1
vote
1answer
331 views

stationary phase method in analytic number theory

I hope someone can tell me something about the error term in the formula calculating the oscillatory integral like $\int_a^b g(x)e(f(x))d x$. Specially, the exact formula on page 114 of M. Huxley's ...
1
vote
1answer
159 views

Does $L(-1+it,f)\ll_f \log^c q(f)t$ hold ture?

Let $f$ be a holomorphic or Maass cusp form for $SL(2,Z)$. Define $L(s,f)=\sum_{n\ge 1}\frac{a_f(n)}{n^s}$, for $\Im s$ sufficiently large. Then $$L(-1+it,f)\ll_f \log^c q(f)t$$ holds, for some ...
4
votes
1answer
305 views

subconvexity problem for $GL(3) × GL(2)$ $L$-function without involving in symmetric lift

A question in study of subconvexity topic puzzles me for a long time, which mabe a stupid question for many experts. I really wish someone to help me out, and any advice will be highly appreciated. ...
5
votes
0answers
200 views

Average of Fourier coefficients of a cusp form of half integral weight

Suppose $f$ is a cusp form of half integral weight $k$ w.r.t. the group $\Gamma_0(4)$ ($k$ is not very low, can assume $k \ge 11/2$), and $a_n$ is its Fourier coefficient. The Linnik bound says that ...
4
votes
2answers
676 views

Kronecker's Jugendtraum for real quadratic fields?

Kronecker's Jugendtraum (or Hilbert's 12'th problem) is to find abelian extensions of arbitrary number fields by adjoining `special' values of transcendental functions. The Kronecker-Weber theorem was ...
12
votes
3answers
810 views

Density of a set of integers

EDIT: this question of mine has received little attention, perhaps in part because it was stated in a too general and complicated way. So let me give it a second chance: Fix an integer $r \geq 0$. ...
2
votes
1answer
286 views

Dual Maass form for level=N in GL(2)

Let $\Gamma=\Gamma_o(N)$ be the congruence subgroup. Let $f\in C^\infty(\Gamma\backslash GL(2,R)/SO(2,R)R^*)$ be a Maass form. How shall we define its dual(contragredient) Maass form $f'$? If ...
7
votes
3answers
745 views

Effective detection of CM modular forms

Say $f$ is a newform of weight $k$ and level $\Gamma_1(N)$. $f$ is called CM if, for example, there is an imaginary quadratic field $K$ such that for all $p\nmid N$ which are inert in $K$, the $p$th ...
5
votes
0answers
345 views

Exponential sums related to cusp forms

Let $$ f(z)=\sum_{n\geq 1} a_f(n) e^{2\pi n i z}$$ be a holomorphic newform on the upper half-plane of weight $k$ for $\Gamma_0(N)$ and of trivial character which is normalized so that $a_f(1)=1$. ...
12
votes
3answers
639 views

There is no lattice in PSL(2,R) which contains PSL(2,Z) properly?

How can I see that there is no lattice in $G=\mathrm{PSL}_2( \mathbb{R})$ which contains $\Gamma_1=\mathrm{PSL}_2( \mathbb{Z})$ properly, or equivalently, that $X_1 =\mathrm{PSL}_2(\mathbb{Z}) ...
4
votes
2answers
641 views

Fourier coefficients of newforms

I apologize in advance for what is probably a very naive question: I'd like to understand the Fourier coefficients of newforms, and so I was wondering what exactly was known about them (I do know that ...
18
votes
4answers
2k views

Modular forms and the Riemann Hypothesis

Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions? What I'm thinking of is this: under the Mellin transform, the Riemann zeta function ...
7
votes
5answers
1k views

Very strong multiplicity one for Hecke eigenforms

In Invent. math. 116, 645-649 (1994) Dinakar Ramakrishnan proves a theorem which I understand to imply that the following statement (in light of the fact that elliptic curves over Q are modular): ...