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Frobenius intertwiners for q-difference equations

Andrey Smirnov

TL;DR

This work establishes a $q$-deformed Frobenius framework for the quantum difference equations attached to $X=T^{*}\mathbb{P}^{n-1}$ over $\mathbb{Q}_p$, constructing an explicit Frobenius intertwiner $\mathsf{U}(h,\boldsymbol{a},q,z)$ with unit-root behavior and a constant term governed by the $p$-adic $q$-gamma function $\Gamma_{p,q}$. In the limit $q\to 1$ the intertwiners converge to Dwork-style Frobenius structures for $p$-adic hypergeometric and Bessel equations, with constant terms expressed via the Morita gamma function $\Gamma_p$ and, in general, yielding $p$-adic zeta-values as coefficients in the cohomological degenerations. The constant term is diagonal in the fixed-point basis for generic equivariant parameters, and the results extend to an enumerative interpretation where the intertwiners arise from partition functions counting equivariant quasimaps, linking $q$-difference analogues of $p$-curvature to geometric and arithmetic data. The paper also provides numerical evidence and a detailed treatment of unit-root specializations, establishing a coherent picture that unites quantum $K$-theory, $p$-adic analysis, and classical $p$-adic differential equations in a single Frobenius-friendly framework.

Abstract

We consider a class of $q$-hypergeometric equations describing the quantum difference equation for the cotangent bundles over projective spaces $X=T^{*}\mathbb{P}^{n-1}$ . We show that over $\mathbb{Q}_p$ these equations are equipped with the Frobenius action $(q,z)\to (q^p,z^p)$. We obtain an explicit formula for the constant term of the Frobenius intertwiner in terms of the $p$-adic $q$-gamma function of Koblitz. In the limit $q\to 1$ we arrive at the Frobenius structures for the $p$-adic hypergeometric and Bessel differential equations studied by Dwork. In particular, we find closed formulas for $p$-adic constants appearing in works of Dwork and Sperber in terms of $p$-adic zeta functions.

Frobenius intertwiners for q-difference equations

TL;DR

This work establishes a -deformed Frobenius framework for the quantum difference equations attached to over , constructing an explicit Frobenius intertwiner with unit-root behavior and a constant term governed by the -adic -gamma function . In the limit the intertwiners converge to Dwork-style Frobenius structures for -adic hypergeometric and Bessel equations, with constant terms expressed via the Morita gamma function and, in general, yielding -adic zeta-values as coefficients in the cohomological degenerations. The constant term is diagonal in the fixed-point basis for generic equivariant parameters, and the results extend to an enumerative interpretation where the intertwiners arise from partition functions counting equivariant quasimaps, linking -difference analogues of -curvature to geometric and arithmetic data. The paper also provides numerical evidence and a detailed treatment of unit-root specializations, establishing a coherent picture that unites quantum -theory, -adic analysis, and classical -adic differential equations in a single Frobenius-friendly framework.

Abstract

We consider a class of -hypergeometric equations describing the quantum difference equation for the cotangent bundles over projective spaces . We show that over these equations are equipped with the Frobenius action . We obtain an explicit formula for the constant term of the Frobenius intertwiner in terms of the -adic -gamma function of Koblitz. In the limit we arrive at the Frobenius structures for the -adic hypergeometric and Bessel differential equations studied by Dwork. In particular, we find closed formulas for -adic constants appearing in works of Dwork and Sperber in terms of -adic zeta functions.
Paper Structure (20 sections, 22 theorems, 184 equations, 1 figure)

This paper contains 20 sections, 22 theorems, 184 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $q$-difference equations and enumerative geometry
  3. $q$-hypergeometric case
  4. $q$-deformed Frobenius intertwiner
  5. The constant term of the Frobenius intertwiner
  6. Limits and specializations
  7. Enumerative definition of Frobenius intertwiner
  8. $q$-hypergeometric functions
  9. Vertex function of $X=T^{*}\mathbb{P}^{n-1}$
  10. $p$ - adic completions and norms
  11. Frobenius intertwiner for $q$-hypergeometric equation
  12. Proof of Theorem \ref{['mainthm']}
  13. Limit to cohomology
  14. Limit to $X=\mathbb{P}^{n-1}$: Frobenius structure for generalized Bessel equations
  15. Numerical examples of Theorem \ref{['mainthm']}
  16. ...and 5 more sections

Key Result

Theorem 3.1

The normalized fundamental solution matrix (psitildef) satisfies the following system of difference equations where $A_i(\boldsymbol{a},h,q,z)$ and $H(\boldsymbol{a},h,q,z)$ denote matrices whose entries are rational functions in the equivariant parameters ${\boldsymbol u}$, $\hbar$, $q$ and $z$: $A_i(\boldsymbol{a},h,q,z), H(\boldsymbol{a},h,q,z) \in \mathsf{GL}_{n}(\mathbb{Q}({\boldsymbol u},\

Figures (1)

  • Figure 1: Frobenius intertwiner as a partition function of relative quasimaps

Theorems & Definitions (34)

  • Theorem 3.1: Theorem 8.2.20, Oko17
  • Lemma 4.1
  • proof
  • Definition 5.1
  • Theorem 5.2
  • Proposition 5.3
  • proof
  • Corollary 5.4
  • Proposition 5.5
  • Proposition 5.6
  • ...and 24 more