**40**

votes

**1**answer

1k views

### “Gross-Zagier” formulae outside of number theory

The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...

**37**

votes

**15**answers

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 ...

**27**

votes

**3**answers

1k views

### Underlying idea for (automorphic) L-function?

Edit: So with a few more months of math under my belt, I recognize some of the issues with this question. I still hope for an answer, so let me say a few things.
Within the Langlands philosophy, ...

**23**

votes

**1**answer

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 ...

**22**

votes

**2**answers

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 ...

**22**

votes

**4**answers

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 ...

**21**

votes

**3**answers

974 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 ...

**21**

votes

**1**answer

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 ...

**18**

votes

**1**answer

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 ...

**16**

votes

**2**answers

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 ...

**16**

votes

**1**answer

722 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 ...

**16**

votes

**1**answer

880 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 ...

**15**

votes

**1**answer

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 ...

**14**

votes

**1**answer

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 ...

**14**

votes

**4**answers

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 ...

**14**

votes

**1**answer

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 ...

**13**

votes

**3**answers

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 ...

**13**

votes

**0**answers

826 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 ...

**12**

votes

**1**answer

729 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 ...

**11**

votes

**3**answers

620 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 ...

**10**

votes

**2**answers

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

**2**answers

306 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 ...

**10**

votes

**0**answers

169 views

### Hodge Decompositions and Gamma Factors of Hasse--Weil L-Functions

Let $X$ be a projective variety over a number field $K$. If $\mathfrak{p}\unlhd\mathcal{O}_K$ is a prime ideal we can regard the Euler factor of its $m$th Hasse-Weil $L$-Function at $\mathfrak{p}$ as ...

**10**

votes

**0**answers

270 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 ...

**10**

votes

**0**answers

389 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 ...

**10**

votes

**0**answers

517 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 ...

**10**

votes

**0**answers

940 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 ...

**9**

votes

**1**answer

631 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')} ...

**9**

votes

**1**answer

280 views

### Is the adjoint L-function on GL(m) holomorphic?

Let $\pi$ be an automorphic representation on GL($m$)/$\mathbb{Q}$.
Define $$L(s,\pi,Ad):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an L-function with euler products of degree $m^2-1$.
...

**9**

votes

**1**answer

244 views

### Primer on Eisenstein series

My apologies if this question is a duplicate. I seached, and the closest I could locate is this question, which has very intriguing and intractable (for me) responses.
In my continuing journey of ...

**9**

votes

**0**answers

445 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!

**8**

votes

**2**answers

384 views

### Averages over integer points of the sphere

A paper of William Duke proves that integer points on the sphere are equidistributed:
$$ V_n = \{ (x,y,z) \in \mathbb{Z}^2 : x^2 + y^2 + z^2 = n \}. $$
Up to reflections across the $x$, $y$ and $z$ ...

**8**

votes

**2**answers

426 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
...

**8**

votes

**3**answers

641 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 ...

**8**

votes

**1**answer

468 views

### Would Elliott-Halberstam conjecture follow from GRH?

The Wikipedia article about Elliott-Halberstam (EH for short) conjecture says that the so-called Bombieri-Vinogradov theorem, which is a weaker form of EH conjecture, is in some sense an averaged form ...

**8**

votes

**1**answer

366 views

### Is this theorem on $L$-functions known?

Notations For $f$ a meromorphic function on a domain $\Omega\subseteq \textbf{C}$, we shall say for convenience that $f$ is represented by an Ordinary Dirichlet Series (ODS) if $f$ can be written ...

**8**

votes

**2**answers

262 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 ...

**8**

votes

**0**answers

214 views

### Symmetric Fifth Power Lift of GL(2) Automorphic Form

Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that
$$L(s, \pi, ...

**7**

votes

**5**answers

812 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 ...

**7**

votes

**1**answer

437 views

### Tate's thesis for Artin L-functions

As far as I know, Tate's thesis has been successfully applied in two fronts:
Hecke L-functions, by Tate and Iwasawa (and Teichmüller, Witt, Schmid)
Automorphic L-functions, by Jacquet, Shalika, ...

**7**

votes

**1**answer

456 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 ...

**7**

votes

**1**answer

576 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 ...

**7**

votes

**1**answer

568 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 $L(X,D(\mathcal{F}),T)$?

**7**

votes

**1**answer

712 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 ...

**7**

votes

**0**answers

288 views

### Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?

(General) Lindelof Hypothesis which says for any $L$-function we have $$L(1/2+it)\ll Q(t)^{\epsilon}$$ for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $t$.
For a Maass form $\phi$ ...

**7**

votes

**0**answers

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 ...

**7**

votes

**0**answers

536 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, ...

**6**

votes

**3**answers

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 ...

**6**

votes

**1**answer

438 views

### Generalization of Watson's triple product

In Watson's thesis (page 51) we can find his beautiful triple product formula. My question is that does there exist a generalization of this formula? By generalization, I mean:
If $\phi_n$'s are ...

**6**

votes

**1**answer

714 views

### Main conjecture for elliptic curves

Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ ...