We evaluate the first three moments of central values of a family of qudratic Hecke $L$-functions in the Gaussian field with power saving error terms. In particular, we obtain asymptotic formulas for the first two moments with error terms of size $O(X^{1/2+\varepsilon})$. We also study the first and second mollified moments of the same family of $L$-functions to show that at least $87.5\%$ of the members of this family have non-vanishing central values.
In this article, we follow Hida's approach to establish an analogue of Washington's theorem on the non-vanishing modulo p of Hecke L-values for CM fields with anticyclotomic twists.
We obtain a closed form polynomial expression for certain coefficients of Drinfeld-Goss double-cuspidal modular forms which are eigenforms for the degree one Hecke operators with power eigenvalues, and we use those formulas to prove vanishing results for an infinite family of those coefficients.
We prove that, for fixed tame level (N,p) = 1, there are only finitely many Hecke eigenforms f of level Gamma_1(N) and even weight with a_p(f) = 0 which are not CM.
We study the first and second mollified moments of central values of a quadratic family of Hecke $L$-functions of prime moduli to show that more than nine percent of the members of this family do not vanish at the central values.
Let F(z) be a newform of weight 2k and level one with a trivial character, and assume that F(z) is a non-zero eigenform of all Hecke operators. In this paper, we study nonvanishing for central values of twisted modular L-function of F.
We introduce a new trace formula of Kuznetsov type involving the central standard L-values and the Whittaker periods of cuspidal automorphic representations of PGL_n(Q) which are spherical at the archimedean place. As an application, we show a simultaneous non-vanishing of standard L-values at n-1 points on the critical strip for infinitely many Hecke-Maass cuspidal newforms.
In this paper, we show that half of non-zero coefficients of the spinor zeta function of a Siegel cusp form of genus 2 are positive and half are negative.
We prove a non-vanishing result for families of $\GL_n\times\GL_n$ Rankin-Selberg $L$-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on $\GL_n$. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.
In this paper, we study the moments of central values of Hecke $L$-functions associated with quadratic characters in $\mathbb{Q}(i)$ and $\mathbb{Q}(ω)$ with $ω= exp(2πi/3)$ and establish some quantitative non-vanishing result for these $L$-values.
We study real zeros of a family of quadratic Hecke $L$-functions in the Gaussian field to show that more than twenty percent of the members have no zeros on the interval $(0, 1]$.
Let $χ$ be a Hecke character of finite order of a totally real number field $F$. By using Hill's Shintani cocyle we provide a cohomological construction of the $p$-adic $L$-series $L_p(χ, s)$ associated to $χ$. This is used to show that $L_p(χ, s)$ has a trivial zero at $s=0$ of order at least equal to the number of places of $F$ above $p$ where the local component of $χ$ is trivial.
Since the study by Jacobi and Hecke, Hecke-type series have received a lot of attention. Unlike such series associated with indefinite quadratic forms, identities on Hecke-type series associated with definite quadratic forms are quite rare in the literature. Motivated by the works of Liu, we first establish many parameterized identities with two parameters by employing different $q$-transformation formulas and then deduce various Hecke-type identities associated with definite quadratic forms by specializing the choice of these two parameters. As applications, we utilize some of these Hecke-typ…
In this paper, we evaluate explicitly certain quadratic Hecke Gauss sums of $\mathbb{Q}(ω), ω=\exp \left( \frac {2πi}{3}\right)$. As applications, we study the moments of central values of quadratic Hecke $L$-functions of $\mathbb{Q}(ω)$, and establish quantitative non-vanishing result for the $L$-values. We also establish an one level density result for the low-lying zeros of quadratic Hecke $L$-functions of $\mathbb{Q}(ω)$.
We relate non-critical special values $p$-adic $L$-functions associated to algebraic Hecke characters of an imaginary quadratic number field with class number one to $p$-adic Coleman function called the $p$-adic Eisenstein-Kronecker series, when the conductors of the algebraic Hecke characters are divisible by $p$.
The purpose of this article is to give an overview of the series of papers [BK1], [BK2] concerning the $p$-adic Beilinson conjecture of motives associated to Hecke characters of an imaginary quadratic field $K$, for a prime $p$ which splits in $K$.
In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke $L$-functions of imaginary quadratic number fields of class number one. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27... \%$ of the $L$-functions under consideration do not vanish at $1/2$.
We obtain an asymptotic formula for the fourth moment of central values of a family of quadratic Hecke $L$-functions in the Gaussian field under the generalized Riemann hypothesis (GRH). We also establish lower bounds unconditionally and upper bounds under GRH for higher moments of the same family.
For integers $k\geq 2$, we study two differential operators on harmonic weak Maass forms of weight $2-k$. The operator $ξ_{2-k}$ (resp. $D^{k-1}$) defines a map to the space of weight $k$ cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are "dual" under $ξ_{2-k}$ to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, w…
We present a theory of reduction of binary quadratic forms with coefficients in Z[lambda], where lambda is the minimal translation in a Hecke group. We generalize from the modular group Gamma(1) = SL(2,Z) to the Hecke groups and make extensive use of modified negative continued fractions.
