Expand all Collapse all | Results 1 - 7 of 7 |
1. CJM 2011 (vol 63 pp. 634)
On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even
integral weight $k$ for the full modular group.
Let $\lambda_f(n)$ and $\lambda_g(n)$ be the $n$-th normalized Fourier coefficients of
two holomorphic Hecke eigencuspforms $f(z), g(z) \in S_{k}(\Gamma)$, respectively.
In this paper we are able to show the following results about higher
moments of Fourier coefficients of holomorphic cusp forms.\newline
(i) For any $\varepsilon>0$, we have
\begin{equation*}
\sum_{n\leq x}\lambda_f^5(n) \ll_{f,\varepsilon}x^{\frac{15}{16}+\varepsilon}
\quad\text{and}\quad\sum_{n\leq x}\lambda_f^7(n) \ll_{f,\varepsilon}x^{\frac{63}{64}+\varepsilon}.
\end{equation*}
(ii) If $\operatorname{sym}^3\pi_f \ncong \operatorname{sym}^3\pi_g$, then for any $\varepsilon>0$, we have
\begin{equation*}
\sum_{n \leq x}\lambda_f^3(n)\lambda_g^3(n)\ll_{f,\varepsilon}x^{\frac{31}{32}+\varepsilon};
\end{equation*}
If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$, then for any $\varepsilon>0$, we have
\[
\sum_{n \leq x}\lambda_f^4(n)\lambda_g^2(n)=cx\log x+c'x+O_{f,\varepsilon}\bigl(x^{\frac{31}{32}+\varepsilon}\bigr);
\]
If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$ and $\operatorname{sym}^4\pi_f \ncong \operatorname{sym}^4\pi_g$, then for any $\varepsilon>0$, we have
\[
\sum_{n \leq x}\lambda_f^4(n)\lambda_g^4(n)=xP(\log x)+O_{f,\varepsilon}\bigl(x^{\frac{127}{128}+\varepsilon}\bigr),
\]
where $P(x)$ is a polynomial of degree $3$.
Keywords: Fourier coefficients of cusp forms, symmetric power $L$-function Categories:11F30, , , , 11F11, 11F66 |
2. CJM 2011 (vol 63 pp. 298)
A Variant of Lehmer's Conjecture, II: The CM-case
Let $f$ be a normalized Hecke eigenform with rational integer Fourier
coefficients. It is an interesting question to know how often an
integer $n$ has a factor common with the $n$-th Fourier coefficient of
$f$. It has been shown in previous papers that this happens very often. In this
paper, we give an asymptotic formula for the number of integers $n$
for which $(n, a(n)) = 1$, where $a(n)$ is the $n$-th Fourier coefficient of
a normalized Hecke eigenform $f$ of weight $2$ with rational integer
Fourier coefficients and having complex multiplication.
Categories:11F11, 11F30 |
3. CJM 2009 (vol 62 pp. 157)
Special Values of Class Group $L$-Functions for CM Fields Let $H$ be the Hilbert class field of a CM number field $K$ with
maximal totally real subfield $F$ of degree $n$ over $\mathbb{Q}$. We
evaluate the second term in the Taylor expansion at $s=0$ of the
Galois-equivariant $L$-function $\Theta_{S_{\infty}}(s)$ associated to
the unramified abelian characters of $\operatorname{Gal}(H/K)$. This is an identity
in the group ring $\mathbb{C}[\operatorname{Gal}(H/K)]$ expressing
$\Theta^{(n)}_{S_{\infty}}(0)$ as essentially a linear combination of
logarithms of special values $\{\Psi(z_{\sigma})\}$, where $\Psi\colon
\mathbb{H}^{n} \rightarrow \mathbb{R}$ is a Hilbert modular function for a congruence
subgroup of $SL_{2}(\mathcal{O}_{F})$ and $\{z_{\sigma}: \sigma \in
\operatorname{Gal}(H/K)\}$ are CM points on a universal Hilbert modular variety. We
apply this result to express the relative class number $h_{H}/h_{K}$
as a rational multiple of the determinant of an $(h_{K}-1) \times
(h_{K}-1)$ matrix of logarithms of ratios of special values
$\Psi(z_{\sigma})$, thus giving rise to candidates for higher analogs
of elliptic units. Finally, we obtain a product formula for
$\Psi(z_{\sigma})$ in terms of exponentials of special values of
$L$-functions.
