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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5
votes
1answer
621 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
436 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
709 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
votes
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 ...
2
votes
1answer
297 views

How do you calculate the Euler factors of the imprimitive symmetric square at primes with bad reduction?

The reference for this question is Coates and Schmidt, Iwasawa theory for the symmetric square. Let $G = \textrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}))$ and let $D_r \supseteq I_r$ be a ...
16
votes
2answers
1k views

Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field.

Let $M$ be the splitting field of x^8 + 3*x^7 + 13*x^6 + 17*x^5 + 45*x^4 + 37*x^3 + 11*x^2 + 112*x + 108 over the rationals. If I've understood some tables ...
1
vote
2answers
666 views

Functional equations relating to p-adic L-functions

Let f be a modular form of weight k for $\Gamma_0(N)$. Let us assume that $p\not\vert$N. Then we can construct 2 p-adic L-functions corresponding to the 2 roots $\alpha$ and $\beta$ of the equation ...
3
votes
0answers
349 views

Non-vanishing of twists of L functions for GL(4)

Hello, This is a question in the spirit of Nonvanishing of central L-values of quadratic twists? and the application I have in mind is to p-adic L-functions a la Ash-Ginzburg. The question is ...
13
votes
3answers
1k views

Nonvanishing of central L-values of quadratic twists?

Let $\pi$ be a cuspidal automorphic representation of GL(2) over a number field (if you want, assume it's $\mathbb Q$ and $\pi$ comes from a holomorphic modular form). In the case $\pi$ has trivial ...
10
votes
0answers
385 views

Is the Gouvea-Mazur problem related to symmetric square $L$-functions?

Here's an idea that I've found appealing but have never been able to get anywhere with. One way to frame the Gouvea-Mazur question (for lack of a better term, since the original conjecture by the ...
23
votes
1answer
1k views

Special values of L-functions as periods

If $M$ is a pure motive over $\mathbb{Q}$, one cas define its $L$-function $L(M,s)$ which conjecturaly is a meromorphic function over $\mathbb{C}$ with finitely many poles. For example, when ...
7
votes
1answer
453 views

Zeroes of complete L-functions

Hello, Let $F$ and $G$ be two functions belonging in the Selberg class, $\Lambda_{F}$ and $\Lambda_{G}$ the complete L-functions associated to $F$ and $G$. I would like to know whether this assertion ...
5
votes
1answer
763 views

Generalizing Eichler-Shimura to higher dimension, again

This question is related to Intuition behind the Eichler-Shimura relation? and L-functions and higher-dimensional Eichler-Shimura relation Answering the first question above, Matt Emerton gives a ...
7
votes
1answer
709 views

Abelian varieties and Selberg class

Hello everyone, I would like to know whether, assuming Selberg's orthonormality conjecture, it would be possible to establish a "natural" correspondence between abelian varieties and functions ...
21
votes
1answer
2k views

Iwasawa main conjectures vs Bloch-Kato conjectures

Let $p$ be a prime, $K$ be a number field, $S$ a finite set of finite places of $K$ containing the set $S_p$ of places above $p$ and the places at infinity, $G:=G_{K,S}$ the Galois group of the ...
10
votes
0answers
506 views

On the relation of special values of motivic L functions and partial zetas

Let $K$ be a number field, $L$ a finite abelian extension and $\chi \in \widehat{Gal(L/K)}$ a (non-trivial) character. If we multiply out the associated Artin L-function $L(\chi,s)$ we can write this ...
0
votes
0answers
265 views

Does it exist a p-adic L function which interpolates the values of the complex one at positive integers?

I known that there are classical ways to construct $p$-adic $L$ functions for Dirichlet characters through $p$-adic integrals. We fix a character $\chi$ modulo $Np^r$ with $N$ and $p$ coprime and an ...
5
votes
1answer
931 views

Principal L-functions on GL(n)

What does the principal L-functions on GL(n), $n \geq 3, n \in \mathbb{Z}$, look like? Where can I find materials about principal L-functions on GL(n)?
22
votes
2answers
2k views

Why are Tamagawa numbers equal to Pic/Sha?

For a connected algebraic group $G$ over a global field $K$ with adeles $A$, the Tamagawa number of $G$ is the volume of $G(A)/G(K)$. It is conjectured (and often known) to be rational, namely the ...
7
votes
1answer
565 views

impact of Poincaré duality on functional equation

Given a variety $X/\mathbf{F}_q$ and a sheaf $\mathcal{F}$ on it, what is the relation of $L(X,\mathcal{F},T)$ and $L(X,D(\mathcal{F}),T)$?
13
votes
0answers
788 views

Special values of Artin L-functions

This question might be naive and might carry the heuristic that we are living in the best possible world a little too far. If so, I appreciate being told so. Background: Stark's conjecture interprets ...
8
votes
1answer
260 views

reference help needed on a fact about poles of L-functions

Suppose $\pi$ and $\rho$ are cuspidal automorphic representations on $GL(n)$ and $GL(m)$ respectively. Then the L-function $L(s,\pi \times \rho)$ has a pole iff and $m=n$ and $\pi$ is isomorphic to ...
1
vote
2answers
900 views

Arithmetic geometry from a bird's-eye view

Is ist true that Arithmetic Geometry can roughly be separated into two areas: 1) Showing that motivic $L$-functions are automorphic. 2) Calculating special values of these $L$-functions.
17
votes
1answer
1k views

What is a path in K-theory space?

