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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154 views

### holomorphic continuation of motivic $L$-functions

The question is rather easy to formulate: when is the $L$-function of a pure motive over $\mathbb{Q}$ expected to have a holomorphic (as opposed to simply meromorphic) continuation to the complex ...

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**0**answers

33 views

### Mean value estimates for general number fields

Results are known in many different cases to bound powers of L-functions on average over a wide enough family. I am interested in results for general number fields, not only for the rationals, for ...

**7**

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**1**answer

276 views

### Relation between Fourier coefficients and Satake parameters

Let $L(s)$ be an automorphic L-function (attached to a self contragredient automorphic representation on $GL(3)$), according to the following notations for $s$ of sufficiently large real part:
$$L(s) =...

**4**

votes

**1**answer

73 views

### Corollary for Casselman-Shalika formula

Assume $\pi$ is an unramified representation of $GL_n(F)$, where $F$ is a p-adic field. And $\phi$ is an unramified vector for $\pi$. Assume $W_{\phi}$ is a Whittaker function associated to $\phi$. ...

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**1**answer

457 views

### Euler factors of L-function at bad primes

This is of course a very-well known problem, but still let me ask the questions my way. Let $L(s)$ be a "motivic" $L$-function, whatever that means: in particular, it has an Euler product (including ...

**8**

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**1**answer

198 views

### Analogue of the original Birch–Swinnerton-Dyer conjecture for abelian varieties

$\newcommand{\Q}{\Bbb Q}
\newcommand{\N}{\Bbb N}
\newcommand{\R}{\Bbb R}
\newcommand{\Z}{\Bbb Z}
\newcommand{\C}{\Bbb C}
\newcommand{\F}{\Bbb F}
\newcommand{\p}{\mathfrak{p}}
$
Let $A$ be an abelian ...

**13**

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**1**answer

389 views

### BSD conjecture for rank 1 elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that
$$\text{ord}_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$
Thanks to the work of Gross-Zagier and ...

**4**

votes

**1**answer

162 views

### Local L-function $L(s,\pi_p\times \chi_p)=1$

Let $\pi_p$ be a ramified representation of $GL(n,\mathbb{Q}_p)$.
Let $\chi_p$ be a ramified representation of $GL(1,\mathbb{Q}_p)$.
Is it generally known that
$L(s,\pi_p\times \chi_p)=1$ if $\...

**15**

votes

**0**answers

664 views

### Why arithmetic Langlands?

In trying to understand the import of Akshay Venkatesh' most recent work I found myself wondering anew about that old gnawing mystery: why Langlands? Why should arithmetic of polynomial equations over ...

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**0**answers

67 views

### Summation formula for twisted L-function

Does any expert here know something about the summation formula of the Voronoi type for the sum $$\sum_{n\le X} a_{f}(n)\chi(n) e\left(\frac{an}{c}\right)?$$
Here $f$ is a newform of level $N$, $\chi$...

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**0**answers

215 views

### Strengthened supercongruences for Ramanujan-type formulas for $1/\pi^k$

The question below is again a follow-up of an old question.
Motivation: Zhi-Wei Sun listed a number of supercongruences attached to Ramanujan-type $1/\pi$ formulas in the arXiv paper which can be ...

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125 views

### Local Langlands Correspondence for unramified principal series representations

Let $G$ be a connected, reductive group over a $p$-adic field $k$. Assume $G$ is quasi-split with Borel subgroup $B = TU$. Consider those irreducible admissible representations $\pi$ of $G(k)$ which ...

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111 views

### A mixed of the Dedekind zeta function and the L-function

I have recently come across the following function, which seems like a "mix" between the Dedekind zeta function and the L-function:
$\sum_I\frac{\chi_k(N(I))}{N(I)^s}$
where $\chi_k(n)$ is the ...

