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, ...

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2
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0answers
249 views

Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve

Edited after Noam Elkies' comment: From what I understand (very few actually), there exist elliptic curves defined over some number fields $\mathbb{K}$ Galois over $\mathbb{Q}$ which are isogenous to ...
2
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0answers
190 views

Automorphicity of L-Factors of Zeta Functions

Associated to a variety over a number field $K$, one has a family of ``Hasse--Weil'' L-functions, which can be combined (as an alternating product) to give the Hasse--Weil zeta function of the ...
1
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1answer
187 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 ...
10
votes
0answers
245 views

L-Functions of Varieties, Zeta Functions of Their Models

Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...
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0answers
250 views

Does the property (P) holds true for the derivatives of $L$?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. As $s$ takes on real negative values, there are trivial zeros ...
3
votes
1answer
320 views

On link between Riemann hypothesis and partial GRH

Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character $\chi(1)$ which could be qualified of ...
1
vote
1answer
122 views

On properties of coefficients of Selberg Class L-function

The coefficient of Selberg Class L-function satisfy: $a_n <M_{\epsilon} n^{\epsilon}$ (for any $\epsilon >0$) and the $a_n$ are multiplicative. So I would like to know if it can be shown that ...
5
votes
1answer
170 views

Periods of Twists of Modular Forms

Let $f \in S_2(\Gamma_1(N))$ be an eigenform. By a theorem of Shimura, there are associated "periods" $\Omega_f^\pm$ such that, after normalizing by these periods, the L-function associated to $f$ ...
4
votes
1answer
137 views

Are supercuspidal reps of GL(2) uniquely determined by the rootnumber

Let $\pi, \pi'$ be a unitary, irreducible, supercuspidal representations of $GL_2(F)$. Does an equality of roots numbers $\epsilon(\pi, \psi, s) = \epsilon(\pi', \psi, s)$ for all $s \in \mathbb{C}$ ...
1
vote
1answer
98 views

Has a universality theorem been proved for the Davenport-Heilbronn L function?

The question is in the title: has a universality theorem in the sense of Voronin been proved for the Davenport-Heilbronn function, or do we expect such a theorem to hold true only for L functions that ...
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votes
1answer
174 views

Poles of the L-series of the elliptic curve which has CM

Let $E/\mathbb{Q}$ be the elliptic curve which has CM. It is well known that the L-series of the elliptic curve which has CM is Hecke L-series. On the other hand, it is pointed out that Hecke ...
6
votes
1answer
608 views

Sufficient condition for Riemann Hypothesis?

Is there an L-function ($L_s=\sum_{n =1}^{\infty} \frac{a_n}{n^s}$) having a functional equation coming from a relation of the type [1]: $\sum_{n =1}^{\infty} a_n \; e^{-2\pi nx}= \frac{A}{x^k} ...
4
votes
1answer
422 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. ...
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vote
0answers
60 views

Sign of the functional equation of L function and Shimura lift

I would like to know what happens to the root number of a half integral weight automorphic form (holomorphic or not), i.e. the sign of the functional equation of its L-function when we apply the ...
5
votes
0answers
125 views

What is the analogy between the Hilbert function and L-functions?

In his book Projective Varieties and Modular Forms, M. Eichler uses the notation $L(\lambda, M)$ for the Hilbert function of a finite graded $R=k[x_0, \dots, x_n]$-module $M$. So, $L(\lambda, M) = ...
2
votes
2answers
415 views

Summation of certain series

Suppose $f(n)$ is a periodic function with period $q$. Now from this paper we get that if $\displaystyle\sum_{n=1}^{q}f(n)=0$ then ...
3
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0answers
117 views

Do local L-functions/epsilon factors vary continuously with the Fell topology?

