arXiv subjects

Fernando Rodriguez-Villegas

Publications and source records attributed to Fernando Rodriguez-Villegas.

13 recordsLinked to original sources

arXiv2024

Ennola duality for decomposition of tensor products

The aim of this paper is to investigate Ennola duality for decomposition of tensor products of irreducible characters of finite general linear groups and finite unitary groups. We prove that Ennola duality holds generically and give a geometric interpretation using the cohomology of quiver varieties. For non-generic characters (like unipotent characters), Ennola duality does not work just by replacing q by -q. We construct two-variable polynomials that interpolate multiplicities for finite general linear groups and finite unitary groups in the unipotent case (which can be considered as Ennola duality).

math.RT↗

arXiv2022

E-series of character varieties of non-orientable surfaces

In this paper we are interested in two kinds of (stacky) character varieties associated to a compact non-orientable surface. (A) We consider the quotient stack of the space of representations of the fundamental group of this surface to GL(n). (B) We choose a set of k-punctures on the surface and a generic k-tuple of semisimple conjugacy classes of GL(n), and we consider the stack of anti-invariant local systems on the orientation cover of the surface with local monodromies around the punctures given by the prescribed conjugacy classes. We compute the number of points of these spaces over finite fields from which we get a formula for their E-series (a certain specialization of the mixed Poincaré series). In case (B), we discuss the mixed Poincaré series when the surface is the real projective plane and k=1.

math.RT↗

arXiv2019

The Tutte polynomial and toric Nakajima quiver varieties

For a quiver $Q$, we take $\mathcal{M}$ an associated toric Nakajima quiver variety and $Γ$ the underlying graph. In this article, we give a direct relation between a specialisation of the Tutte polynomial of $Γ$, the Kac polynomial of $Q$ and the Poincaré polynomial of $\mathcal{M}$. We do this by giving a cell decomposition of $\mathcal{M}$ indexed by spanning trees of $Γ$ and `geometrising' the deletion and contraction operators on graphs. These relations have been previously established by Sturmfels-Hausel and (Crawley-Boovey)-Van den Bergh, however the methods here are more hands-on.

math.AG↗

arXiv2019

A case of the Rodriguez Villegas conjecture

Let L be a number field and let E be any subgroup of the units O_L^* of L. If rank(E) = 1, Lehmer's conjecture predicts that the height of any non-torsion element of E is bounded below by an absolute positive constant. If rank(E) = rank(O_L^*), Zimmert proved a lower bound on the regulator of E which grows exponentially with [L:Q]. Fernando Rodriguez Villegas made a conjecture in 2002 that "interpolates" between these two extremes of rank. Here we prove a high-rank case of this conjecture. Namely, it holds if L contains a subfield K for which [L:K] >> [K:Q] and E contains the kernel of the norm map from O_L^* to O_K^*.

math.NT↗

arXiv2016

Vertex operators and character varieties

We prove some combinatorial conjectures extending those proposed in [13, 14]. The proof uses a vertex operator due to Nekrasov, Okounkov, and the first author [4] to obtain a "gluing formula" for the relevant generating series, essentially reducing the computation to the case of complex projective space with three punctures.

math.AG↗

arXiv2013

On the bilinear structure associated to Bezoutians

This paper is partly a survey of known results on quadratic forms that are hard to find in the literature. Our main focus is a twisted form of a construction due to Bezout. This skew Bezoutian is a symplectic (resp. quadratic) space associated to a pair of reciprocal (or skew reciprocal) coprime polynomials of same degree. The isometry group of this space turns out to contain a certain associated hypergeometric group. Using the skew Bezoutian we construct explicit isometries of bilinear spaces with given invariants (such as the characteristic polynomial or Jordan form and, in the quadratic case, the spinor norm).

math.AC↗

arXiv2012

Positivity of Kac polynomials and DT-invariants for quivers

We give a cohomological interpretation of both the Kac polynomial and the refined Donaldson-Thomas- invariants of quivers. This interpretation yields a proof of a conjecture of Kac from 1982 and gives a new perspective on recent work of Kontsevich-Soibelman. This is achieved by computing, via an arithmetic Fourier transform, the dimensions of the isoytpical components of the cohomology of associated Nakajima quiver varieties under the action of a Weyl group. The generating function of the corresponding Poincaré polynomials is an extension of Hua's formula for Kac polynomials of quivers involving Hall-Littlewood symmetric functions. The resulting formulae contain a wide range of information on the geometry of the quiver varieties.

