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Results tagged with automorphic-forms
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user 111215
An automorphic form is a well-behaved function from a topological group $G$ to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.
2
votes
Accepted
Question of pole and zeros of symmetric or exterior global $L$-function of $\mathrm{GL}_n(\m...
For simplicity, let $F$ be a number field, and assume that $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_{n}(\mathbb{A}_F)$ with trivial conductor. Miller and Schmid proved that $L(s …
- 5,089
3
votes
Smallest Fourier coefficient divisible by a prime
An answer that applies when $f$ is a non-CM newform with trivial nebentypus and integral Fourier coefficients can be found in Theorem 1.5 of a paper by Thorner and Zaman (arXiv version). As you sugge …
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7
votes
The error term for the second moment of Fourier coefficients of cusp forms with the level ex...
If $f$ is a $\mathrm{GL}(2)$ newform over $\mathbb{Q}$ with prime level $p$ and $\zeta^{(p)}(s)$ denotes the Riemann zeta function $\zeta(s)$ with the Euler factor at $p$ removed, then
$$L(s,f\times \ …
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3
votes
Relation between Fourier coefficients and Satake parameters
Let $s_{k}(\alpha_1(p),\ldots,\alpha_n(p))$ be the complete homogeneous symmetric polynomial of degree $k$ in variables $\{\alpha_1(p),\ldots,\alpha_n(p)\}$. If $\mathrm{Re}(s)$ is sufficiently large …
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7
votes
Is the Sato-Tate conjecture known for Bianchi modular forms?
Let $K/\mathbb{Q}$ be CM extension. Suppose that a primitive Bianchi cusp form $F$ corresponds with a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{A}_K)$ that is not the automorphic …
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0
votes
Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms
The prime number theorem for $L(s,f\times\bar{f})$ and the truth of Hypothesis H for $\mathrm{GL}_3$ automorphic forms yields the asymptotic
$\sum_{p\leq x}|A(1,p)|^2\sim \frac{x}{\log x}$.
(See htt …
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4
votes
Bound for $GL(3)$ symmetric square
Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_m$ with unitary central character. In work of Takeda, it is shown that the (unramified part of the) $L$-function $L(s,\pi,\mathrm{Sy …
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4
votes
Accepted
Does Rankin-Selberg convolution preserve primitivity?
Let $L(s,F)$ be the $L$-function of a self-dual $\mathrm{GL}(2)$ holomorphic cuspidal newform without complex multiplication and with trivial nebentypus. Let $G=\mathrm{Sym}^2 F$ be the symmetric squ …
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6
votes
Accepted
Rankin-Selberg convolution and product of degrees as of Christmas 2019
Newton and Thorne proved that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$ corresponding with a holomorphic cuspidal newform of even integral weight $k …
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2
votes
On the square mean of Fourier coefficients of cusp forms
Something much stronger is true. There exists an effective constant $c_f>0$ such that
$$\sum_{X<n\leq 2X}\lambda_f(n)^2 \sim c_f L(1,\mathrm{Ad}\,f) X.$$
If the level $M_f$ of $f$ is squarefree, then …
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3
votes
Lower bound of Hecke eigenvalues of Maass form
Following up on @Idoneal's comment, one can prove something a bit more precise. In particular, $S(x)\asymp\frac{x}{\log x}$, provided that $\log x\gg\log(\lambda N)$, where $f$ has level $N$ and Lap …
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8
votes
Accepted
Behaviour of a certain $L$ function at $s=1$
Let $n\geq 1$. The $L$-function of an automorphic representation of $\mathrm{GL}(n)$ is either (1) entire, or (2) holomorphic away from a pole of order $\leq n$ at $s=1+i\tau$ for some fixed $\tau\in …
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6
votes
A lower-bound for the square-mean of Fourier coefficients of cusp forms at primes argument
Assuming that the spectral parameter is absolutely bounded (which you seem to implicitly state), the best that one can achieve with existing tools is the following: There exist absolute and effective …
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10
votes
Symmetric powers of Ramanujan tau-function
According to this exciting new preprint of Newton and Thorne (https://arxiv.org/abs/1912.11261), the answer to both Questions 1 and 2 is "all $k\geq 1$".
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0
votes
Average bounds on Rankin-Selberg coefficients for modular forms
In Mertens' original result, he proves that
$$\sum_{p\leq x}\frac{1}{p}= \log\log x+B+O\Big(\frac{1}{\log x}\Big),$$
where
$$B = \sum_p\Big(\log\Big(1-\frac{1}{p}\Big)-\frac{1}{p}\Big) = - \sum_p\sum_ …
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