Multiplicity one bound for cohomological automorphic representations with a fixed level

Let $F$ be a totally real field, and $mathbb{A}_F$ be the adele ring of $F$. Let us fix $N$ to be a positive integer. Let $pi_1=otimespi_{1,v}$ and $pi_2=otimespi_{2,v}$ be distinct cohomological cuspidal automorphic representations of $mathrm{GL}_n(mathbb{A}_{F})$ with levels less than or equal to $N$. Let $mathcal{N}(pi_1,pi_2)$ be the minimum of the absolute norm of $v mid infty$ such that $pi_{1,v} ot simeq pi_{2,v}$ and that $pi_{1,v}$ and $pi_{2,v}$ are unramified. We prove that there exists a constant $C_N$ such that for every pair $pi_1$ and $pi_2$, $$mathcal{N}(pi_1,pi_2) leq C_N.$$ This improves known bounds $$ mathcal{N}(pi_1,pi_2)=O(Q^A) ;;; (text{some } A text{ depending only on } n), $$ where $Q$ is the maximum of the analytic conductors of $pi_1$ and $pi_2$. This result applies to newforms on $Gamma_1(N)$. In particular, assume that $f_1$ and $f_2$ are Hecke eigenforms of weight $k_1$ and $k_2$ on $mathrm{SL}_2(mathbb{Z})$, respectively. We prove that if for all $p in {2,7}$, $$lambda_{f_1}(p)/sqrt{p}^{(k_1-1)} = lambda_{f_2}(p)/sqrt{p}^{(k_2-1)},$$ then $f_1=cf_2$ for some constant $c$. Here, for each prime $p$, $lambda_{f_i}(p)$ denotes the $p$-th Hecke eigenvalue of $f_i$.

$p$-adic Limit of Weakly Holomorphic Modular Forms of Half Integral Weight

Serre obtained the p-adic limit of the integral Fourier coefficient of modular forms on $SL_2(mathbb{Z})$ for $p=2,3,5,7$. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on $Gamma_{0}(4N)$ for $N=1,2,4$. A proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications we obtain congruences of Borcherds exponents, congruences of quotient of Eisentein series and congruences of values of $L$-functions at a certain point are also studied. Furthermore, the congruences of the Fourier coefficients of Siegel modular forms on Maass Space are obtained using Ikeda lifting.

Distribution of integral Fourier Coefficients of a Modular Form of Half Integral Weight Modulo Primes

Recently, Bruinier and Ono classified cusp forms $f(z) := sum_{n=0}^{infty} a_f(n)q ^n in S_{lambda+1/2}(Gamma_0(N),chi)cap mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this paper, using Rankin-Cohen Bracket, we extend this result to modular forms of half integral weight for primes $p geq 5$. As applications of our main theorem we derive distribution properties, for modulo primes $pgeq5$, of traces of singular moduli and Hurwitz class number. We also study an analogue of Newmans conjecture for overpartitions.