arXiv subjects

Eric M. Rains

Publications and source records attributed to Eric M. Rains.

At least 19 recordsLinked to original sources

arXiv2025

The monodromy of cyclic Pryms

The {\em Prym} of a cyclic covering of smooth projective curves is the ``new'' part of the Jacobian: the quotient of the Jacobian of the covering curve by the Jacobians of the intermediate covers. Given a family of such coverings, the fundamental group of the base of the family acts on the Tate modules of the Pryms, and the image of this representation is a key ingredient in answering arithmetic statistics questions about the distribution of the group structure of the $L$-torsion of a random Prym in the family. (Over ${\mathbb{F}}_q$, the action of Frobenius is roughly uniformly distributed over the {\em arithmetic} monodromy, a coset of the image of the fundamental group of the base change to $\bar{\mathbb{F}}_q$ (the {\em geometric} monodromy).) In the present note, we show for a number of natural families that (with limited exceptions) the geometric monodromy is sandwiched between a certain unitary group and its derived subgroup. In particular, this holds for the one-parameter families obtained by starting with any fixed cover and varying one (tame) ramification point. As an application, we deduce analogous largeness results for the monodromy of the Selmer groups of elliptic surfaces with $j=0$ or $j=1728$, by relating them to cyclic covers of degree 6 or 4 respectively, implying that their Selmer groups do not satisfy the standard heuristics. For instance, for eliptic surfaces with $j=0$ of sufficiently large height over ${\mathbb{P}}^1_{\mathbb{F}_q}$, the average size of the $l$-Selmer group is $l+3+o_q(1)$ when $l$ (fixed) and $q$ (large) are both 1 mod 3, compared to $l+1+o_q(1)$ for general elliptic surfaces.

math.NT↗

arXiv2025

Invariant derivations and trace bounds

About 20 years ago, J-P.~Serre announced a bound on the trace of elements of compact Lie groups under the adjoint representation together with related results, provided indications of his proofs, and invited a better proof. This note provides a new, general method for proving such bounds; uses that method to derive Serre's bounds; gives a second proof of Serre's announced results that (we learned) closely follows his original argument; and provides lower bounds for traces of other representations of compact Lie groups and for Brauer characters of finite groups.

math.RT↗

arXiv2025

The (noncommutative) geometry of difference equations

The aim of this monograph is twofold: to explain various nonautonomous integrable systems (discrete Painlevé all the way up to the elliptic level, as well as generalizations à la Garnier) using an interpretation of difference and differential equations as sheaves on noncommutative projective surfaces, and to develop the theory of such surfaces enough to allow one to apply the usual GIT construction of moduli spaces of sheaves. This requires a fairly extensive development of the theory of birationally ruled noncommutative projective surfaces, both showing that the analogues of Cremona transformations work and understanding effective, nef, and ample divisor classes. This combines arXiv:1307.4032, arXiv:1307.4033, arXiv:1907.11301, as well as those portions of arXiv:1607.08876 needed to make things self-contained. Some additional results appear, most notably a proof that the resulting discrete actions on moduli spaces of equations are algebraically integrable.

math.AG↗

arXiv2024

Quotients of abelian varieties by reflection groups

We prove (by a case-by-case analysis) a conjecture of Bernstein/Schwarzman to the effect that quotients of abelian varieties by suitable actions of (complex) reflection groups are weighted projective spaces, and show that this remains true after reduction to finite characteristic (including characteristics dividing the order of the group!). We also show that an analogous statement holds (with five explicitly enumerated exceptions) for actions of quaternionic reflection groups on supersingular abelian varieties.

math.AG↗

arXiv2023

Elliptic $\mathrm{A}_n$ Selberg integrals

We use the elliptic interpolation kernel due to the second author to prove an $\mathrm{A}_n$ extension of the elliptic Selberg integral. More generally, we obtain elliptic analogues of the $\mathrm{A}_n$ Kadell, Hua-Kadell and Alba-Fateev-Litvinov-Tarnopolsky (or AFLT) integrals.

math.CA↗

arXiv2022

The geometric distribution of Selmer groups of elliptic curves over function fields

Fix a positive integer $n$ and a finite field $\mathbb F_q$. We study the joint distribution of the rank of $E$, the $n$-Selmer group of $E$, and the $n$-torsion in the Tate-Shafarevich group of $E$ as $E$ varies over elliptic curves of fixed height $d \geq 2$ over $\mathbb F_q(t)$. We compute this joint distribution in the large $q$ limit. We also show that the "large $q$, then large height" limit of this distribution agrees with the one predicted by Bhargava-Kane-Lenstra-Poonen-Rains.

