Bounds on the multiplicity of the Hecke eigenvalues

Fix an integer $N$ and a prime $p\nmid N$ where $p\geq 5$. We show that the number of newforms $f$ (up to a scalar multiple) of level $N$ and even weight $k$ such that $\mathcal{T}_p(f)=0$ is bounded independently of $k$, where $\mathcal{T}_p(f)$ is the Hecke operator.Comment: Comments are always very welcome

The least prime number represented by a binary quadratic form

Let $D<0$ be a fundamental discriminant and $h(D)$ be the class number of $\mathbb{Q}(\sqrt{D})$. Let $R(X,D)$ be the number of classes of the binary quadratic forms of discriminant $D$ which represent a prime number in the interval $[X,2X]$. Moreover, assume that $\pi_{D}(X)$ is the number of primes, which split in $\mathbb{Q}(\sqrt{D})$ with norm in the interval $[X,2X].$ We prove that $$\Big(\frac{\pi_D(X)}{\pi(X)}\Big)^2 \ll \frac{R(X,D)}{h(D)}\Big(1+\frac{h(D)}{\pi(X)}\Big),$$ where $\pi(X)$ is the number of primes in the interval $[X,2X]$ and the implicit constant in $\ll$ is independent of $D$ and $X$

Complexity of strong approximation on the sphere

By assuming some widely-believed arithmetic conjectures, we show that the task of accepting a number that is representable as a sum of $d\geq2$ squares subjected to given congruence conditions is NP-complete. On the other hand, we develop and implement a deterministic polynomial-time algorithm that represents a number as a sum of 4 squares with some restricted congruence conditions, by assuming a polynomial-time algorithm for factoring integers and Conjecture~\ref{cc}. As an application, we develop and implement a deterministic polynomial-time algorithm for navigating LPS Ramanujan graphs, under the same assumptions.Comment: Submitted to Mathematics of Computatio

Quadratic forms and semiclassical eigenfunction hypothesis for flat tori

Let $Q(X)$ be any integral primitive positive definite quadratic form with discriminant $D$ and in $k$ variables where $k\geq4$. We give an upper bound on the number of integral solutions of $Q(X)=n$ for any integer $n$ in terms of $n$, $k$ and $D$. As a corollary, we give a definite answer to a conjecture of Rudnick and Lester on the small scale equidistribution of orthonormal basis of eigenfunctions restricted to an individual eigenspace on the flat torus $\mathbb{T}^d$ for $d\geq 5$. Another application of our main theorem gives a sharp upper bound on $A_{d}(n,t)$, the number of representation of the positive definite quadratic form $Q(x,y)=nx^2+2txy+ny^2$ as a sum of squares of $d\geq 5$ linear forms where $n- n^{\frac{1}{(d-1)}-o(1)}< t < n$. This upper bound allows us to study the local statistics of integral points on sphere.Comment: Submitted to CM

Diameter of Ramanujan Graphs and Random Cayley Graphs

We study the diameter of LPS Ramanujan graphs $X_{p,q}$. We show that the diameter of the bipartite Ramanujan graphs is greater than $(4/3)\log_{p}(n) +O(1)$ where $n$ is the number of vertices of $X_{p,q}$. We also construct an infinite family of $(p+1)$-regular LPS Ramanujan graphs $X_{p,m}$ such that the diameter of these graphs is greater than or equal to $\lfloor (4/3)\log_{p}(n) \rfloor$. On the other hand, for any $k$-regular Ramanujan graph we show that the distance of only a tiny fraction of all pairs of vertices is greater than $(1+\epsilon)\log_{k-1}(n)$. We also have some numerical experiments for LPS Ramanujan graphs and random Cayley graphs which suggest that the diameters are asymptotically $(4/3)\log_{k-1}(n)$ and $\log_{k-1}(n)$, respectively