In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form $\mathfrak{f}$ with half-integral weight:\begin{itemize}\item[--]The first negative coefficient of the sequence $\{\mathfrak{a}\_{\mathfrak{f}}(tn^2)\}\_{n\in \N}$,\item[--]The number of coefficients $\mathfrak{a}\_{\mathfrak{f}}(tn^2)$ of same signs,\item[--]Non-vanishing of coefficients $\mathfrak{a}\_{\mathfrak{f}}(tn^2)$ in short intervals and in arithmetic progressions,\end{itemize}where $\mathfrak{a}\_{\mathfrak{f}}(n)$ is the $n$-th Fourier coefficient of $\mathfrak{f}$ and $t$ is a square-f…
Let $p$ and $q$ be two distinct odd primes. Let $K$ be an imaginary quadratic field over which $p$ and $q$ are both split. Let $Ψ$ be a Hecke character over $K$ of infinity type $(k,j)$ with $0\le-j< k$. Under certain technical hypotheses, we show that for a Zariski dense set of finite-order characters $κ$ over $K$ which factor through the $\mathbb{Z}_q^2$-extension of $K$, the $p$-adic valuation of the algebraic part of the $L$-value $L(\overline{κΨ},k+j)$ is a constant independent of $κ$. In addition, when $j=0$ and certain technical hypothesis holds, this constant is zero.
In this paper, we prove some one level density results for the low-lying zeros of famliies of quadratic and quartic Hecke $L$-functions of the Gaussian field. As corollaries, we deduce that, respectively, at least $94.27 \%$ and $5\%$ of the members of the quadratic family and the quartic family do not vanish at the central point.
We study smoothed character sums involving $\sum_{m,n} ( \frac{m}{n} )_2$, where $( \frac{m}{n} )_2$ denotes the quadratic symbol in the Gaussian field. We extend previously known results to obtain asymptotic formulas for the sums considered to larger ranges of $m$ and $n$.
In this paper, we study the one level density of low-lying zeros of a family of quadratic Hecke $L$-functions to prime moduli over the Gaussian field under the generalized Riemann hypothesis (GRH) and the ratios conjecture. As a corollary, we deduce that at least $75 \%$ of the members of this family do not vanish at the central point under GRH.
We consider the analogue of the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke eigenvalues summed over quadratic progressions. Our reduction provides an analogue for the compact case of a criterion established by Luo--Sarnak for the case of the non-compact modular surface. The novelty is that known proofs of such criteria have depended crucially upon Fourier expansions, which are not available in the compact case. Unconditionally, we establish a twisted varian…
In this paper, we apply the ratio conjecture of $L$-functions to derive the lower order terms of the $1$-level density of the low-lying zeros of a family quadratic Hecke $L$-functions in the Gaussian field. Up to the first lower order term, we show that our result is consistent with that obtained from previous work under the generalized Riemann hypothesis, when the Fourier transforms of the test functions are supported in $(-2, 2)$.
Let K/F be a quadratic extension of number fields. After developing a theory of the Eisenstein series over F, we prove a formula which expresses a partial zeta function of K as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker's type which relates the 0-th Laurent coefficients at s=1 of zeta functions of K and F.
First Moment of Quadratic Hecke $L$-Functions with Lower Order Term
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）
We present a collection of results on a conjecture of Jannsen about the $p$-adic realizations associated to Hecke characters over an imaginary quadratic field $K$ of class number 1. The conjecture is easy to check for Galois groups purely of local type. We prove the conjecture under a geometric regularity condition for the imaginary quadratic field $K$ at $p$, which is related to the property that a global Galois group is purely of local type. Without this regularity assumption at $p$, we present a review of the known situations in the critical case and in the non-critical case for the realiza…
Let $f$ and $g$ be two distinct newforms which are normalized Hecke eigenforms of weights $k_1, k_2 \ge 2$ and levels $N_1, N_2 \ge 1$ respectively. Also let $a_f(n)$ and $a_g(n)$ be the $n$-th Fourier-coefficients of $f$ and $g$ respectively. In this article, we investigate the first sign change of the sequence $\{a_f(p^α)a_g(p^α) \}_{p^α \in \N, α\le 2}$, where $p$ is a prime number. We further study the non-vanishing of the sequence $\{a_f(n)a_g(n) \}_{n \in \N}$ and derive bounds for first non-vanishing term in this sequence. We also show, using ideas of Kowalski-Robert-Wu and Murty-Murty,…
In this paper, we study lower order terms of the $1$-level density of low-lying zeros of quadratic Hecke L-functions in the Gaussian field. Assuming the Generalized Riemann Hypothesis, our result is valid for even test functions whose Fourier transforms are supported in $(-2, 2)$. Moreover, we apply the ratios conjecture of $L$-functions to derive these lower order terms as well. Up to the first lower order term, we show that our results is consistent with each other, when the Fourier transforms of the test functions are supported in $(-2, 2)$.
