# Questions tagged [l-functions]

Questions about generalizations of the Riemann Zeta function of arithmetic interest whose definition relies on meromorphic continuation of special kinds of Dirichlet series, such as Dirichlet L-functions, Artin L-functions, elements of the Selberg class, automorphic L-functions, Shimizu L-functions, p-adic L-functions, etc.

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### What are the unsolved problems in Formal groups and $L$-functions?

In the 1st page of the introduction of Hazewinkel's Formal Groups and Applications book, there are two ways of constructing formal groups (law): $\bullet$ Given a Lie group $G$, one can define a ...
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### Relating the multiplicative Fourier transform and the derived characteristic polynomial

(Tuesday, Sept 5:) For a number field $Fˣ$ and a number ring $Oˣ$ it is common to define: $Z(f,χ) = ʃ_{Fˣ} f(x) χ(x) dˣ x$ $g(ω,ψ) = ʃ_{Oˣ} ω(x) ψ(x) dˣ x$ where $dˣx$ is the multiplicative Haar ...
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### Zero dimensional varieties and the L-function $1/(1-p^{-n})$

I am interested in positive characteristic varieties which produce an L-function of the form $\frac{1}{1-χ} = \frac{1}{1-p^{-s}} = \sum_{n = 0}^\infty p^{-ns}$. It seems related to the positive ...
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### Question on automorphic $L$-functions

Let $\pi$ be an automorphic representation of $\textrm{GL}_n$. Associated to $\pi$, we can define the standard $L$-function $L(s, \pi)$. My question is: what is the difference between $L(s, \pi)$ and ...
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1 vote
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### What are the best known upper bounds for $\frac{1}{L(s, \chi)}$?

Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding Dirichlet L-function. What are the best known bounds for $\frac{1}{L(s, \chi)}$ in the half-plane of convergence? I'm aware of ...
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### Waldspurger's formula and toric periods — classical and adelic versions

As far as I know, there are two versions of Waldspurger's formula (classical and adelic), which can be vaguely stated as follows (Classical version) Let $f$ be a half-integral weight modular form of ...
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### Basic results concerning the intertwining operator in the $\mathrm{SL}_2$ case

I am reading [Ikeda, Tamotsu, On the location of poles of the triple L-functions]. On page 194, the author recalled some known results concerning $\operatorname{SL}_2$. I would like to know any ...
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### Meaning of the meromorphic continuation of intertwining operators

I am trying to make sure the meaning of the meromorphic continuation of the intertwining operators. Assume we deal with a non-Archimedean field $F$ and just consider $G= SL_2$, for simplicity. We fix ...
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### Positivity of partial Dirichlet series for a quadratic character?

Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet ...
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### Langlands-Shahidi method in classical language

The Langlands-Shahidi method says that the $L$-functions of automorphic representations appear in the constant terms of Eisenstein series. Since those Eisenstein series have analytic continuation and ...
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### Which automorphic L-functions have an integral representation?

Is there a list of which automorphic L-functions are known to have an integral representation?
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### What is the Perrin-Riou logarithm (or regulator)?

Recently I've been rewatching some recordings of old talks on L-functions and explicit reciprocity laws (in particular, the series of talks by Loeffler and Zerbes given at this workshop at the CRM in ...
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### What are the modularity conjectures for Artin motives?

Classically, singular cohomology is an important tool for studying topological spaces, in particular, complex varieties. In the mid-twentieth century it was realized that there are many analogues of ...
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### Second moment of $S(T)$ for Dirichlet L-functions

Let $S(T)$ denote the argument of the Riemann zeta function. Selberg established that $$\int_0^T |S(t)|^2 \text{d}t\sim\frac{T}{2\pi^2}\log \log T.$$ Let now $\chi$ be a Dirichlet character modulo $q$,...
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### Why $p$-adic measures?

I'm currently learning about the Kubota–Leopoldt $p$-adic $L$-function and I'm noticing that many people view the Kubota–Leopoldt $p$-adic $L$-function as a measure as opposed to a $p$-adic analytic ...
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### Special values of non-cm $L$-functions

For the sake of simplicity, assume $f$ is a non-cm eigenform of weight $k$ on the group $\mathrm{SL}(2, \mathbb{Z})$. Are there any known results or conjectures regarding any special values of the ...
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### Spacings of Satake parameters under Ramanujan conjecture

I would like to know if, under Ramanujan conjecture, the following three distributions are known or conjectured to match: the distribution of spacings between Satake parameters of an L-function $F$ ...
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### Additivity of Elliptic Curve Rank over Compositum of Fields

Assume that BSD holds for number fields. Let $E/\mathbf{Q}$ be an elliptic curve. For simplicity, let's assume it has Mordell-Weil rank zero. Let $F_1/\mathbf{Q}$ and $F_2/\mathbf{Q}$ be finite, ...
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### Does asymptotic Goldbach imply GRH?

It seems to me that a proof of $\alpha_{n}=o(n)$ where the quantity $\alpha_{n}$ is defined in About Goldbach's conjecture together with the main result of https://kyushu-u.pure.elsevier.com/en/...
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### Special value of Hecke $L$ function

Let $E:y^2＝x^3-x/ \Bbb{Q}(i)$ be elliptic curve and $L(E,1)$ be a special value of $L$ function of $E$ at $1$. Let $L(ψ,1)$ be value at $1$ of Hecke $L$ function with respect to Hecke character $ψ$, ...
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### Do Artin L functions have polynomial growth in in the critical strip?

Given an irreducible representation $\rho$ of the Galois group $G$ of a number field $K$ over $\mathbb{Q}$, we have the associated Artin $L$ function which we denote by $L(s, \rho)$. By Brauer ...
1 vote
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### Explanation about Lapid-Rallis iductive argument (doubling method)

I am reading Lapid-Rallis "On the local factors of representations of classical groups" and I am completely stuck with the proof of Proposition 3. In the case $\mathcal V$ is not anisotropic,...
166 views