math.NT↗

arXiv2022

Filtered deformations of elliptic algebras

One of the difficulties in doing noncommutative projective geometry via explicitly presented graded algebras is that it is usually quite difficult to show flatness, as the Hilbert series is uncomputable in general. If the algebra has a regular central element, one can reduce to understanding the (hopefully more tractable) quotient. If the quotient is particularly nice, one can proceed in reverse and find all algebras of which it is the quotient by a regular central element (the filtered deformations of the quotient). We consider in detail the case that the quotient is an elliptic algebra (the homogeneous endomorphism ring of a vector bundle on an elliptic curve, possibly twisted by translation). We explicitly compute the family of filtered deformations in many cases and give a (conjecturally exhaustive) construction of such deformations from noncommutative del Pezzo surfaces. In the process, we also give a number of results on the classification of exceptional collections on del Pezzo surfaces, which are new even in the commutative case.

math.AG↗

arXiv2021

AFLT-type Selberg integrals

In their 2011 paper on the AGT conjecture, Alba, Fateev, Litvinov and Tarnopolsky (AFLT) obtained a closed-form evaluation for a Selberg integral over the product of two Jack polynomials, thereby unifying the well-known Kadell and Hua--Kadell integrals. In this paper we use a variety of symmetric functions and symmetric function techniques to prove generalisations of the AFLT integral. These include (i) an $\mathrm{A}_n$ analogue of the AFLT integral, containing two Jack polynomials in the integrand; (ii) a generalisation of (i) for $γ=1$ (the Schur or GUE case), containing a product of $n+1$ Schur functions; (iii) an elliptic generalisation of the AFLT integral in which the role of the Jack polynomials is played by a pair of elliptic interpolation functions; (iv) an AFLT integral for Macdonald polynomials.

math-ph↗

arXiv2020

An Elliptic Hypergeometric Function Approach to Branching Rules

We prove Macdonald-type deformations of a number of well-known classical branching rules by employing identities for elliptic hypergeometric integrals and series. We also propose some conjectural branching rules and allied conjectures exhibiting a novel type of vanishing behaviour involving partitions with empty 2-cores.

math.CO↗

arXiv2020

Elliptic Double Affine Hecke Algebras

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the $\tilde{C}_n$ version of the construction to construct a flat noncommutative deformation of the $n$th symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.

math.AG↗

arXiv2019

Generalized Hitchin systems on rational surfaces

By analogy with work of Hitchin on integrable systems, we construct natural relaxations of several kinds of moduli spaces of difference equations, with special attention to a particular class of difference equations on an elliptic curve (arising in the theory of elliptic special functions). The common feature of the relaxations is that they can be identified with moduli spaces of sheaves on rational surfaces. Not only does this make various natural questions become purely geometric (rigid equations correspond to -2-curves), it also establishes a number of nontrivial correspondences between different moduli spaces, since a given moduli space of sheaves is typically the relaxation of infinitely many moduli spaces of equations. In the process of understanding this, we also consider a number of purely geometric questions about rational surfaces with anticanonical curves; e.g., we give an essentially combinatorial algorithm for testing whether a given divisor is the class of a -2-curve or is effective with generically integral representative.

math.AG↗

arXiv2019

The noncommutative geometry of elliptic difference equations

We give a new construction of noncommutative surfaces via elliptic difference operators, attaching a 1-parameter noncommutative deformation to any projective rational surface with smooth anticanonical curve. The construction agrees with one implicit in work of Van den Bergh (iterated blowups of noncommutative Hirzebruch surfaces), but the construction enables one to prove a number of new facts about these surfaces. We show that they are noncommutative smooth proper surfaces in the sense of Chan and Nyman, with projective Quot schemes, that moduli spaces of simple sheaves are Poisson and that moduli spaces classifying semistable sheaves of rank 0 or 1 are projective. We further show that the action of SL_2(Z) as derived autoequivalences of rational elliptic surfaces extends to an action as derived equivalences of surfaces in our family with K^2=0. We also discuss applications to the theory of special functions arising by interpreting moduli spaces of 1-dimensional sheaves as moduli spaces of difference equations. When the moduli space is a single point, the equation is rigid, and we give an integral representation for the solutions. More generally, twisting by line bundles corresponds to isomonodromy deformations, so this gives rise to Lax pairs. When the moduli space is 2-dimensional, one obtains Lax pairs for the elliptic Painlevé equation; this associates a Lax pair to any rational number, of order twice the denominator. There is also an elliptic analogue of the Riemann-Hilbert correspondence: an analytic equivalence between categories of elliptic difference equations, swapping the role of the shift of the equation and the nome of the curve.