This is essentially the author's thesis submited to The University of Chicago (May 1997). I prove the validity of Tamagawa number conjecture of Bloch-Kato for certain Hecke characters. I study the exponential map and local Tamagawa number for all odd primes (both ordinary and supersingular), using Kato's explicit reciprocity law for one dimensional Lubin-Tate formal group. I also study p-part of Shafarevich-Tate group for motives associated to Hecke characters, using Rubin's Main Conjecture in Iwasawa theory. Interestingly a congruence property between p-adic periods of elliptic curve and weig…
We study simultaneous non-vanishing of $L(\tfrac{1}{2},\di)$ and $L(\tfrac{1}{2},g\otimes \di)$, when $\di$ runs over an orthogonal basis of the space of Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ and $g$ is a fixed $SL(2,\mathbb{Z})$ Hecke cusp form of weight $k\equiv 0 \pmod 4$.
For certain algebraic Hecke characters chi of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL_2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-value L(0,chi). We further prove that its index is bounded from above by the order of the Selmer group of the p-adic Galois character associated to chi^{-1}. This uses the work of R. Taylor et al. on a…
We investigate non-vanishing properties of $L(f,s)$ on the real line, when $f$ is a Hecke eigenform of half-integral weight $k+{1\over 2}$ on $Γ_0(4).$
Integrality of Hecke eigenvalues and the growth of Hecke fields
Faculty of Education, Shinshu University（信州大学教育学部）；Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University（金泽大学理工学域数理系）
In the early 1980s, Rohrlich began a study of canonical Hecke characters, which are closely related to the simplest examples of CM elliptic curves. He and Montgomery showed the non-vanishing of the central value when the L-function has an even functional equation, and we now show the non-vanishing of the central derivative when the functional equation is odd. Using the results of Gross-Zagier and Kolyvagin-Logachev, we can apply the non-vanishing to the ranks and Shafarevitch-Tate groups of the Q-curves "A(p)" studied by Gross in his thesis. In particular, their rank is determined by a congrue…
Let $-D < -4$ denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of $\Bbb Q(\sqrt{-D})$ exists. Let $d$ be a fundamental discriminant prime to $D$. Let $2k-1$ be an odd natural integer prime to the class number of $\Bbb Q(\sqrt{-D})$. Let $χ$ be the twist of the $(2k-1)$th power of a canonical Hecke character of $\Bbb Q(\sqrt{-D})$ by the Kronecker's symbol $n\mapsto(\frac{d}{n})$. It is proved that the order of the Hecke $L$-function $L(s,χ)$ at its central point $s=k$ is determined by its root number when $|d| \leq c(ε)D^{{1/24}-ε}$…
We prove a mean value theorem for the traces of Hecke operators acting on cusp forms of GL(n) over imaginary quadratic number fields together with an upper bound for the error term depending explicitly on the Hecke operator.
Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \leq X} a(q(n)) = cX + O(X^{6/7+eps}) for some constant c depending on f and q. The constant vanishes in many cases, for example if k is even. On the way a Kuznetsov formula for half-integral weight and entries having different sign is derived.
In this paper, we study moments of central values of cubic Hecke $L$-functions in $\mathbb{Q}(i)$, and establish quantitative non-vanishing result for those values.
Let F be a real quadratic field with ring of integers O and with class number 1. Let Gamma be a congruence subgroup of GL_2 (O). We describe a technique to compute the action of the Hecke operators on the cohomology H^3 (Gamma; C). For F real quadratic this cohomology group contains the cuspidal cohomology corresponding to cuspidal Hilbert modular forms of parallel weight 2. Hence this technique gives a way to compute the Hecke action on these Hilbert modular forms.
For a family of real quadratic fields $\{K_n=\FQ(\sqrt{f(n)})\}_{n\in \FN}$, a Dirichlet character $χ$ modulo $q$ and prescribed ideals $\{\fb_n\subset K_n\}$, we investigate the linear behaviour of the special value of partial Hecke's L-function $L_{K_n}(s,χ_n:=χ\circ N_{K_n},\fb_n)$ at $s=0$. We show that for $n=qk+r$, $L_{K_n}(0,χ_n,\fb_n)$ can be written as $$\frac{1}{12q^2}(A_χ(r)+kB_χ(r)),$$ where $A_χ(r),B_χ(r)\in \FZ[χ(1),χ(2),..., χ(q)]$ if a certain condition on $\fb_n$ in terms of its continued fraction is satisfied. Furthermore, we write precisely $A_χ(r)$ and $B_χ(r)$ using values…
We formulate and prove a large sieve inequality for quadratic characters over a number field. To do this, we introduce the notion of an n-th order Hecke family. We develop the basic theory of these Hecke families, including versions of the Poisson summation formula.
