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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2answers
878 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.
15
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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
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2answers
638 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
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1answer
593 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
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1answer
705 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 ...
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13answers
4k 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
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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 ...
9
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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 ...
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0answers
856 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 ...
5
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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
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3answers
585 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
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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
508 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
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1answer
798 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 ...
18
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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 ...
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 ...
20
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3answers
892 views

Universality of zeta- and L-functions

VoroninĀ“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
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3answers
627 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
775 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 ...
12
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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 ...