Motivated by the result of Rankin for representations of integers as sums of squares, we use a decomposition of a modular form into a particular Eisenstein series and a cusp form to show that the number of ways of representing a positive integer n as the sum of k triangular numbers is asymptotically equivalent to the modified divisor function σ 2k−1 (2n + k). 2 for 0 (4) to study the functions r k (n). Ono, Robins, and Wahl [1995] defined an analogous modular form to study triangular numbers. We begin by defining triangular numbers.

Abstract. Let p be an odd prime number, and F a nonarchimedean local field of residual characteristic p. We classify the smooth, irreducible, admissible genuine mod-p representations of the twofold metaplectic cover Sp 2n (F ) of Sp 2n (F ) in terms of genuine supercuspidal (equivalently, supersingular) representations of Levi subgroups. To do so, we use results of Henniart-Vignéras as well as new technical results to adapt Herzig's method to the metaplectic setting. As consequences, we obtain an irreducibility criterion for principal series representations generalizing the complete irreducibility of principal series representations in the rank 1 case, as well as the fact that irreducibility is preserved by parabolic induction from the cover of the Siegel Levi subgroup.

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