Optimal strong approximation for quadratic forms

For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in $d\geq5$ variables, we prove an optimal strong approximation theorem. Let $\Omega$ be a fixed compact subset of the affine quadric $F(x_1,\dots,x_d)=1$ over the real numbers. Take a small ball $B$ of radius $0<r<1$ inside $\Omega$, and an integer $m$. Further assume that $N$ is a given integer which satisfies $N\gg_{\delta,\Omega}(r^{-1}m)^{4+\delta}$ for any $\delta>0$. Finally assume that an integral vector $(\lambda_1, \dots, \lambda_d)$ mod $m$ is given. Then we show that there exists an integral solution $X=(x_1,\dots,x_d)$ of $F(X)=N$ such that $x_i\equiv \lambda_i \text{ mod } m$ and $\frac{X}{\sqrt{N}}\in B$, provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a non-degenerate integral quadratic form in 4 variables we prove the same result if $N$ is odd and $N\gg_{\delta,\Omega} (r^{-1}m)^{6+\epsilon}$. Based on our numerical experiments on the diameter of LPS Ramanujan graphs and the expected square root cancellation in a particular sum that appears in Remark~\ref{evidence}, we conjecture that the theorem holds for any quadratic form in 4 variables with the optimal exponent $4$.Comment: Comments are always very welcome

The Siegel variance formula for quadratic forms

We introduce a smooth variance sum associated to a pair of positive definite symmetric integral matrices $A_{m\times m}$ and $B_{n\times n}$, where $m\geq n$. By using the oscillator representation, we give a formula for this variance sum in terms of a smooth sum over the square of a functional evaluated on the $B$-th Fourier coefficients of the vector valued holomorphic Siegel modular forms which are Hecke eigenforms and obtained by the theta transfer from $O_{A_{m\times m}}$. By using the Ramanujan bound on the Fourier coefficients of the holomorphic cusp forms, we give a sharp upper bound on this variance when $n=1$. As applications, we prove a cutoff phenomenon for the probability that a unimodular lattice of dimension $m$ represents a given even number. This gives an optimal upper bound on the sphere packing density of almost all even unimodular lattices. Furthermore, we generalize the result of Bourgain, Rudnick and Sarnak~\cite{Bourgain}, and also give an optimal bound on the diophantine exponent of the $p$-integral points on any positive definite $d$-dimensional quadric, where $d\geq 3$. This improves the best known bounds due to Ghosh, Gorodnik and Nevo~\cite{GGN} into an optimal bound

Asymptotic trace formula for the Hecke operators

Given integers $m$, $n$ and $k$, we give an explicit formula with an optimal error term (with square root cancelation) for the Petersson trace formula involving the $m$-th and $n$-th Fourier coefficients of an orthonormal basis of $S_k(N)^*$ (the weight $k$ newforms with fixed square-free level $N$) provided that $|4 \pi \sqrt{mn}- k|=o(k^{\frac{1}{3}})$. Moreover, we establish an explicit formula with a power saving error term for the trace of the Hecke operator $\mathcal{T}_n^*$ on $S_k(N)^*$ averaged over $k$ in a short interval. By bounding the second moment of the trace of $\mathcal{T}_{n}$ over a larger interval, we show that the trace of $\mathcal{T}_n$ is unusually large in the range $|4 \pi \sqrt{n}- k| = o(n^{\frac{1}{6}})$. As an application, for any fixed prime $p$ with $\gcd(p,N)=1$, we show that there exists a sequence $\{k_n\}$ of weights such that the error term of Weyl's law for $\mathcal{T}_p$ is unusually large and violates the prediction of arithmetic quantum chaos. In particular, this generalizes the result of Gamburd, Jakobson and Sarnak~\cite[Theorem 1.4]{Gamburd} with an improved exponent.Comment: Comments are always very welcome

Quantum Chaos on random Cayley graphs of SL2[Z/pZ]{\rm SL}_2[\mathbb{Z}/p\mathbb{Z}]

We investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of ${\rm SL}_2[\mathbb{Z}/p\mathbb{Z}]$ %and the Symmetric group $S_n$ as the prime number $p$ goes to infinity. We prove a density theorem for the number of exceptional eigenvalues of random Cayley graphs i.e. the eigenvalues with absolute value bigger than the optimal spectral bound. Our numerical results suggest that random Cayley graphs of ${\rm SL}_2[\mathbb{Z}/p\mathbb{Z}]$ and the explicit LPS Ramanujan projective graphs of $\mathbb{P}^1(\mathbb{Z}/p\mathbb{Z})$ have optimal spectral gap and diameter as the prime number $p$ goes to infinity

The diophantine exponent of the Z/qZ\mathbb{Z}/q\mathbb{Z} points of Sd−2⊂SdS^{d-2}\subset S^d

Assume a polynomial-time algorithm for factoring integers, Conjecture~\ref{conj}, $d\geq 3,$ and $q$ and $p$ are prime numbers, where $p\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\log(q)$ that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d-2}\subset S^{d}$ to a $\mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$.Comment: Comments are always very welcome