Weber modular curves and modular isogenies

Abstract: We study the modular curves defined by Weber functions, and associated modular polynomials, action of $\mathrm{SL}_2(\mathbb{Z})$, and parametrizations of elliptic curves with a view to the study of the isogeny graphs that they determine, particularly for supersingular elliptic curves. In addition to applications to efficient isogeny computation in cryptographic applications, we present an application to explicit Galois representations.

• The paper’s main contribution is the explicit construction of Weber modular curves and associated modular polynomials with symmetry properties that dramatically reduce coefficient complexity.
• It employs precise parametrizations via Weber functions to enable efficient isogeny computations and detailed analysis of the supersingular locus for cryptographic relevance.
• The work presents broad implications for arithmetic geometry and Galois representations while advancing computational methods in isogeny-based cryptosystems.

Weber Modular Curves and Modular Isogenies: An Expert Analysis

Introduction

The paper "Weber modular curves and modular isogenies" (2603.29802) provides an in-depth analysis of modular curves defined by Weber functions, their associated modular polynomials, the symmetries induced by $SL_2(\mathbb{Z})$, and parametrizations of elliptic curves. It systematically explores the utility of these objects, emphasizing applications for efficient isogeny computation, with notable focus on the supersingular locus and implications for computational number theory and cryptographic protocols. Additionally, the paper investigates the explicit construction of Galois representations arising from these frameworks.

Weber Functions, Modular Polynomials, and Their Structure

The authors provide explicit models for modular curves determined by Weber functions, which originate from Weber’s classic work as modular functions of level $48$. Each function constructs a degree-$72$ cover of the $j$-line via a genus-$0$ modular curve, which currently represents the maximal known degree for such a cover.

The central technical result is that modular polynomials $\Phi_\ell(x, y)$ associated with the Weber function not only yield substantial reduction in both their coefficient heights and their sparsity, but also admit a symmetry:

$$\Phi_\ell(\zeta_{24}x, \zeta_{24}^\ell y) = \zeta_{24}^{\ell+1} \Phi_\ell(x,y),$$

where $\zeta_{24}$ is a $24$-th root of unity. This restricts nonzero monomials to those for which $i + \ell j \equiv \ell+1 \pmod{24}$, reducing the asymptotic number of coefficients by a factor of $48$0 compared to traditional modular polynomials. This structural result is fundamental for efficient computation in both CM theory and isogeny-based settings.

Further, the explicit formulas for the modular polynomials for small primes (e.g., $48$1) are provided, with a detailed analysis of their irreducibility properties and compatibility with isogeny chains, especially under the $48$2- and $48$3-level structures. The explicit descent to functions $48$4 and $48$5 underpins efficient arithmetic for small-degree isogeny computations.

Modular Group Actions and Curve Symmetries

A significant portion of the work is devoted to the action of $48$6 (and its quotients) on the triples of Weber functions. The authors construct projective embeddings of Weber modular curves $48$7 parameterized by the functions $48$8 (normalized Weber conjugates) and rigorously derive the action of $48$9 generators on these triples. In particular, identified automorphism groups for covers $72$0 (with $72$1 dividing $72$2) are computed, and explicit relations to generalized Fermat curves are established, notably $72$3 for $72$4.

In the modular curve tower context, congruence subgroups $72$5 associated to each Weber modular curve are characterized. These identifications leverage subgroup calculations among $72$6, various Cartan subgroups, and their intersections with non-split Cartan subgroups, which relates the modular interpretation of these curves directly to the structure of the moduli spaces.

Supersingular Isogeny Graphs and Cryptographic Applications

The theoretical advances culminate with detailed analysis of the supersingular locus on the Weber modular curves and associated isogeny graphs. The authors prove that for $72$7 arbitrary, supersingular invariants on $72$8 and its covers (including the high-level Weber curves) are defined over $72$9, with no further field extensions required to split the relevant modular correspondences. This has direct practical consequence for cryptographic protocols such as SIDH and CSIDH, particularly for the implementation of isogeny-based cryptosystems where efficient traversal and computation on supersingular isogeny graphs is critical.

The explicit parametrizations provided for these modular curves allow for the construction of streamlined isogeny chains and computation of Hecke operators, which are critical for both cryptographic and arithmetic applications. The modular polynomials’ sparseness and height minimality directly translate into improved performance in end-to-end protocols.

Modular Forms, Hecke Modules, and Galois Representations

Beyond cryptography, the paper systematically explores the role of modular isogeny graphs in the study of modular forms and Galois representations. Using Mestre's method of graphs, the authors demonstrate how the combinatorial structure of isogeny graphs on Weber and Fermat curves yields explicit constructions of supersingular Hecke modules, providing examples where modular abelian varieties can be computed even at composite levels far exceeding prior computational datasets.

The enumeration of pseudo-elliptic Hecke module factors corresponding to various levels demonstrates computational tractability of instances with large cofactors (e.g., $j$0 or $j$1) often beyond the reach of standard LMFDB datasets. The paper highlights that, especially for levels with high powers of 2 and 3, one can obtain modular data for isogeny classes well beyond conventional conductor bounds.

Implications and Future Directions

The results have substantial implications:

• Cryptography: The explicit structure, efficiency, and field-of-definition properties of Weber modular curves directly influence the practicality and security analyses of isogeny-based cryptosystems, potentially enabling new protocols leveraging high-level modular functions for compactness and performance.
• Arithmetic Geometry: The connections established between Weber curves, modular polynomials, and Fermat curves enrich the toolkit for both explicit CM theory and the study of modular forms over nontrivial level structures, especially for composite and high-power levels.
• Computational Representation Theory: The detailed modular descriptions assist in systematic Galois representation computations, extending the class of modular forms and abelian varieties accessible for explicit analysis.

Going forward, several research avenues appear promising. The maximality conjecture for class polynomial height reduction remains open. Further, an explicit modular interpretation for certain high-level twisted Fermat curves (e.g., $j$2) could yield new insights. Finally, the construction of efficient algorithms harnessing the full symmetry group and automorphism structures identified herein would further advance computational capabilities in both arithmetic and cryptographic contexts.

Conclusion

This work rigorously constructs and analyzes the algebraic and arithmetic structure of Weber modular curves, their modular polynomials, and associated isogeny graphs. Through explicit models, symmetry analysis, and applications to supersingular isogeny graphs, it delivers results of direct consequence for computational number theory, arithmetic geometry, and isogeny-based cryptography. The explicit interplay between modular functions, curve symmetries, and Galois actions opens new directions both for efficient computation and for theoretical explorations in modern arithmetic geometry and cryptographic protocol design.