**6**

votes

**1**answer

192 views

### p-adic L-functions for (dual of) fine Selmer Groups

If $R(E/\Bbb Q_{\infty})$ is the fine Selmer group and $Y(E/\Bbb Q_{\infty})$ is its dual, then we know that $Y(E/\Bbb Q_{\infty})$ is a finitely generated $\Lambda$-module and by a theorem of Kato, ...

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**0**answers

103 views

### L-functions for the Weil group over short exact sequences

Let $(\rho,V)$ be a continuous finite dimensional representation of the Weil group $W_F$ over a local field $F$. If $V$ decomposes as a direct sum $V_1 \oplus V_2$ of representations, then
$$L(s,\...

**8**

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**1**answer

271 views

### Definitions of $\pi_1 \times \pi_2, \pi_1 \boxplus \pi_2, \pi_1 \boxtimes \pi_2$

Let $\pi_i$ be a smooth, admissible (possibly irreducible) representation of $\operatorname{GL}_{n_i}(k)$ for $k$ a $p$-adic field. I have seen the following representations defined in terms of $\...

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**0**answers

156 views

### Generalization of Weil's theorem for L-functions

I have a question reffering to a theorem by Weil, which gives sufficient conditions that a given L-series $$ L(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}$$ which is convergent somewhere comes from a ...

**2**

votes

**0**answers

62 views

### Definition of Local L-function for a representation of a torus?

Let $G$ be a connected, reductive group over a $p$-adic field $k$. Let $\pi$ be an irreducible, admissible representation of $G(k)$, and $r$ a finite dimensional continuous representation of the $L$-...

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**2**answers

354 views

### Some questions on the $p$-adic properties of special $L$-values

Warning: Some naive, speculative questions from a total non expert.
Let $\rho$ be a p-adic representation of the Galois group $Gal(\overline K/K)$ for a number field $K$. We can consider the Artin L-...

**3**

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**2**answers

136 views

### What is known about gaps between zeros of L-functions?

In many different settings, it is possible to determine statistics about spacings (pair correlation, small gaps, large gaps, champions, etc.), for instance
prime numbers
Laplacian eigenvalues on a ...

**1**

vote

**1**answer

219 views

### A question about Kato's explicit reciprocity law

In the paper Iwasawa Theory and F-analytic Lubin-tate $(\phi,\Gamma)$-modules
Prop 3.4.2 says that for any $x\in{S}$, there exists (not uniuqely) $f(T)\in{B_{rig,F}^+}$ such that
$f(u_n)=\log_{LT}(...

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votes

**1**answer

193 views

### References on Erdos conjecture on arithmetic progressions

Erdos conjectured that any set $ A $ of positive integers such that $ \sum_{n\in A}\dfrac{1}{n} $ diverges contains arbitrary long arithmetic progressions. The celebrated Green-Tao theorem is a ...

**14**

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**2**answers

645 views

### What are zeta functions good for?

I know a couple of answers to the above question:
They can be used for point counting over finite fields/estimating the distribution of primes in characteristic 0.
There are various conjectures/...

**10**

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**1**answer

221 views

### Critical points of Dirichlet L functions

Let $L(s,\chi)$ denote a Dirichlet $L$-function for a real-valued non-principal
character $\chi$. This has limiting value $L(\infty,\chi) = 1$ and we are interested in how this limit is approached ...

**6**

votes

**1**answer

229 views

### Herbrand-Ribet and Mazur-Wiles for function fields

Is there a version of Herbrand-Ribet or Mazur-Wiles (relating divisibility of class groups to special values of L-functions) for functions fields (over finite fields)?
Probably the proofs would have ...

**3**

votes

**0**answers

116 views

### L-functions and Hecke theory for general number fields

I often read about analytic number theory on $\mathbf{Q}$ when it turns to explicit computations, so I wonder how much results generalize as they stand and the use of $\mathbf{Q}$ is merely made for ...

**3**

votes

**1**answer

116 views

### Consistency of the notion of conductor of a representation

The notion of analytic conductor of a generic representation of $\mathrm{GL}(n)$ has been defined by Iwaniec and Sarnak, and since then is at the heart of many works in analytic number theory and used ...

