SMFs

SMFs is a Wolfram Language / Mathematica package for working with truncated genus-2 Siegel modular forms.

It provides two complementary ways to work:

• directly in nome variables through truncated Fourier expansions,
• symbolically in geometric variables (T, Z, U) through replacement rules.

The package includes:

• truncated genus-2 Eisenstein series via Ek,
• truncated expansions of the Igusa cusp forms via Chi10 and Chi12,
• symbolic modular-form heads E4, E6, E10, E12, \[Chi]10, \[Chi]12,
• precomputed replacement rules via PrecomputedRules[n],
• support for derivatives of the symbolic modular forms.

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Files

• SMFs.m — the package
• examples.nb — example notebook illustrating the main workflows

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Loading the package

In a notebook located in the same directory as SMFs.m, load the package with

SetDirectory[NotebookDirectory[]];
Get["SMFs.m"];

To inspect the exported symbols, use

Names["SMFs`*"]

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Main workflows

1. Direct expansion in nome variables

The functions

• Ek[k, n, q1, r, q2]
• Chi10[n, q1, r, q2]
• Chi12[n, q1, r, q2]

return truncated Fourier expansions in the nome variables

• q1 = Exp[2 Pi I T]
• r = Exp[2 Pi I Z]
• q2 = Exp[2 Pi I U]

Examples:

Chi10[2, q1, r, q2]
Chi12[2, q1, r, q2]
Ek[4, 2, q1, r, q2]

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2. Symbolic replacement in (T, Z, U)

The symbolic heads

• E4[T, Z, U]
• E6[T, Z, U]
• E10[T, Z, U]
• E12[T, Z, U]
• \[Chi]10[T, Z, U]
• \[Chi]12[T, Z, U]

can be replaced by truncated Fourier expansions using

PrecomputedRules[n]

For example:

rules = PrecomputedRules[2];
\[Chi]10[T, Z, U] /. rules

or for a composite expression:

expr = E4[T, Z, U]^3/\[Chi]12[T, Z, U];
expr /. PrecomputedRules[2]

This is especially useful for numerical evaluation, plotting, and symbolic experimentation.

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Derivatives

The precomputed rules also support derivatives with respect to T, Z, and U.

Examples:

Derivative[1, 0, 0][E4][T, Z, U] /. PrecomputedRules[2]
Derivative[0, 1, 0][\[Chi]12][T, Z, U] /. PrecomputedRules[2]
Derivative[1, 1, 0][\[Chi]10][T, Z, U] /. PrecomputedRules[2]

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Fixed points of the Siegel upper half space

The example notebook also includes a collection of distinguished fixed points of the Siegel upper half space, stored as replacement rules

• pt1, pt2, ..., pt6

and collected into the list

• allpts

These can be used to evaluate modular forms and their derivatives numerically at special points.

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Exported symbols

Direct nome expansions

• Ek
• Chi10
• Chi12

Symbolic modular forms

• E4
• E6
• E10
• E12
• \[Chi]10
• \[Chi]12

Replacement rules

• PrecomputedRules

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Notes

• All expansions are truncated, so results depend on the chosen order n.
• The symbolic variable order is consistently (T, Z, U).
• Chi10 and Chi12 are the nome-series generators.
• \[Chi]10 and \[Chi]12 are symbolic modular-form heads intended to be replaced using PrecomputedRules[n].

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Example notebook

The file examples.nb provides a minimal introduction to the package and demonstrates:

• how to load SMFs.m,
• how to inspect the exported symbols,
• how to compute direct truncated expansions in nome variables,
• how to replace symbolic modular forms in (T, Z, U),
• how to evaluate derivatives,
• how to test expressions at fixed points of the Siegel upper half space.

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Citation

If you use SMFs in work that leads to a publication, please cite the companion paper:

@article{Leedom:2026nby,
    author = "Leedom, Jacob M. and Righi, Nicole and Westphal, Alexander",
    title = "{Automorphic Structures of Heterotic Vacua}",
    eprint = "2605.05322",
    archivePrefix = "arXiv",
    primaryClass = "hep-th",
    reportNumber = "DESY-26-063",
    month = "5",
    year = "2026"
}

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Author

Nicole Righi