Edit due to the comment. Consider $G=GL(2)$ over a local field $F$. The Fell topology on the unitary dual of $G(F)$ is seperable. Given a sequence of irreducible unitary representations $(\pi_n)$ of ...
5
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0answers
365 views

a generalization of a formula of Shimura

Let $\phi$ be a $GL(2)$ automorphic form with Fourier coefficients $a(n)$ and $a(1)=1$. Obviously we have $L(s,\phi)=\sum \frac{a(n)}{n^s}$. Shimura have the following formula $L(s, Ad\; ...
7
votes
1answer
555 views

On Deligne's determinant of motives

This is a question about Deligne's conjecture on special values of L-functions. I have to confess that I've never understood the definition of the determinant which is supposed to give the right ...
4
votes
0answers
99 views

cluster variables and L-functions

There is something in common between cluster variables in the theory of cluster algebras, L-functions in number theory, namely the fact that both map direct sums to products, just like ...
3
votes
1answer
210 views

Extending the Shimura Lift to Non-Cuspidal Classical Modular Forms of Higher Level

The definition of the Shimura lift of a classical cusp form is well documented. Zagier and Kohnen define a modified version of the lift for a cusp form $g(z)=\sum a(n)q^n \in S_{k+1/2}^{+}(4)$ in the ...
2
votes
0answers
98 views

Theta lift to 1-dimensional vector space.

Hi! My question is very simple and it is about the theta lift of unitary group in global situation. Let $E/F$ be a quadratic number fields and $V,W$ be an $n$-dimensional and $1$-dimensional ...
8
votes
2answers
375 views

central/critical/special values of L-functions terminology

I have a question about the terminology for special values of L-functions. Is the following a correct description of standard usage: Suppose L(s) is an L-function which satisfies a functional ...
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0answers
113 views

Off critical line zeros for half integer weight $L$-functions

Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put $$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s} $$ to be the $L$-function. Further ...
2
votes
1answer
138 views

On the absolute convergence of the local-zeta integral.

Though I am in a situation considering only local-zeta integral, to explain my question briefly, let me ask it in quite general form. Let $f(s,g)$ be a two variable smooth good (in a suitable sense) ...
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0answers
77 views

Non vanishing and cuspidality of the theta lift of trivial representation.

Hi! Let E/F be a quadratic number field extension. Then we make some hermitian and skew hermition vector spaces and define unitary group on it.(namely U(1) and U(3)) Then, I am wondering whether the ...
0
votes
0answers
95 views

On the explicit formula of the height function occuring on the doubled Weil representation.

Hi! I am wondering the exact formula of height function of $GL(n)$ which occurs in the doubling Weil representation. To be more precise, let me introduce the basic setting for this. Let $F$ be the ...
2
votes
2answers
300 views

A question on twisted L-function

Hi! This is my first question here. In studying automorphic form, I am wondering the relation of critical L-values of some representation and its twisted representation by a character. For example, ...
0
votes
0answers
86 views

Can GRH for complex primitive Dirichlet characters fail with a single non-trivial zero off the critical line?

Can GRH for complex primitive Dirichlet character fail with a single non-trivial zero off the critical line? For real characters this is impossible because the non-trivial zeros are in ...
3
votes
2answers
498 views

About equivalent statements of the Birch and Swinnerton-Dyer Conjecture [closed]

The Birch and Swinnerton-Dyer Conjecture is well known in the current literature http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture My question is about the possible equivalent ...
6
votes
1answer
564 views

A stupid question on automorphic l-function

This may be a silly question for experts in this area. But I am really suffering for not being able to compute local-L function of some automorphic representation. So, I post it hoping some ...
8
votes
2answers
257 views

Is the following the right definition of $L$-functions (on the Galois side)?

This question may be too elementary for this forum, but I have asked it on math stackexchange and didn't get an answer. I have now deleted it so there wouldn't be duplicates... Here is the question as ...
6
votes
1answer
1k views

derivatives of Artin L-functions

This is a vague question: I'm sorry for that. Let's start with $\chi$ a (primitive odd) Dirichlet character modulo $n$ and look at the corresponding L-function $$ L(s, \chi)=\sum ...
4
votes
0answers
191 views

Do infinite and ramified local factors of the Dedekind zeta function of a tame number field characterize its local root numbers?

Let say you have two number fields, that are tamely ramified, and suppose that the $p$-part of their Dedekind zeta functions coincide for all prime $p$ which is ramified in either field. Suppose ...
3
votes
0answers
182 views

Is this extension of the Selberg class trivial?

