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 $f_i$.

Congruences involving the $U_{ell}$ operator for weakly holomorphic modular forms

Let $lambda$ be an integer, and $f(z)=sum_{ngg-infty} a(n)q^n$ be a weakly holomorphic modular form of weight $lambda+frac 12$ on $Gamma_0(4)$ with integral coefficients. Let $ellgeq 5$ be a prime. Assume that the constant term $a(0)$ is not zero modulo $ell$. Further, assume that, for some positive integer $m$, the Fourier expansion of $(f|U_{ell^m})(z) = sum_{n=0}^infty b(n)q^n$ has the form [ (f|U_{ell^m})(z) equiv b(0) + sum_{i=1}^{t}sum_{n=1}^{infty} b(d_i n^2) q^{d_i n^2} pmod{ell}, ] where $d_1, ldots, d_t$ are square-free positive integers, and the operator $U_ell$ on formal power series is defined by [ left( sum_{n=0}^infty a(n)q^n right) bigg| U_ell = sum_{n=0}^infty a(ell n)q^n. ] Then, $lambda equiv 0 pmod{frac{ell-1}{2}}$. Moreover, if $tilde{f}$ denotes the coefficient-wise reduction of $f$ modulo $ell$, then we have [ biggl{ lim_{m rightarrow infty} tilde{f}|U_{ell^{2m}}, lim_{m rightarrow infty} tilde{f}|U_{ell^{2m+1}} biggr} = biggl{ a(0)theta(z), a(0)theta^ell(z) in mathbb{F}_{ell}[[q]] biggr}, ] where $theta(z)$ is the Jacobi theta function defined by $theta(z) = sum_{ninmathbb{Z}} q^{n^2}$. By using this result, we obtain the distribution of the Fourier coefficients of weakly holomorphic modular forms in congruence classes. This applies to the congruence properties for traces of singular moduli.

Independence between coefficients of two modular forms

Let $k$ be an even integer and $S_k$ be the space of cusp forms of weight $k$ on $SL_2(ZZ)$. Let $S = oplus_{kin 2ZZ} S_k$. For $f, gin S$, we let $R(f, g) = { (a_f(p), a_g(p)) in mathbb{P}^1(CC) | text{$p$ is a prime} }$ be the set of ratios of the Fourier coefficients of $f$ and $g$, where $a_f(n)$ (resp. $a_g(n)$) is the $n$th Fourier coefficient of $f$ (resp. $g$). In this paper, we prove that if $f$ and $g$ are nonzero and $R(f,g)$ is finite, then $f = cg$ for some constant $c$. This result is extended to the space of weakly holomorphic modular forms on $SL_2(ZZ)$. We apply it to studying the number of representations of a positive integer by a quadratic form.

Schneider-Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits

Let $j(z)$ be the modular $j$-invariant function. Let $tau$ be an algebraic number in the complex upper half plane $mathbb{H}$. It was proved by Schneider and Siegel that if $tau$ is not a CM point, i.e., $[mathbb{Q}(tau):mathbb{Q}] eq2$, then $j(tau)$ is transcendental. Let $f$ be a harmonic weak Maass form of weight $0$ on $Gamma_0(N)$. In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of $f$ on Hecke orbits of $tau$. For a positive integer $m$, let $T_m$ denote the $m$-th Hecke operator. Suppose that the coefficients of the principal part of $f$ at the cusp $i infty$ are algebraic, and that $f$ has its poles only at cusps equivalent to $i infty$. We prove, under a mild assumption on $f$, that for any fixed $tau$, if $N$ is a prime such that $ Ngeq 23 text{ and } N ot in {23, 29, 31, 41, 47, 59, 71},$ then $f(T_m.tau)$ are transcendental for infinitely many positive integers $m$ prime to $N$.

Values of Harmonic Weak Maass forms on Hecke orbits

Let $q:=e^{2 pi iz}$, where $z in mathbb{H}$. For an even integer $k$, let $f(z):=q^hprod_{m=1}^{infty}(1-q^m)^{c(m)}$ be a meromorphic modular form of weight $k$ on $Gamma_0(N)$. For a positive integer $m$, let $T_m$ be the $m$th Hecke operator and $D$ be a divisor of a modular curve with level $N$. Both subjects, the exponents $c(m)$ of a modular form and the distribution of the points in the support of $T_m. D$, have been widely investigated. When the level $N$ is one, Bruinier, Kohnen, and Ono obtained, in terms of the values of $j$-invariant function, identities between the exponents $c(m)$ of a modular form and the points in the support of $T_m.D$. In this paper, we extend this result to general $Gamma_0(N)$ in terms of values of harmonic weak Maass forms of weight $0$. By the distribution of Hecke points, this applies to obtain an asymptotic behaviour of convolutions of sums of divisors of an integer and sums of exponents of a modular form.

