a Conformal Field…
We solve a long standing open problem concerning the structure of finite cycles in the category mod A of finitely generated modules over an arbitrary artin algebra A, that is, the chains of homomorphisms $M_0 \stackrel{f_1}{\rightarrow} M_1…
Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a…
We study a certain cycle map defined on finite dimensional modules for the W-algebra with regular integral central character. Via comparison with the theory in postive characteristic, we show that this map injects into the top Borel-Moore…
We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.
With the help of hypergeometric functions over finite fields, we study some arithmetic properties of cyclotomic matrices involving characters and binary quadratic forms over finite fields. Also, we confirm some related conjectures posed by…