**6**

votes

**1**answer

215 views

### Symmetric powers of Ramanujan tau-function

Let $\Delta(z)$ be the modular form associated with Ramanujan $\tau$-function.
For any $k=2,3,...$, $Sym^k\Delta$ is conjectured to be an automorphic form on $\mathrm{GL}(k+1)$ and $L(s, Sym^k\Delta)$...

**6**

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**1**answer

158 views

### Regulator of abelian extensions of Q

Let $K = \mathbb Q(\mu_m)$ and $\zeta_K$ it's Dedekind zeta function. We know from the class number formula that, around $0$:
$$\zeta_K(s) \sim s^{r_1+r_2-1}h(K)R(K)/w(K) $$
where $h,R,w$ stand for ...

**6**

votes

**1**answer

202 views

### The L-function of Q(-1/2) and the “number of prime $p\equiv 3$ divisors” function

In the framework of classical motives, there is no such thing as a motive $\mathbb Q(-\tfrac 12)$, i.e. a tensor root of $\mathbb Q(-1)$. There is one, however, in a more general setting of "...

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290 views

### Infinite extensions such that every elliptic curve has finite rank

The comments to this answer seem to make the following claim.
Claim. Let $K$ be the maximal abelian extension of $\mathbb Q$ that is unramified away from $p$ (more generally, away from a finite set $...

**5**

votes

**1**answer

1k views

### Does $ M(x)=O(\sqrt{x}) $ if and only if the De Bruijn-Newman constant is negative?

The Riemann hypothesis is equivalent to the assertion that the De Bruijn-Newman constant $ \Lambda $ , as defined in https://www.sciencedirect.com/science/article/pii/S0001870809001133/pdf?md5=...

**5**

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**0**answers

85 views

### Relation between Gross-Koblitz and Chowla-Selberg formulas

The Chowla-Selberg formula relates the eta function with values of the gamma function at rational numbers. The eta function appears, at least in the proofs I have seen, related to values of $L$-...

**2**

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**0**answers

328 views

### Questions about the exceptional zeros of Dirichlet $L$-functions

I have couple questions regarding the exceptional zeros of Dirichlet $L$-functions. We have the following result:
There is a constant $c_1 > 0$ such that $L(\sigma, \chi) \not = 0$ whenever
$$
\...

**6**

votes

**1**answer

264 views

### Difference of Beilinson conjecture and equivariant Tamagawa number conjecture

As stated in the title, I am wondering the main difference between Beilinson conjecture and eTNC. If I read correctly, I can see that there are many literature treating both conjectures in the same ...

**5**

votes

**2**answers

190 views

### Can the relative degree and ramification index can be read off the characteristic polynomial?

Let $L/K$ be a Galois extension of global fields with Galois group $G$. Assume that for a prime $p$ of $K$ we are given the datum $(L_{p}(s,\rho))_{\rho}$, where $\rho$ varies over the irreducible ...

**3**

votes

**0**answers

103 views

### Expression of the root number for Maass forms

Take a holomorphic cusp newform, say $f \in S_k(N)^\mathrm{new}$, for a squarefree level $N$. It is an eigenvalue of the Atkin-Lehner operator, and this feature allows to express its root number as
$$\...

**-1**

votes

**1**answer

188 views

### Which properties of L-functions can be proven assuming they are objects of a symmetric bimonoidal category?

The title says it all : assuming all L-functions are objects of a symmetric bimonoidal category $ (\mathcal{C},\oplus,\otimes,s\mapsto 1,\zeta) $ , where $ \oplus $ stands for the usual product, ...

**8**

votes

**1**answer

174 views

### Special values of adjoint $L$-functions of automorphic representations of $\mathrm{GSp}(4)$ as Petersson norms

Here I consider cuspidal automorphic representations $\pi$ over the similitude group $\mathrm{GSp}(4,\mathbb{A}_\mathbb{Q})$. Let $f$ be a non-zero vector in the representation $\pi$. I want to know ...