In a comment on Tom Goodwillie's question about relating the Alexander polynomial and the Iwasawa polynomial, Minhyong Kim makes the cryptic but tantalizing statement: In brief, the current view is ...
6
votes
2answers
663 views

Rankin-Selberg convolutions of motivic L-series

Background: Let $M_{f_i}, i=1,2$ be two modular motives associated to cusp forms $f_i \in S_{w_i}(\Gamma_0(N_i))$ of weight $w_i$ and level $N_i$ respectively. The Rankin-Selberg convolution ...
6
votes
1answer
629 views

$L$-functions for $\Theta$-lifts

Let $E/F$ be a quadratic extension of number fields. Let $W$ be a hermitian space over $E$ of dimension $2,$ and let $V$ be a skew-hermitian space of dimension $3$ over $E.$ Consider the associated ...
12
votes
1answer
723 views

P-adic L-functions of nonabelian twists of elliptic curves

Let $E$ be an elliptic curve and $\rho$ an Artin representation of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. Then there is a "twisted L-function" $L(E, \rho, s)$, corresponding to the ...
36
votes
15answers
5k views

Shortest/Most elegant proof for $L(1,\chi)\neq 0$

Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$? Background: The non-vanishing ...
7
votes
0answers
1k views

What are “fractional motives”?

Kirti Joshi's musings mention "fractional motives". Do you know what are they good for and what the current state of constructions is for them? Edit: Further cases of "fractional motives" as ...
10
votes
2answers
2k views

p-adic L-functions

For modular forms, it is known that you can construct p-adic L-functions by interpolating (p-power conductor) twists of their associated L-functions at special values. Similarly, Kubota-Leopoldt's ...
10
votes
0answers
921 views

Are there analogues of Beilinson's conjectures for motives with coefficients?

There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...
6
votes
3answers
1k views

Why are they called L-functions?

I was hoping to see this pop up on the recent big list question about etymology or terms and symbols. Since it has not, and I can't find an answer, I will ask: What is the reason for the $L$ in ...
11
votes
3answers
614 views

Decomposition of Tate-Shafarevich groups in field extensions

Suppose $E/\mathbb{Q}$ is an elliptic curve with rank zero. According to the conjecture of Birch and Swinnerton-Dyer, the special value $L(1,E_{/\mathbb{Q}})$ should be equal (up to some harmless ...
14
votes
1answer
1k views

Stark's conjecture and p-adic L-functions

Not long back I asked a question about the existence of p-adic L-functions for number fields that are not totally real; and I was told that when the number field concerned has a nontrivial totally ...
7
votes
0answers
525 views

Existence of multi-variable p-adic L-functions

What's the "state of the art" in constructing multi-variable p-adic L-functions for number fields? More precisely: if K is a number field, and $K_{\infty} / K$ is an infinite Galois extension, ...
15
votes
1answer
860 views

constants in Gamma factors in functional equation for zeta functions.

Usually the Riemann zeta function $\zeta(s)$ gets multiplied by a "gamma factor" to give a function $\xi(s)$ satisfying a functional equation $\xi(s)=\xi(1-s)$. If I changed this gamma factor by a ...
22
votes
4answers
3k 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 ...
14
votes
4answers
2k views

Special values of $p$-adic $L$-functions.

This is a very naive question really, and perhaps the answer is well-known. In other words, WARNING: a non-expert writes. My understanding is that nowadays there are conjectures which essentially ...
21
votes
3answers
962 views

Universality of zeta- and L-functions

Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
8
votes
3answers
635 views

How many L-values determine a modular form?

Suppose $f$ and $g$ are two newforms of certain levels, weights etc. If we know that L(f,n)=L(g,n) for all sufficiently large $n$, can we conclude that $f=g$? Same question when the forms have the ...
6
votes
5answers
801 views

L-functions and random matrices

I am curious about the connection between properties of L-functions and random matrices, and about (if existent) function field versions of that. Do you know a survey or an other article where one ...
13
votes
1answer
3k views

Beilinson conjectures

Continuing an amazingly interesting chain of answers about motivic cohomology, I thought I should learn about the Beilinson conjectures, referred there. I have found some references, and they seem to ...