I came across the following modification of the Selberg class in some of my work (see below), and while I've moved on in some sense -- I submitted the paper in question -- I can't get it off of my ...
1
vote
1answer
184 views

Understanding the determinant of the action of Frobenius on the character group of the toric part of the reduction of the Jacobian of a curve.

I am trying to understand a certain sentence in a paper that I am reading. Let me start with some notation/background. (For a motivation of why this should be interesting, see below, under the ...
4
votes
2answers
456 views

Blueprint of L-functions and need for introducing them ( Hasse-Weil L-functions )

Dear All, This question may appear elementary to all the experts in number theory , but forgive me. I really wanted to know how did the $L$-functions came into existence, especially the Hasse-Weil ...
0
votes
0answers
334 views

Linking L function dynamics with behavior close to s = 1 ?

A division, found on a sample set of semi-stable elliptic curves, calls for interpretation regarding the Birch and Swinnerton-Dyer conjecture and the dynamic behavior of the L functions involved. ...
0
votes
1answer
657 views

Interplay between Riemann and Swinnerton-Dyer

Hello everyone, After reading RH ( Riemann's Hypothesis ) and Swinnerton-Dyer conjecture, I asked myself why can't RH hold for $L$-Functions ( Hasse-Weil L-function ). In particular the GRH imposes ...
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0answers
321 views

P-Adic poly Bernoulli numbers

we can define p-adic Bernoulli polynomials by using q-integral on $Z_p$ and T.Kim's method, But how can we define p-adic poly-Bernoulli numbers and polynomials by using integral on $Z_p$ ?
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0answers
126 views

Non-holomorphic L-functions

In some old notes, I found the following conjecture: Let $\mathbb T$ denote the unit circle and let $\chi:{\mathbb N} \to {\mathbb T}$ be fully multiplicative. Then the L-series $$ ...
9
votes
1answer
598 views

Montgomery's pair correlation function without RH?

In the theory of the Riemann zeta function, Montgomery's Pair correlation function is defined as $$ F(\alpha) = \frac{1}{N(T)} \sum_{T < \gamma, \gamma' < 2T} T^{i \alpha (\gamma - \gamma')} ...
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0answers
246 views

Automorphism group of algebraic function fields

Let $K$ be a finite field and let $F/K$ be a function field. Is it possible to deduce the genus of $F/K$ from the automorphism group of $G=Aut(F/K)$? Is it possible to do so if we know that $|G|$ is ...
9
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0answers
434 views

Applications of Artin L-functions

Does anybody know a good reference which gives examples of applications of Artin L-functions to "elementary" number theory? Many thanks!
3
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4answers
765 views

Values of Dirichlet L-funcions at natural numbers

I want to know about reference of formulas for $$ L(s,D)=\sum_{n=1}^\infty \left(\frac{D}{n}\right)\,n^{-s} $$ for $s$ a positive integer number and $D$ a fundamental discriminant. For $s=1$ we have ...
5
votes
1answer
609 views

The Correlation of the Mobius Function and Dirichlet Characters.

Let $\chi$ be a Dirichlet character, and define $\phi_\chi (n)$ so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$ In other words ...
1
vote
1answer
432 views

What are non-abelian $L$-functions?

I have heard people discussing the utility of $L$-functions, claiming that since they are essentially cohomological entities, they are "abelian" and therefore lack force. From looking around on the ...
16
votes
1answer
696 views

Distinct simple zeros of Dirichlet L-functions

Given a finite set of distinct primitive Dirichlet characters, $\chi_1, \dots, \chi_r$, is it known that the product of the L-functions, $$L(s):=\prod_{i=1}^r L(s,\chi_i),$$ has a simple zero? It's ...
4
votes
1answer
1k views

Corvallis 1979 proceedings

These proceedings have long been freely available on the AMS website, but now it seems we can't even find them anymore (e.g. http://www.ams.org/publications/online-books/pspum331-index and ...
15
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1answer
1k views

Two-variable p-adic L-functions of elliptic curves

Suppose $K$ is an imaginary quadratic field (with class number 1, for simplicity), $p \ne 2$ a prime split in $K$, and $K_\infty$ the $\mathbb{Z}_p^2$-extension of $K$. If $E / \mathbb{Q}$ is an ...