Limits of traces of singular moduli

Let $f$ and $g$ be weakly holomorphic modular functions on $Gamma_0(N)$ with the trivial character. For an integer $d$, let $Tr_d(f)$ denote the modular trace of $f$ of index $d$. Let $r$ be a rational number equivalent to $iinfty$ under the action of $Gamma_0(4N)$. In this paper, we prove that, when $z$ goes radially to $r$, the limit $Q_{hat{H}(f)}(r)$ of the sum $H(f)(z) = sum_{d>0}Tr_d(f)e^{2pi idz}$ is a special value of a regularized twisted $L$-function defined by $Tr_d(f)$ for $dleq0$. It is proved that the regularized $L$-function is meromorphic on $mathbb{C}$ and satisfies a certain functional equation. Finally, under the assumption that $N$ is square free, we prove that if $Q_{hat{H}(f)}(r)=Q_{hat{H}(g)}(r)$ for all $r$ equivalent to $i infty$ under the action of $Gamma_0(4N)$, then $Tr_d(f)=Tr_d(g)$ for all integers $d$.

Pairings of harmonic Maass-Jacobi forms involving special values of partial $L$-functions

In this paper, considering the Eichler-Shimura cohomology theory for Jacobi forms, we study connections between harmonic Maass-Jacobi forms and Jacobi integrals. As an application we study a pairing between two Jacobi integrals, which is defined by special values of partial $L$-functions of skew-holomorphic Jacobi cusp forms. We obtain connections between this pairing and the Petersson inner product for skew-holomorphic Jacobi cusp forms. This result can be considered as analogue of Haberland formula of elliptic modular forms for Jacobi forms.

Cohomological relation between Jacobi forms and skew-holomorphic Jacobi forms

Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass work on the Saito-Kurokawa conjecture. Later Skoruppa introduced skew-holomorphic Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi forms. In this paper, we explain a relation between holomorphic Jacobi forms and skew-holomorphic Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from the direct sum of the space of Jacobi cusp forms on $Gamma^J$ and the space of skew-holomorphic Jacobi cusp forms on $Gamma^J$ with the same half-integral weight to the Eichler cohomology group of $Gamma^J$ with a coefficient module coming from polynomials.

Structures for pairs of mock modular forms with the Zagier duality

Zagier introduced special bases for weakly holomorphic modular forms to give the new proof of Borcherds theorem on the infinite product expansions of integer weight modular forms on $SL_2(ZZ)$ with a Heegner divisor. These good bases appear in pairs, and they satisfy a striking duality, which is now called the Zagier duality. After the result of Zagier, this type duality was studied broadly in various view points including the theory of a mock modular form. In this paper, we consider this problem with the Eichler cohomology theory, especially the supplementary function theory developed by Knopp. Using holomorphic Poincare series and their supplementary functions, we construct a pair of families of vector-valued harmonic weak Maass forms satisfying the Zagier duality with integer weights $-k$ and $k+2$ respectively, $k>0$, for a $H$-group. We also investigate the structures of them such as the images under the differential operators $D^{k+1}$ and $xi_{-k}$ and quadric relations of the critical values of their $L$-functions.

The Eichler cohomology theorem for Jacobi forms

Let $Gamma$ be a finitely generated Fuchsian group of the first kind which has at least one parabolic class. Eichler introduced a cohomology theory for Fuchsian groups, called as Eichler cohomology theory, and established the $CC$-linear isomorphism from the direct sum of two spaces of cusp forms on $Gamma$ with the same integral weight to the Eichler cohomology group of $Gamma$. After the results of Eichler, the Eichler cohomology theory was generalized in various ways. For example, these results were generalized by Knopp to the cases with arbitrary real weights. In this paper, we extend the Eichler cohomology theory to the context of Jacobi forms. We define the cohomology groups of Jacobi groups which are analogues of Eichler cohomology groups and prove an Eichler cohomology theorem for Jacobi forms of arbitrary real weights. Furthermore, we prove that every cocycle is parabolic and that for some special cases we have an isomorphism between the cohomology group and the space of Jacobi forms in terms of the critical values of partial $L$-functions of Jacobi cusp forms.