math.RT↗

arXiv2011

Arithmetic harmonic analysis on character and quiver varieties II

We study connections between the topology of generic character varieties of fundamental groups of punctured Riemann surfaces, Macdonald polynomials, quiver representations, Hilbert schemes on surfaces, modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups.

math.RT↗

arXiv2008

Mixed Hodge polynomials of character varieties

We calculate the E-polynomials of certain twisted GL(n,C)-character varieties M_n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n,F_q) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,C)-character variety. The calculation also leads to several conjectures about the cohomology of M_n: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n = 2.

math.AG↗

arXiv2006

Computation of central value of quadratic twists of modular L-functions

Let f be a newform of weight two, prime level p. If D is a fundamental discriminant, define the twisted L-function L(f,D,s) to be the L-function associated to the twist of f by the quadratic character of conductor D. In this paper we consider the question of computing the family of twisted central values {L(f,D,1) : |D| <= x} for some x, by using an explicit version of Waldspurger's formula relating the central values L(f,D,1) to the |D|-th Fourier coefficient of weigth 3/2 modular forms in Shimura correspondence with f.

math.NT↗

arXiv2005

Mahler's Measure and the Dilogarithm (II)

We continue to investigate the relation between the Mahler measure of certain two variable polynomials, the values of the Bloch--Wigner dilogarithm $D(z)$ and the values $ζ_F(2)$ of zeta functions of number fields. Specifically, we define a class $\A$ of polynomials $A$ with the property that $πm(A)$ is a linear combination of values $D$ at algebraic arguments. For many polynomials in this class the corresponding argument of $D$ is in the Bloch group, which leads to formulas expressing $πm(A)$ as a linear combination with unspecified rational coefficients of $V_F$ for certain number fields $F$ ($V_F := c_Fζ_F(2)$ with $c_F>0$ an explicit simple constant). The class $\A$ contains the $A$-polynomials of cusped hyperbolic manifolds. The connection with hyperbolic geometry often provides means to prove identities of the form $πm(A)= r V_F$ with an explicit value of $r\in \Q^*$. We give one such example in detail in the body of the paper and in the appendix.

math.NT↗

arXiv2004

Calabi-Yau Manifolds Over Finite Fields, II

We study zeta-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The zeta-function for the quintic family involves factors that correspond to a certain pair of genus 4 Riemann curves. The appearance of these factors is intriguing since we have been unable to `see' these curves in the geometry of the quintic. Having these zeta-functions to hand we are led to comment on their form in the light of mirror symmetry. That some residue of mirror symmetry survives into the zeta-functions is suggested by an application of the Weil conjectures to Calabi-Yau threefolds: the zeta-functions are rational functions and the degrees of the numerators and denominators are exchanged between the zeta-functions for the manifold and its mirror. It is clear nevertheless that the zeta-function, as classically defined, makes an essential distinction between Kahler parameters and the coefficients of the defining polynomial. It is an interesting question whether there is a `quantum modification' of the zeta-function that restores the symmetry between the Kahler and complex structure parameters. We note that the zeta-function seems to manifest an arithmetic analogue of the large complex structure limit which involves 5-adic expansion.

hep-th↗

arXiv2000

Calabi-Yau Manifolds Over Finite Fields, I

We study Calabi-Yau manifolds defined over finite fields. These manifolds have parameters, which now also take values in the field and we compute the number of rational points of the manifold as a function of the parameters. The intriguing result is that it is possible to give explicit expressions for the number of rational points in terms of the periods of the holomorphic three-form. We show also, for a one parameter family of quintic threefolds, that the number of rational points of the manifold is closely related to as the number of rational points of the mirror manifold. Our interest is primarily with Calabi-Yau threefolds however we consider also the interesting case of elliptic curves and even the case of a quadric in CP_1 which is a zero dimensional Calabi-Yau manifold. This zero dimensional manifold has trivial dependence on the parameter over C but a not trivial arithmetic structure.

hep-th↗