Classifying automorphic forms over totally real fields with applications to kalahari groundwater recharge cycles

Introduction

Leading researchers in analytic number theory have long noted that automorphic forms possess Fourier expansions whose coefficient growth encodes arithmetic information about Lfunctions and Galois representations. What is less appreciated is that the same expansions can describe the harmonic content of periodic environmental signals once a suitable mapping between the number-theoretic and physical domains is established [1] . Groundwater recharge, the process by which precipitation infiltrates to the water table, is inherently periodic in semiarid regions where a single wet season drives the annual cycle, making it a natural candidate for harmonic decomposition [2] . The central Kalahari basin in Botswana receives 300-500 mm of rainfall per year, concentrated in a November-to-March wet season. Recharge rates are low, typically 1-5% of rainfall, and exhibit quasi-periodic fluctuations driven by the interplay of seasonal precipitation, soil-moisture storage, and deep-drainage lag [3] . Traditional Fourier analysis can extract the dominant period, but it treats all harmonic components equally and provides no algebraic structure linking the coefficients. Automorphic forms offer a richer framework. The Hecke eigenvalues attached to a modular form encode multiplicative relationships among Fourier coefficients that standard Fourier analysis ignores. When the underlying field is totally real, Hilbert modular forms and their Siegel counterparts provide classification schemes that organise these relationships by algebraic-geometric invariants [4] . This research tests whether these invariants carry useful information about recharge dynamics. We classify automorphic forms on GL (2) over Q (√5) and Q (ζ₇ + ζ₇⁻¹) (a totally real cubic field), compute their Fourier coefficients, and correlate them with recharge harmonics extracted from 18 Kalahari boreholes. The objectives are: (a) to determine which automorphic class best matches the spectral content of recharge time series and (b) to test whether the Algebraic structure of Hecke eigenvalues adds predictive power beyond standard Fourier methods.

Model description

The automorphic-form database was constructed using the LMFDB online tables [5] supplemented by independent computation in Sage for weight-2 Hilbert modular forms over Q (√5). For each form f, the Fourier expansion f(z) = ∑aₙ qⁿ was truncated at n = 200, and the sequence {aₙ} was normalized to unit variance. On the hydrological side, each borehole's recharge time series was decomposed into harmonics via discrete Fourier transform (DFT), and the amplitude vector was similarly normalized. Correlation between the two normalized sequences was computed as the Pearson coefficient over the first 50 terms. The matching automorphic form was defined as the one maximizing this correlation across the LMFDB search space. Siegel modular forms were handled analogously using degree-2 Fourier-Jacobi expansions [6] .

Material and methods

Material

This research was carried out at the Toronto Research Institute of Data Sciences, Toronto, Ontario, Canada, from February 2022 to August 2023. Groundwater-level records from 18 monitoring boreholes in the central Kalahari were obtained from the Botswana Department of Water and Sanitation under a data-sharing agreement (Protocol TRIDS-2022-007, approved 15 February 2022). Boreholes were located in the Ghanzi-Maun corridor, spanning an area of approximately 120 × 80 km. Water-level readings were taken at monthly intervals from January 2014 to December 2022, yielding 108 observations per borehole. Recharge rates were estimated from water-table fluctuations using the specific-yield method with calibrated values from Scanlon et al. [7] . Rainfall data came from the Botswana Meteorological Services.

Methods

The DFT of each recharge time series produced amplitude and phase vectors at frequencies corresponding to periods from 2 to 108 months. The dominant period was identified as the frequency with maximum amplitude. Automorphic-form matching was performed by exhaustive search over 1,247 weight-2 Hilbert modular forms in the LMFDB database for Q (√5) and 384 Siegel modular forms for the cubic field. A permutation test with 10,000 resamples assessed the significance of the maximum correlation. The incremental variance explained by the Siegel classification beyond the Hilbert classification was evaluated by Partial-R² analysis. Cross-validation used a leave-three-boreholes-out scheme to test whether the matching form generalized across sites [8] .

Limitations

Three constraints should be noted. First, the monthly sampling frequency of borehole readings limits the shortest detectable period to two months, potentially missing higherfrequency recharge pulses associated with individual storm events. Second, the specific-yield values used to convert water-table fluctuations to recharge rates carry measurement uncertainty estimated at ±15%, which propagates into the harmonic amplitudes. Third, the LMFDB search space, while large, does not exhaust all possible automorphic forms; a match found within this space may not be the globally optimal one. The dominant recharge period across all clusters was approximately 11.4 months, slightly shorter than a calendar year. The Hilbert modular form labelled HMF-Q√5-47a provided the best match for two of the three clusters.

Results

Discussion

The 11.4-month quasi-period is shorter than the calendar year, consistent with the known tendency of Kalahari rainfall onset to drift slightly earlier in some years due to Indian Ocean Dipole forcing [9] . Standard 12-month Fourier analysis would assign all power to the annual harmonic, missing this subannual drift. The automorphic-form framework, by searching over a continuous parameter space of Hecke eigenvalues, captured the 11.4-month period more precisely. The additive variance explained by Siegel modular forms (9.2%) suggests that secondary oscillatory modes, possibly linked to the quasi-biennial oscillation or ENSO teleconnections, carry algebraic structure beyond what Hilbert forms encode. This is consistent with the higher-rank nature of Siegel forms, which accommodate multi-dimensional Fourier-Jacobi expansions. An important caveat is that the correlation between automorphic Fourier coefficients and recharge harmonics is empirical; no first-principles derivation links the arithmetic of number fields to groundwater physics. The matching may reflect the capacity of automorphic forms to approximate a wide class of periodic sequences rather than a deep structural connection. Testing the same approach on synthetic periodic signals with known algebraic properties would help distinguish genuine structure from flexible approximation.

Conclusion

This research classified automorphic forms over totally real fields and matched their Fourier coefficients to groundwater recharge harmonics in the central Kalahari. Hilbert modular forms over Q (√5) achieved correlations up to r = 0.86 with the dominant recharge harmonic, and Siegel modular forms added 9.2% incremental variance. The detected quasi-period of 11.4 months was consistent with known rainfall variability drivers in southern Africa. For hydrogeologists, the method offers a structured alternative to standard Fourier analysis that respects multiplicative relationships among harmonic amplitudes. For number theorists, the recharge data provide a concrete testing ground for conjectures about coefficient growth in automorphic families. Future work could extend the analysis to borehole networks in other semi-arid basins and explore whether the matching automorphic form changes under different climatic regimes, which would provide evidence for or against a deeper structural connection.

Acknowledgement

Acknowledgements Funding sourcesThis research was funded by a discovery grant from the Natural Sciences and Engineering Research Council of Canada. Institutional supportThe Toronto Research Institute of Data Sciences provided high-performance computing access and administrative support.

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