In this paper, we study the Mellin transform of the weight 2 level \(N\,\)polar harmonic Maass form \(H_N,z^*(τ)\,\)and analyze this (generalized) \(L\)-function as \(Im(z) \to ∞\). On the way, we also calculate the Fourier expansion of \(H_N,z^*(τ)\,\)at arbitrary cusps of \(\Gamma_0(N)\), and we give a functional equation and factorization into local factors of the \(L\)-function for the weight 2 level \(N\,\)Eisenstein series at the cusps \(i∞\,\)and 0.
@misc{polar_harmonic_maass_forms,title={The Completed \(L\)-function attached to the Weight 2 Polar Harmonic Maass Form \(H\_{N,z}^{*}(\tau)\)},author={Singhal, Kush},year={2022},eprint={2201.03146},archiveprefix={arXiv},primaryclass={math.NT},}
Near-miss Identities and Spinor Genus Classification of Ternary Quadratic Forms with Congruence Conditions
In this paper, near-miss identities for the number of representations of some integral ternary quadratic forms with congruence conditions are found and proven. The genus and spinor genus of the corresponding lattice cosets are then classified. Finally, a complete genus and spinor genus classification for all conductor 2 lattice cosets of 2-adically unimodular lattices is given. Published in the International Journal of Number Theory.
@misc{near_miss_quad_forms_identity,title={Near-miss Identities and Spinor Genus Classification of Ternary Quadratic Forms with Congruence Conditions},author={Singhal, Kush},year={2021},eprint={2104.08798},archiveprefix={arXiv},primaryclass={math.NT},}