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Numerical Calculation of Periods on Schoen's Class of Calabi-Yau Threefolds

Azur Đonlagić

TL;DR

This work develops a robust numeric framework for computing periods of the holomorphic 3‑form on Schoen’s class of Calabi–Yau threefolds, built from small resolutions of fiber products of elliptic surfaces. It combines a two‑stage approach: first, partial period data are obtained by solving Gauss–Manin/Picard–Fuchs equations via a Frobenius method along paths between singular fibers; second, these partial results are assembled into full period lattices using vanishing cycles and monodromy data. The authors provide explicit computations across three Schoen types (I, II, III), connect the resulting period lattices to specific modular forms, and validate high‑precision agreement with expected arithmetic structures. This approach enables precise numerical verification of period–modularity correspondences and yields rich data sets for comparing periods against L‑values of modular forms, with detailed Maple implementations and meticulous internal consistency checks.

Abstract

Through classical modularity conjectures, the period integrals of a holomorphic $3$-form on a rigid Calabi-Yau threefold are interesting from the perspective of number theory. Although the (approximate) values of these integrals would be very useful for studying such relations, they are difficult to calculate and generally not known outside of the rare cases in which we can express them exactly. In this paper, we present an efficient numerical method to compute such periods on a wide class of Calabi-Yau threefolds constructed by small resolutions of fiber products of elliptic surfaces over $\mathbf{P}^1$, introduced by C. Schoen in his 1988 paper. Many example results are given, which can easily be calculated with arbitrary precision. We provide tables in which each result is written with precision of 30 decimal places and then compared to period integrals of the appropriate modular form, to confirm accuracy.

Numerical Calculation of Periods on Schoen's Class of Calabi-Yau Threefolds

TL;DR

This work develops a robust numeric framework for computing periods of the holomorphic 3‑form on Schoen’s class of Calabi–Yau threefolds, built from small resolutions of fiber products of elliptic surfaces. It combines a two‑stage approach: first, partial period data are obtained by solving Gauss–Manin/Picard–Fuchs equations via a Frobenius method along paths between singular fibers; second, these partial results are assembled into full period lattices using vanishing cycles and monodromy data. The authors provide explicit computations across three Schoen types (I, II, III), connect the resulting period lattices to specific modular forms, and validate high‑precision agreement with expected arithmetic structures. This approach enables precise numerical verification of period–modularity correspondences and yields rich data sets for comparing periods against L‑values of modular forms, with detailed Maple implementations and meticulous internal consistency checks.

Abstract

Through classical modularity conjectures, the period integrals of a holomorphic -form on a rigid Calabi-Yau threefold are interesting from the perspective of number theory. Although the (approximate) values of these integrals would be very useful for studying such relations, they are difficult to calculate and generally not known outside of the rare cases in which we can express them exactly. In this paper, we present an efficient numerical method to compute such periods on a wide class of Calabi-Yau threefolds constructed by small resolutions of fiber products of elliptic surfaces over , introduced by C. Schoen in his 1988 paper. Many example results are given, which can easily be calculated with arbitrary precision. We provide tables in which each result is written with precision of 30 decimal places and then compared to period integrals of the appropriate modular form, to confirm accuracy.

Paper Structure

This paper contains 10 sections, 9 theorems, 160 equations, 5 figures.

Table of Contents

  1. Introduction and Notation
  2. Schoen's Construction and the Problem Statement
  3. Picard-Fuchs Equations and the Partial Results
  4. An Illustrative Example of the Computation
  5. Determining the $\mathbf Z$-Module of Periods
  6. Computing the Vanishing Cycles
  7. Final Results
  8. Type I
  9. Type II
  10. Type III

Key Result

Proposition 3.3

Given an elliptic surface $E\rightarrow B$ in Weierstrass form, for every nonsingular fiber $E_b$ such that $J(E_b)\neq 0,1$, there is some primitive (that is, not a multiple of any other) element $[\gamma_b]\in\mathrm{H}_r(E_b,\mathbf Z)\simeq\mathbf Z^2$ such that the following identity holds up to a choice of branch of the $4$th root (a factor of a $4$th root of unity).

Figures (5)

  • Figure 1: A schematic representation of the real parts of root triples $(r_{1}^{\textcolor{gray}{\#} j},r_{2}^{\textcolor{gray}{\#} j},r_{3}^{\textcolor{gray}{\#} j}), j = 1,2,$ of the surfaces $E^{\textcolor{gray}{\#} 1}$ (top) of $E^{\textcolor{gray}{\#} 2}$ (bottom) from the example above, on the intervals (in order) $(-5,a),(a,b),(b,c),(c,d),(d,5)$. Roots with coinciding real parts are colored red.
  • Figure 2: Two possible extensions from $b$ to $c$ (red dots). We start in a disk of convergence centered at $b$, and end in one at $c$. On the left picture, we can imagine we're integrating along the green curve as we pass through an intermediate disk (we extend $Q^b$ by comparing its $n$-th derivatives at the green points, $n\in\{0,1,2,3,4\}$). We avoid expanding a series at the point of degeneracy $-1/4$ (cross), as the convergence is slow there, hence the results get imprecise. On the right picture, we integrate through the lower half-plane. The result need not be different from the one obtained on the left; it remains the same if we compare at the green points. However, if we perform the last comparison at the purple point instead (with negative imaginary part), we pass under the line (red) marking the end of a branch of the logarithm function $z\mapsto\log(z-c)$ (we let the values on this line continue the branch on $\mathbf{H}_+$, which is consistent with the convention used in Maple) and the effect is as if winding around $c$ by integrating along the purple line, giving a different result. This is avoided by restricting ourselves to $\overline{\mathbf{H}_+}$.
  • Figure 3: (images a,b,c; from left to right) Up to homeomorphism, the points in $\Sigma\sqcup\{p,q\}$ and paths $ps$ look as in image (a). The second, image (b), shows a possible cover of $\mathbf P^1$ by $\mathcal{V}$ and $\mathcal{U}_s$ as described above. In particular, it is clear that such a cover exists (homeomorphically to this image). Finally, image (c) shows a cover of the set $\mathcal{U}_a$, as below.
  • Figure 4: An $8$-pinched torus.
  • Figure 5: First row: A schematic representation of $X_0$ with $(n,m) = (6,8)$. Pinched tori have one (nondegenerate) cycle as a dominant feature, so their product naturally resembles a torus. The light areas are those at which one of the two components degenerates to a point, so they are 2-dimensional varieties. The dark areas are locally 4-dimensional and we make no attempt to represent them. The second and third picture highlight natural representatives of $uv$ and $vu$, which are locally $1$-dimensional and whose classes thus clearly vanish. Second row: A schematic representation of $\widehat{X}_0$, which differs by the replacement of each node by a sphere. The second and third picture again highlight representatives of $uv$ and $vu$, but this time the classes do not vanish since they respectively contain $m$ and $n$ different spheres.

Theorems & Definitions (40)

  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Example 3.1
  • Example 3.2
  • Proposition 3.3
  • Definition 3.4
  • Remark 3.5
  • Example 3.6
  • Remark 3.7
  • ...and 30 more