arXiv subjects
Publications and source records attributed to Tianyu Ni.
Let $S_k$ be the space of cusp forms of weight $k$ for the full modular group ${\rm SL}_2(\mathbb{Z})$. In this paper, we will exhibit an explicit rational spanning subset of $S_k$ constructed from Eisenstein series at prime levels. The main ingredient of our proof is to show a result on determining cusp forms by noncentral quadratic-twist $L$-values.
Let $S_{\kappa}$ be the space of cusp forms of weight $\kappa$ and level one, and let $S_{\kappa}^{\ast}$ denote its dual space. In this paper, we find explicit spanning subsets of $S_{\kappa}$ consisting of Rankin-Cohen brackets of Eisenstein series and explicit subsets of periods that span $S_{\kappa}^{\ast}$.
We present an infinite family of identities that represent Ramanujan's tau function in terms of convolution sums of twisted divisor functions. Our method involves explicitly constructing non-vanishing level $1$ cusp forms from modular forms of higher levels.
Let $S_k$ denote the space of cusp forms of weight $k$ and level one. For $0\leq t\leq k-2$ and primitive Dirichlet character $χ$ mod $D$, we introduce twisted periods $r_{t,χ}$ on $S_k$. We show that for a fixed natural number $n$, if $k$ is sufficiently large relative to $n$ and $D$, then any $n$ periods with the same twist but different indices are linearly independent. We also prove that if $k$ is sufficiently large relative to $D$ then any $n$ periods with the same index but different twists mod $D$ are linearly independent. These results are achieved by studying the trace of the products and Rankin-Cohen brackets of Eisenstein series of level $D$ with nebentypus. Moreover, we give two applications of our method. First, we prove certain identities that evaluate convolution sums of twisted divisor functions. Second, we show that Maeda's conjecture implies a non-vanishing result on twisted central $L$-values of normalized Hecke eigenforms.
For $S_k$, the space of cusp forms of weight $k$ for the full modular group, we first introduce periods on $S_k$ associated to symmetric square $L$-functions. We then prove that for a fixed natural number $n$, if $k$ is sufficiently large relative to $n$, then any $n$ such periods are linearly independent. With some extra assumption, we also prove that for $k\geq e^{12}$, we can always pick up to $\frac{\log k}{4}$ arbitrary linearly independent periods.
Let $T_m(N,2k)$ denote the $m$-th Hecke operator on the space $S_{2k}(Γ_0(N))$ of cuspidal modular forms of weight $2k$ and level $N$. In this paper, we study the non-repetition of the second coefficient of the characteristic polynomial of $T_m(N,2k)$. We obtain results in the horizontal aspect (where $m$ varies), the vertical aspect (where $k$ varies), and the level aspect (where $N$ varies). Finally, we use these non-repetition results to extend a result of Vilardi and Xue on distinguishing Hecke eigenforms.
In this paper, we define a version of the arithmetic-geometric mean (AGM) function for arbitrary finite fields $\mathbb{F}_q$, and study the resulting AGM graph with points $(a,b) \in \mathbb{F}_q \times \mathbb{F}_q$ and directed edges between points $(a,b)$, $(\frac{a+b}{2},\sqrt{ab})$ and $(a,b)$, $(\frac{a+b}{2},-\sqrt{ab})$. The points in this graph are naturally associated to elliptic curves over $\mathbb{F}_q$ in Legendre normal form, with the AGM function defining a 2-isogeny between the associated curves. We use this correspondence to prove several results on the structure, size, and multiplicity of the connected components in the AGM graph.
In this paper, we construct explicit spanning sets for two spaces. One is the subspace generated by integral-weight Hecke eigenforms with nonvanishing quadratic twisted central $L$-values. The other is a subspace generated by half-integral weight Hecke eigenforms with certain nonvanishing Fourier coefficients. Along the way, we show that these subspaces are isomorphic via the Shimura lift.
The price-anticipating Kelly mechanism (PAKM) is one of the most extensively used strategies to allocate divisible resources for strategic users in communication networks and computing systems. The users are deemed as selfish and also benign, each of which maximizes his individual utility of the allocated resources minus his payment to the network operator. However, in many applications a user can use his payment to reduce the utilities of his opponents, thus playing a misbehaving role. It remains mysterious to what extent the misbehaving user can damage or influence the performance of benign users and the network operator. In this work, we formulate a non-cooperative game consisting of a finite amount of benign users and one misbehaving user. The maliciousness of this misbehaving user is captured by his willingness to pay to trade for unit degradation in the utilities of benign users. The network operator allocates resources to all the users via the price-anticipating Kelly mechanism. We present six important performance metrics with regard to the total utility and the total net utility of benign users, and the revenue of network operator under three different scenarios: with and without the misbehaving user, and the maximum. We quantify the robustness of PAKM against the misbehaving actions by deriving the upper and lower bounds of these metrics. With new approaches, all the theoretical bounds are applicable to an arbitrary population of benign users. Our study reveals two important insights: i) the performance bounds are very sensitive to the misbehaving user's willingness to pay at certain ranges; ii) the network operator acquires more revenues in the presence of the misbehaving user which might disincentivize his countermeasures against the misbehaving actions.