**4**

votes

**1**answer

294 views

### Poles of $L$-functions associated to Maass forms

Let $\pi$ be an automorphic representation of $GL_2$ over a number field. What can I say concerning the order of the pole at $1$ of the $L$-function $L(s, \pi)$? Can we say more about $L(s, \mathrm{...

**9**

votes

**1**answer

753 views

### How does Riemann hypothesis implies estimates?

In Iwaniec, Luo and Sarnak article (precisely (4.23)), it is said that GRH for $L(s, \mathrm{sym}^2(f))$, for a holomorphic cusp newform $f$ of level $N$ and weight $k$, implies
$$\sum_{p \nmid N} \...

**5**

votes

**0**answers

83 views

### Local L-factors for automorphic representations

For Hecke L-functions associated to a holomorphic cusp form $f$ of level $N$, the local factors can be decomposed into
$$L_p(s, f) = (1-\lambda_f(p)p^{-s} + \chi_N(p)p^{-2s})^{-1}$$
where $\chi_N$ is ...

**3**

votes

**1**answer

147 views

### On computing the periods for $L$-function of a primitive form for $\Gamma_0(N)$ and of weight $k > 2$

I am interested in computing the algebraic parts of $L(f, n, \chi)$ for a primitive form $f \in S_k(\Gamma_0(N))$ with wieght $k > 2$ twisted by a primitive Dirichlet character $\chi$ of conductor $...

**36**

votes

**1**answer

2k views

### A mysterious connection between Ramanujan-type formulas for $1/\pi^k$ and hypergeometric motives

The question below is the follow-up of this question on MathOverflow.
Motivation: As is stated in the former question, those identities(formula (35)-(44)) of $1/\pi$ attributed to Ramanujan are ...

**8**

votes

**0**answers

203 views

### Residue of Eisenstein Series on GL(n)

Reference: Mœglin, C.; Waldspurger, J.-L.: Le spectre residuel de GL(n)
On GL($n$), the result of Waldspurger shows that if an automorphic representation $\pi$ is non-cuspidal and in the discrete ...

**11**

votes

**1**answer

398 views

### What kind of non-cuspidal automorphic representation are not isobaric sums?

Let's say $\pi$ is an automorphic representation on $GL_3(A_{\mathbb Q})$ (or $GL_n(A_{\mathbb Q})$).
If $\pi$ is not cuspidal, what $\pi$ can be other than isobaric sums?
If there is such a thing, ...

**10**

votes

**0**answers

142 views

### Explicit L-factor for supercuspidals

I am interested in L-functions for the quasi-split unitary group $U$ in three variables, following the construction by zeta integrals of Gelbart-Piatetski-Shapiro. My aim is to understand the ...

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votes

**2**answers

306 views

### Rankin-Selberg integral for GL(3) form with Odd Maass form on GL(2)

Let $F$ be a Hecke-Maass cusp form for $SL_3(\mathbb Z)$.
Let $u$ be a Hecke-Maass cusp form for $SL_2(\mathbb Z)$.
The following integral
$$\mathcal L(s,F\times u)=\int_{{SL}(2,\mathbb{Z})\...

**8**

votes

**3**answers

473 views

### “Extra Euler factors” in one definition of the L-function of a twist of a modular form

Let $(\rho_{f,\lambda})_\lambda$ be the system of Deligne's $\ell$-adic representations attached to a modular newform $f$ (where $\lambda$ runs over the finite places of the number field $K$ generated ...

**4**

votes

**0**answers

102 views

### Examples of conjectural functorial transfer which has $\times GL(1)$ functional equation?

I am look for some conjectural functorial transfer $X$ which
(A)for any $GL(1)$ automorphic representation $\pi$, we have
$L(s, X\times \pi)$ is holomorphic and satisfies certain functional ...