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Atkin polynomials for families of abelian varieties with real multiplication

Gabriele Bogo, Yingkun Li

TL;DR

This work generalizes Atkin polynomials to families of abelian varieties with real multiplication on genus-zero curves inside Hilbert modular varieties, linking the non-ordinary locus to zeros of orthogonal polynomials via Atkin scalar products. It builds a bridge between Padé approximants and the logarithmic derivatives of Picard–Fuchs solutions, enabling a PF-based, modular-agnostic description of the non-ordinary locus that extends Igusa’s classical perspective. The paper provides explicit recurrences and polynomial data for affine triangle curves and the Teichmüller curve $W_{17}$, and establishes a framework in which Atkin polynomials arise as denominators of Padé approximants, with partial Hasse polynomials often exhibiting palindromicity and, conjecturally, supersingular-point-pairing. Overall, the results generalize Kaneko–Zagier’s elliptic-case connections to higher-dimensional RM settings, offering computational and conceptual tools for studying non-ordinary loci in Hilbert modular geometries.

Abstract

Generalizing the work of Atkin and Kaneko-Zagier in the elliptic case, we describe the non-ordinary locus of a genus-zero non-compact curve $Y$ in a Hilbert modular variety in terms of the zeros of generalized Atkin's orthogonal polynomials. The argument relies on the recent construction of lifts of partial Hasse invariants for $Y$. We further describe these orthogonal polynomials as denominators of Padé approximants to the logarithmic derivatives of solutions of the Picard-Fuchs differential equations associated with $Y$. This provides a new link between Padé approximation and the geometry of the non-ordinary locus, extending a classical observation of Igusa for the Legendre family and applying, in particular, to situations where the Picard-Fuchs equations do not admit modular solutions. As applications, we determine the three-term recurrence relations for Atkin polynomials attached to triangle curves via hypergeometric identities, and compute the supersingular locus of a double cover of the Teichmüller curve $W_{17}$. In the latter case, we conjecture that the associated supersingular polynomial is self-reciprocal, implying that supersingular points occur in pairs.

Atkin polynomials for families of abelian varieties with real multiplication

TL;DR

This work generalizes Atkin polynomials to families of abelian varieties with real multiplication on genus-zero curves inside Hilbert modular varieties, linking the non-ordinary locus to zeros of orthogonal polynomials via Atkin scalar products. It builds a bridge between Padé approximants and the logarithmic derivatives of Picard–Fuchs solutions, enabling a PF-based, modular-agnostic description of the non-ordinary locus that extends Igusa’s classical perspective. The paper provides explicit recurrences and polynomial data for affine triangle curves and the Teichmüller curve , and establishes a framework in which Atkin polynomials arise as denominators of Padé approximants, with partial Hasse polynomials often exhibiting palindromicity and, conjecturally, supersingular-point-pairing. Overall, the results generalize Kaneko–Zagier’s elliptic-case connections to higher-dimensional RM settings, offering computational and conceptual tools for studying non-ordinary loci in Hilbert modular geometries.

Abstract

Generalizing the work of Atkin and Kaneko-Zagier in the elliptic case, we describe the non-ordinary locus of a genus-zero non-compact curve in a Hilbert modular variety in terms of the zeros of generalized Atkin's orthogonal polynomials. The argument relies on the recent construction of lifts of partial Hasse invariants for . We further describe these orthogonal polynomials as denominators of Padé approximants to the logarithmic derivatives of solutions of the Picard-Fuchs differential equations associated with . This provides a new link between Padé approximation and the geometry of the non-ordinary locus, extending a classical observation of Igusa for the Legendre family and applying, in particular, to situations where the Picard-Fuchs equations do not admit modular solutions. As applications, we determine the three-term recurrence relations for Atkin polynomials attached to triangle curves via hypergeometric identities, and compute the supersingular locus of a double cover of the Teichmüller curve . In the latter case, we conjecture that the associated supersingular polynomial is self-reciprocal, implying that supersingular points occur in pairs.
Paper Structure (16 sections, 19 theorems, 98 equations)

This paper contains 16 sections, 19 theorems, 98 equations.

Table of Contents

  1. Families of abelian varieties with real multiplication
  2. Twisted modular forms on genus zero curves
  3. Partial Hasse invariants
  4. Serre derivatives
  5. Non-ordinary locus and zeros of orthogonal polynomials
  6. Orthogonal polynomials of one variable
  7. Atkin's polynomials
  8. Positive definiteness of Atkin scalar products
  9. Partial Hasse polynomials and Atkin's polynomials
  10. Hasse invariant and Atkin's polynomials
  11. Partial Hasse invariants from Padé approximants of solutions of Picard–Fuchs differential equations
  12. Examples
  13. Arithmetic cases
  14. Affine triangle curves $\mathbb{H}/\Delta(n,m,\infty)$
  15. The curve $\mathbb{H}/\Delta(2,5,\infty)$
  16. ...and 1 more sections

Key Result

Theorem 1

There exists a positive definite scalar product $\langle\,,\rangle$ on $\mathbb{R}[J]$ and a unique associated family of monic orthogonal polynomials $\{A_n(J)\}_{n} \subset \mathbb{Q}[J]$ with the following property: for every prime $p>3$ there exists $n_p\in\mathbb{Z}_{\ge0}$ such that $A_{n_p}(J

Theorems & Definitions (37)

  • Theorem : Atkin, Kaneko–Zagier
  • Theorem A: Theorem \ref{['thm:main2']}
  • Theorem B: Theorem \ref{['thm:main']}, Corollary \ref{['cor:main']}
  • Theorem C
  • Conjecture
  • Theorem : Goren GHi, Andreatta-Goren AG
  • Example 1
  • Theorem 1: Proposition 3 and Theorem 2 in BPf, Theorem 1 in BLpHi
  • Remark 1
  • Lemma 1
  • ...and 27 more