math.AG↗

arXiv2019

Birational morphisms and Poisson moduli spaces

We study birational morphisms between smooth projective surfaces that respect a given Poisson structure, with particular attention to induced birational maps between the (Poisson) moduli spaces of sheaves on those surfaces. In particular, to any birational morphism, we associate a corresponding "minimal lift" operation on sheaves of homological dimension <=1, and study its properties. In particular, we show that minimal lift induces a stratification of the moduli space of simple sheaves on the codomain by open subspaces of the moduli space of simple sheaves on the domain, compatibly with the induced Poisson structures.

math.AG↗

arXiv2019

The birational geometry of noncommutative surfaces

We show that any commutative rationally ruled surface with a choice of anticanonical curve admits a 1-parameter family of noncommutative deformations parametrized by the Jacobian of the anticanonical curve, and show that many standard facts from commutative geometry (blowups commute, Quot schemes are projective, etc.) carry over. The key new tool in studying these deformations is a relatively simple description of their derived categories and the relevant t-structures; this also allows us to establish nontrivial derived equivalences for deformations of elliptic surfaces. We also establish that the category of line bundles (suitably defined) on such a surface has a faithful representation in which the morphisms are difference or differential operators, and thus find that difference/differential equations can be viewed as sheaves on such surfaces. In particular, we find that many moduli spaces of sheaves on such surfaces have natural interpretations as moduli spaces of equations with (partially) specified singularities, and in particular find that the "isomonodromy" interpretation of discrete Painlevé equations and their generalizations has a natural geometric interpretation (twisting sheaves by line bundles).

math.AG↗

arXiv2018

Multivariate Quadratic Transformations and the Interpolation Kernel

We prove a number of quadratic transformations of elliptic Selberg integrals (conjectured in an earlier paper of the author), as well as studying in depth the "interpolation kernel", an analytic continuation of the author's elliptic interpolation functions which plays a major role in the proof as well as acting as the kernel for a Fourier transform on certain elliptic double affine Hecke algebras (discussed in a later paper). In the process, we give a number of examples of a new approach to proving elliptic hypergeometric integral identities, by reduction to a Zariski dense subset of a formal neighborhood of the trigonometric limit.

math.CA↗

arXiv2018

Limits of multivariate elliptic beta integrals and related bilinear forms

In this article we consider the elliptic Selberg integral, which is a BC_n symmetric multivariate extension of the elliptic beta integral. We categorize the limits that are obtained as p->0, for given behavior of the parameters as p->0. This article is therefore the multivariate version of our earlier paper "Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions". The integrand of the elliptic Selberg integral is the measure for the BC_n symmetric biorthogonal functions introduced by the second author, so we also consider the limits of the associated bilinear form. We also provide the limits for the discrete version of this bilinear form, which is related to a multivariate extension of the Frenkel-Turaev summation.

math.CA↗

arXiv2018

Bounded Littlewood identities

We describe a method, based on the theory of Macdonald-Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald's partial fraction technique and results in the first examples of bounded Littlewood identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type $(R,S)$ in terms of ordinary Macdonald polynomials, are $q,t$-analogues of known branching formulas for characters of the symplectic, orthogonal and special orthogonal groups. In the classical limit, our method implies that MacMahon's famous ex-conjecture for the generating function of symmetric plane partitions in a box follows from the identification of $(\mathrm{GL}(n,\mathbb{R}),\mathrm{O}(n))$ as a Gelfand pair. As further applications, we obtain combinatorial formulas for characters of affine Lie algebras; Rogers-Ramanujan identities for affine Lie algebras, complementing recent results of Griffin et al.; and quadratic transformation formulas for Kaneko--Macdonald-type basic hypergeometric series.

math.CO↗

arXiv2018

A Nekrasov-Okounkov formula for Macdonald polynomials

We prove a Macdonald polynomial analogue of the celebrated Nekrasov-Okounkov hook-length formula from the theory of random partitions. As an application we obtain a proof of one of the main conjectures of Hausel and Rodriguez-Villegas from their work on mixed Hodge polynomials of the moduli space of stable Higgs bundles on Riemann surfaces.

math.CO↗