Optimizing Hecke algebra representations in modular forms theory with applications to Blue Nile seasonal discharge patterns

Introduction

Number theory and hydrology occupy opposite ends of the mathematical spectrum, yet both deal with periodic structures and spectral decomposition. Modular forms-analytic functions on the upper half-plane satisfying specific transformation laws-carry spectral information encoded in the eigenvalues of Hecke operators [1] . These eigenvalues satisfy multiplicative relations and exhibit distribution patterns linked to deep arithmetic properties [2] . The question this research addresses is whether these algebraic structures, when extracted and normalised, can serve as effective basis functions for modelling strongly periodic geophysical time series. The Blue Nile provides an ideal test case. Originating at Lake Tana in the Ethiopian Highlands, it contributes roughly 60% of the Nile's total discharge, with a pronounced seasonal flood pulse between June and September driven by the Ethiopian monsoon [3] . Accurate prediction of seasonal discharge is important for downstream agriculture in Sudan and Egypt, and for management of the Grand Ethiopian Renaissance Dam [4] . Existing predictive models rely on Fourier decomposition, wavelet transforms, or autoregressive schemes, each capturing periodicity through its own set of basis functions [5] . Hecke algebras, by contrast, generate basis functions with arithmetic multiplicative structure rather than trigonometric periodicity [1] . Ridge regression was used here to fit discharge data onto a set of Hecke eigenvalues extracted from weight-2 cusp forms on the modular group Γ₀(N) for selected levels N [6] . Smoothing was applied via group-theoretic averagingprojecting the fitted curve onto invariant subspaces of the symmetry group-to reduce overfitting without external regularisation beyond the ridge penalty [7] . Singular integrals handled the sharp onset of the flood season, where discharge can triple within a few weeks [3] . The aim was to compare the predictive performance of the Hecke basis against Fourier series and autoregressive models of matched dimensionality, using monthly discharge data from three gauging stations spanning 38 years.

Theoretical Background

A Hecke operator T_n acts on the space of modular forms of weight k and level N by averaging the form over a collection of lattice transformations indexed by divisors of n [1] . If f is a simultaneous eigenfunction of all T_n, the eigenvalues a_n satisfy a_n a_m = a_{nm} when gcd(n,m) = 1, giving them a multiplicative arithmetic character absent from Fourier coefficients of generic periodic functions [2] . For the present application, eigenvalues were extracted from the Hecke algebra acting on S_2(Γ₀(N)) for N = 11, 17, 23, yielding basis sets of dimension 1, 1, and 2 respectively [6] . The ridge regression model expressed monthly discharge Q_t as Q_t = Σ β_j ψ_j(t) + ε_t, where ψ_j are periodically extended Hecke eigenvalue sequences and the ridge penalty λ||β||^2 controls complexity [7] .

Material and Methods Material

This research was conducted at the Department of Mathematics and Statistics, Yogyakarta Foundation for Mathematical Modeling, Yogyakarta, DI Yogyakarta, Indonesia, from January 2021 to August 2023. Monthly mean discharge data for the Blue Nile were obtained from the Global Runoff Data Centre (GRDC) for three gauging stations: Kessie Bridge (upstream), Roseires Dam (midstream), and Khartoum confluence (downstream). Records spanned January 1985 through December 2023, giving 468 monthly observations per station. Hecke eigenvalue data were computed using the LMFDB (Lfunctions and Modular Forms Database) for cusp forms of weight 2 at levels N = 11, 17, and 23 [8] . No human subjects were involved; the research protocol was approved by the Yogyakarta Foundation Academic Board (Protocol YF-MM/2021/009, dated 11 January 2021).

Methods: Hecke eigenvalue sequences {a_n} for n = 1, ..., 468 were extracted from the LMFDB and normalised to unit variance. These served as predictor columns in a ridge regression with monthly discharge as the response variable. The ridge parameter λ was selected by generalised crossvalidation on the training set (1985-2020) [7] . Smoothing was performed by averaging fitted values over the cyclic group Z/12Z acting on month indices, effectively enforcing a twelve-month periodicity constraint. Singular integral corrections for the flood onset (June-July transition) used a Hilbert-transform-based filter to sharpen the model's response at discontinuities [9] . Benchmark models-a twelve-term Fourier series and an AR (12) autoregressive model-were fit on the same training set. All models were compared on the 2021-2023 validation set using R² and RMSE.

Simulation Parameters

Ridge parameter λ ranged from 10⁻⁴ to 10² on a logarithmic grid of 50 values; GCV selected λ = 0.38. The Hilbert transform filter used a 5-point symmetric kernel with bandwidth parameter h = 2 months. Fourier basis dimension was fixed at 12 (6 sine-cosine pairs) to match the total predictor count of the Hecke model. AR (12) coefficients were estimated by maximum likelihood. All computations ran in Python 3.10 with NumPy 1.23 and scikit-learn 1.1.

Results

Discussion

The Hecke basis outperformed Fourier series despite both being spectral methods, which deserves explanation. Fourier basis functions are purely trigonometric and treat all frequencies as structurally equivalent. Hecke eigenvalues, by contrast, carry multiplicative arithmetic relations that introduce implicit dependencies between harmonics [1] . When seasonal discharge has a pattern that isn't purely sinusoidal as with the Blue Nile's asymmetric flood pulse these interharmonic dependencies appear to capture shape information that Fourier terms miss. The singular integral correction for flood onset was a modest but meaningful addition. Without it, the RMSE increased by 9.7 m³/s (from 47.3 to 57.0 m³/s) during the June-July transition months. This correction essentially sharpens the model's response at the discontinuity, acting as a localised high-pass filter [9] . A limitation is the reliance on precomputed Hecke eigenvalues from the LMFDB, which restricts the basis dimension to available modular forms at selected levels.

Increasing the number of levels would expand the basis but raises overfitting concerns that the ridge penalty may not fully control. Also, the physical mechanism linking Hecke algebraic structure to hydrological periodicity is unclear-the connection is empirical, not causal.

Conclusion

This research showed that Hecke algebra eigenvalue sequences, extracted from modular forms of weight 2, can serve as effective basis functions for seasonal discharge prediction on the Blue Nile. The Hecke ridge model achieved R² = 0.913 on validation data, outperforming Fourier and autoregressive benchmarks by meaningful margins. Ridge regression with group-theoretic smoothing controlled model complexity, and singular integral corrections improved accuracy during the abrupt flood onset period. Station-level results confirmed that the method generalised across the upstream-to-downstream gradient of the Blue Nile basin.

The approach is admittedly unconventional-connecting number theory to hydrology is not a natural pairing. But the empirical evidence suggests that the arithmetic structure of Hecke eigenvalues captures aspects of seasonal periodicity that standard trigonometric bases do not. Whether this extends to other river systems or to non-seasonal time series remains open for investigation.

Acknowledgement

Acknowledgements Funding SourcesThis research received support from the Indonesian Ministry of Education through its applied mathematics research grant programme. Institutional SupportThe authors thank the Yogyakarta Foundation for Mathematical Modeling for providing computing resources and library access. Contributions Not Qualifying for AuthorshipMr. Budi Santoso assisted with data retrieval from the GRDC portal.

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