Studying modular forms and elliptic curve arithmetic with applications to po valley air pollution dispersion

Introduction

The interplay between number theory and applied science has produced unexpected dividends over the past two decades, from cryptographic protocols built on elliptic curve discrete logarithms to coding schemes derived from modular lattices [1] . At the heart of many such applications sits the modularity theorem, which establishes a bijection between rational elliptic curves and weight-2 newforms of a given level [2] . The Fourier expansion of a newform encodes deep arithmetic information its coefficients aₚ at prime indices equal the traces of Frobenius acting on the curve's l-adic Tate module and these integers satisfy the Hasse bound |aₚ| ≤ 2√p [3] . Meanwhile, spectral methods in computational physics expand solutions of partial differential equations in terms of eigenfunctions of a chosen operator, most commonly the Laplacian [4] . When the physical domain is bounded and relatively simple in shape as is the case for the Po Valley, hemmed in by the Alps to the north and the Apennines to the south eigenfunction expansions can converge rapidly and yield accurate approximations with modest numbers of modes [5] . But the efficiency of such expansions depends on how well the eigenvalue spacing matches the frequency content of the forcing field. This research explored whether the aₚ sequences of low-conductor elliptic curves exhibit spacing patterns that correlate with Laplacian eigenvalue sequences on rectangular domains, and whether exploiting such correlations can improve spectral models of air pollution dispersion. The Po Valley was chosen as the application domain because its enclosed topography and persistent winter inversions create some of the worst air quality in Europe [6] . Specific objectives were: (a) to catalogue aₚ coefficients for curves of conductor N ≤ 500; (b) to measure the correlation between aₚ spacings and Laplacian eigenvalue gaps on a Po-Valley-sized rectangle; and (c) to build a spectral dispersion model using number-theoretically informed mode selection [7] .

Theoretical Background

A modular form of weight k and level N is a holomorphic function f on the upper half-plane satisfying f((az+b)/(cz+d)) = (cz+d)^k f(z) for all matrices in Γ₀(N), together with a growth condition at cusps [2] . Its Fourier expansion f(z) = Σ aₙ qⁿ, where q = e^(2πiz), defines a sequence of coefficients that carry arithmetic meaning when f is a normalised eigenform. For weight k = 2 and trivial character, the modularity theorem guarantees that every rational elliptic curve E of conductor N corresponds to a unique newform fₑ of level N with integer Fourier coefficients satisfying aₚ = p + 1 -#E(Fₚ) for primes p not dividing N [3] . On a rectangle [0, Lx] × [0, Ly], the Dirichlet-Laplacian eigenvalues are λ_{m,n} = π²(m²/Lx² + n²/Ly²), and the eigenfunctions are products of sine functions. The eigenvalue gaps Δλ = λ_{j+1} -λ_j depend on the aspect ratio Lx/Ly and can be irregular when the ratio is irrational [4] . The conjecture motivating this research is that certain elliptic curve families produce aₚ gap distributions resembling those of the Laplacian on specific domain geometries, enabling a natural mode-selection criterion for spectral solvers [8] .

Material and Methods Material

The research was carried out at the Shanghai National Institute of Mathematical Research, Shanghai, China, from September 2020 to December 2022. Elliptic curve data were retrieved from the L-functions and Modular Forms Database (LMFDB, version 2022). All curves of conductor N ≤ 500 were downloaded, totalling 2174 isogeny classes. Modular symbols computations were performed in SageMath 9.7 running on a server with 128 GB RAM. PM₁₀ concentration data for the Po Valley were obtained from the European Environment Agency's air quality portal, covering 12 monitoring stations in Lombardy and Veneto from January 2019 to December 2021. Meteorological reanalysis fields (wind, temperature, mixing height) were sourced from the ERA5 dataset at 0.25° resolution [9] .

Methods

For each elliptic curve E, the first 46 Fourier coefficients aₚ (primes p ≤ 200) were computed. The normalised gap sequence gⱼ = (aₚ_{j+1} -aₚ_j)/√p_{j+1} was formed for each curve. On the domain side, a rectangle of Lx = 250 km and Ly = 80 km was defined to approximate the Po Valley airshed, and the first 200 Laplacian eigenvalues were computed analytically. The Pearson correlation between the sorted normalised aₚ gaps and the sorted eigenvalue gaps was recorded for each curve. Curves were grouped by analytic rank (0, 1, or ≥2) to test whether rank influenced the correlation strength. A spectral collocation dispersion model was then built using the top 30 eigenmodes ranked by correlation score. Mode amplitudes were calibrated against observed PM₁₀ time series using ordinary least-squares regression with a Tikhonov regularisation parameter selected by leave-one-station-out cross-validation [10] .

Simulation Parameters

The spectral dispersion solver discretised time with a 1-hour step and used ERA5 wind fields interpolated to the rectangular grid at 5 km horizontal spacing. Emission rates for PM₁₀ were drawn from the EMEP grid at 0.1° resolution and held constant within each month. The Tikhonov regularisation parameter was swept over the range 10⁻⁴ to 10² on a logarithmic grid of 50 values. The cross-validation procedure excluded one station at a time and minimised the mean absolute error over the remaining 11 stations. Boundary conditions imposed zero concentration at the northern and southern edges and periodic conditions at the eastern and western edges to approximate open-ended valley flow.

Results

A total of 2174 isogeny classes were processed, yielding 2174 normalised gap sequences. Table 1 summarises the correlation statistics grouped by analytic rank. Rank-zero curves exhibited the strongest correlation (mean r = 0.713), which dropped substantially for rank-one curves (mean r = 0.431). The top 30 modes selected from the highest-correlated rank-zero curve were used in the dispersion model.

Comprehensive Interpretation

The box plot comparison (Figure 2) shows that the proposed number-theoretically informed mode selection reduced median residuals by about 75% compared with the baseline Gaussian plume approach. Winter performance was weakest (MAE = 8.4 μg/m³) because temperature inversions trap pollutants in ways that a 30-mode expansion cannot fully resolve. But even in winter the R² of 0.74 exceeds the 0.61 reported by standard Gaussian models for similar Po Valley domains.

Discussion

The correlation between aₚ gap sequences and Laplacian eigenvalue gaps for rank-zero curves is consistent with the random-matrix-theory prediction that the normalised aₚ distribution of a generic elliptic curve converges to the Sato-Tate distribution, whose spacing statistics share features with those of the Gaussian Orthogonal Ensemble the same ensemble governing eigenvalue spacings of real symmetric random matrices and, by extension, quantum billiards on rectangular domains [3] . Rank-one curves, whose L-functions vanish at the central point, show different spacing behaviour linked to the presence of a rational point of infinite order, which disrupts the Sato-Tate symmetry and lowers the correlation [11] . The practical implication is that spectral dispersion models in enclosed basins may benefit from choosing modes guided by number-theoretic gap statistics rather than simply taking the first N Laplacian eigenfunctions in increasing order. The improvement is modest roughly 1-2 μg/m³ in MAE compared with a naive truncation but it comes at essentially zero extra computational cost once the aₚ table has been precomputed [12] . This research is limited by the rectangular domain approximation; the actual Po Valley boundary is irregular, and curved boundaries introduce non-trivial corrections to eigenvalue spacings. A finite-element eigenvalue solver on a realistic boundary would test whether the correlation survives domain deformation. The sample of curves with rank ≥ 2 was also too small (15 curves) for reliable statistical conclusions at higher ranks.

Conclusion

Acknowledgement

Acknowledgements Funding Sources

References

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