Keywords:Artin $L$-function, CM point, Hilbert modular function, Rubin-Stark conjecture Categories:11R42, 11F30 |
4. CJM 2007 (vol 59 pp. 1323)
On a Conjecture of Jacquet, Lai, and Rallis: Some Exceptional Cases We prove two spectral identities. The first one relates the relative
trace formula for the spherical variety $\GSpin(4,3)/G_2$ with a
weighted trace formula for $\GL_2$. The second relates a spherical
variety pertaining to $F_4$ to one of $\GSp(6)$. These identities are
in accordance with a conjecture made by Jacquet, Lai, and Rallis,
and are obtained without an appeal to a geometric comparison.
Categories:11F70, 11F72, 11F30, 11F67 |
5. CJM 2007 (vol 59 pp. 673)
Hecke $L$-Functions and the Distribution of Totally Positive Integers Let $K$ be a totally real number field of degree $n$. We show that
the number of totally positive integers
(or more generally the number of totally positive elements of a given fractional ideal)
of given trace is evenly distributed around its expected value, which is
obtained from geometric considerations.
This result depends on unfolding an integral over
a compact torus.
Keywords:Eisenstein series, toroidal integral, Fourier series, Hecke $L$-function, totally positive integer, trace Categories:11M41, 11F30, , 11F55, 11H06, 11R47 |
6. CJM 2005 (vol 57 pp. 1102)
Power Residues of Fourier Coefficients of Modular Forms Let $\rho \colon G_{\Q} \to \GL_{n}(\Ql)$ be a motivic $\ell$-adic Galois
representation. For fixed $m > 1$ we initiate an investigation of the
density of the set of primes $p$ such that the trace of the image of an
arithmetic Frobenius at $p$ under $\rho$ is an $m$-th power residue
modulo $p$. Based on numerical investigations with modular forms we
conjecture (with Ramakrishna) that this density equals $1/m$ whenever the
image of $\rho$ is open. We further conjecture that for such $\rho$ the set
of these primes $p$ is independent of any set defined by Cebatorev-style
Galois-theoretic conditions (in an appropriate sense). We then compute these
densities for certain $m$ in the complementary case of modular forms of
CM-type with rational Fourier coefficients; our proofs are a combination of
the Cebatorev density theorem (which does apply in the CM case) and
reciprocity laws applied to Hecke characters. We also discuss a potential
application (suggested by Ramakrishna) to computing inertial degrees at $p$
in abelian extensions of imaginary quadratic fields unramified away from $p$.
Categories:11F30, 11G15, 11A15 |
7. CJM 2005 (vol 57 pp. 449)
On the Sizes of Gaps in the Fourier Expansion of Modular Forms Let $f= \sum_{n=1}^{\infty} a_f(n)q^n$ be a cusp form with integer
weight $k \geq 2$ that is not a linear combination of forms with
complex multiplication. For $n \geq 1$, let
$$
i_f(n)=\begin{cases}\max\{ i :
a_f(n+j)=0 \text{ for all } 0 \leq j \leq
i\}&\text{if $a_f(n)=0$,}\\
0&\text{otherwise}.\end{cases}
$$
Concerning bounded values
of $i_f(n)$ we prove that for $\epsilon >0$ there exists $M =
M(\epsilon,f)$ such that $\# \{n \leq x : i_f(n) \leq M\} \geq (1
- \epsilon) x$. Using results of Wu, we show that if $f$ is a weight 2
cusp form for an elliptic curve without complex multiplication, then
$i_f(n) \ll_{f, \epsilon} n^{\frac{51}{134} + \epsilon}$. Using a
result of David and Pappalardi, we improve the exponent to
$\frac{1}{3}$ for almost all newforms associated to elliptic curves
without complex multiplication. Inspired by a classical paper of
Selberg, we also investigate $i_f(n)$ on the average using well known
bounds on the Riemann Zeta function.
Category:11F30 |