arXiv subjects
Publications and source records attributed to Fernando Rodriguez Villegas.
Hypergeometric motives are family of motives associated to hypergeometric local systems. Their special features, in particular their rigidity, makes them more tractable than general motives. In the present article we prove most of the properties that they are expected to satisfy in the rank $2$ case.
In this short note we prove a couple of facts about polynomial count varieties, answering natural questions that they raise. A polynomial count $X$ variety is essentially one for which its number of points over finite fields is given by a polynomial in the field size. Well-known examples include affine or projective space (or more generally the Grassmanian) and other standard varieties. The two questions we address are the following. 1) If $X$ is smooth, polynomial count with $\#X(q)=q^n$ for some $n$, is $X$ isomorphic to $n$-dimensional affine space? 2) If $X$ is a polynomial count, is it true that its Hodge numbers in a given graded piece of fixed weight satisfy~$h^{p,q}=0$ unless $p=q$? We show that in both cases the answer is no.
We consider all irreducible rank-4 hypergeometric local systems defined over $\mathbb{Q}$ that support a rational one-dimensional variation of Hodge structures of weight 3 and Hodge vector $(1,1,1,1)$. Up to a natural equivalence there are only 47 cases. The first 14 cases have maximally unipotent monodromy at one point and have been extensively studied in the literature. We show that all 47 local systems are associated to families of generically smooth threefolds and we analyze the geometry and arithmetic at their conifold point.
In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R[t]/(t^2). Using these results together with results of Geiss, Leclerc and Schroer we give, when k is algebraically closed, a classification of pairs (Q,R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally, when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra F_q[t]/(t^r). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.
We give two constructions of families of elliptic curves over cubic or quartic fields with three, respectively four, `integral' elements in the kernel of the tame symbol on the curves. The fields are in general non-Abelian, and the elements linearly independent. For their integrality, we prove a new criterion that does not ignore any torsion. We also verify Beilinson's conjecture numerically for just over 90 of the curves.
We study a hypergeometric local system that arises from the quantum Chen-Ruan cohomology of a family of weighted del Pezzo hypersurfaces. We prove that it is the anti-invariant variation of a pencil of genus-7 curves with respect to an involution having 4 fixed points.
Survey of hypergeometric motives, with a focus on their source varieties, Hodge numbers, and L-functions.
Let $Γ$ be a simple graph and $I_Γ(x)$ its multivariate independence polynomial. The main result of this paper is the characterization of chordal graphs as the only $Γ$ for which the power series expansion of $I_Γ^{-1}(x)$ is Horn hypergeometric.
This note is an extended version of the slides for my talk with the same title at the {\it Arithmetic, geometry, and modular forms: a conference in honour of Bill Duke} in June 2019 at the ETH in Z"urich. The results presented concern three geometric criteria for the integrality of factorial ratios, numbers such as (30n)!n!/(6n)!(10n)!(15n)!, which are integral in a non-immediate way for all n. This work is an offshoot of an ongoing project on hypergeometric motives joint with D. Roberts and M. Watkins.
We study classical hypergeometric series as a p-adic function of its parameters inspired by a problem in the American Mathematical Monthly solved by D. Zagier. This is an extended abstract of a talk given at the workshop "Hypergeometric motives and Calabi-Yau differential equations" at the Mathematical Research Institute (MATRIX) of The University of Melbourne in Creswick, Australia in January of 2017.
We discuss two related principles for hypergeometric supercongrences, one related to accelerated convergence and the other to the vanishing of Hodge numbers. This is an extended abstract of a talk given at the workshop "Hypergeometric motives and Calabi-Yau differential equations" at the Mathematical Research Institute (MATRIX) of The University of Melbourne in Creswick, Australia in January of 2017.
We study the general properties of certain rank four rigid local systems considered by Goursat. We analyze when they are irreducible, give an explicit integral description as well as the invariant Hermitian form when it exists. By a computer search we find what we expect are all irreducible such systems all whose solutions are algebraic functions and give several explicit examples defined over the rationals. We also exhibit one example with infinite monodromy as arising from a family of genus two curves.
In this paper we survey geometric and arithmetic techniques to study the cohomology of semiprojective hyperkaehler manifolds including toric hyperkaehler varieties, Nakajima quiver varieties and moduli spaces of Higgs bundles on Riemann surfaces. The resulting formulae for their Poincare polynomials are combinatorial and representation theoretical in nature. In particular we will look at their Betti numbers and will establish some results and expectations on their asymptotic shape.
In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients.
We extend and reformulate a result of Solomon on the divisibility of the title. We show, for example, that if $Γ$ is a finitely generated group, then $|G|$ divides $#\Hom(Γ,G)$ for every finite group $G$ if and only if $Γ$ has infinite abelianization. As a consequence we obtain some arithmetic properties of the number of subgroups of a given index in such a group $Γ$.
We study a refinement of the A-polynomial in the case of the g-loop quiver. We give an explicit formula for its value at q=1. Conjecturally this implies a formula for the middle Betti number of the moduli space of Higgs bundles or equivalently of the character variety of a Riemann surface of genus g.
We describe the structure of all codimension-two lattice configurations $A$ which admit a stable rational $A$-hypergeometric function, that is a rational function $F$ all whose partial derivatives are non zero, and which is a solution of the $A$-hypergeometric system of partial differential equations defined by Gel'fand, Kapranov and Zelevinsky. We show, moreover, that all stable rational $A$-hypergeometric functions may be described by toric residues and apply our results to study the rationality of bivariate series whose coefficients are quotients of factorials of linear forms.
Let $Γ$ be a quiver on n vertices $v_1, v_2, ..., v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $α\in \N^n$. Hua gave a formula for $A_Γ(α, q)$, the number of isomorphism classes of absolutely indecomposable representations of $Γ$ over the finite field $\F_q$ with dimension vector $α$. Kac showed that $A_Γ(\bmα, q)$ is a polynomial in q with integer coefficients. Using Hua's formula, we show that for each non-negative integer s, the s-th derivative of $A_Γ(α,q)$ with respect to q, when evaluated at q = 1, is a polynomial in the variables $g_{ij}$, and we compute the highest degree terms in this polynomial. Our formulas for these coefficients depend on the enumeration of certain families of connected graphs.