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Questions tagged [l-functions]

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, p-adic L-functions, etc.

5
votes
1answer
154 views

Functional equation for general number fields

When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, ...
-2
votes
0answers
77 views

Are there proven cases of permutations of the Satake parameters being actually isometries ?

This is a follow-up to that rather old question of mine : Would a proof of Ramanujan Conjecture together with other known results about automorphic L-functions imply the Grand Riemann Hypothesis? I ...
2
votes
0answers
82 views

Function equation over general number fields

Let $\chi$ be a Hecke character on a number field $k$, where could I find a precise reference for the function equation of the $GL(1)$ L-functions $$L(s, \chi)?$$ I only find references for the case ...
4
votes
1answer
179 views

Modular forms and Period Polynomials

1.) What is the importance of special values of L functions in connection to weakly holomorphic modular forms? Why is the study of special values a subject of intense study except the fact it is ...
3
votes
0answers
164 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 ...
1
vote
0answers
36 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
votes
1answer
288 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
1answer
86 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 $\phi$. ...
10
votes
1answer
469 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
votes
1answer
212 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
votes
1answer
408 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
1answer
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 $\...
16
votes
0answers
681 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 ...
0
votes
0answers
70 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 $N$, $\chi$...
4
votes
0answers
218 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 ...
8
votes
0answers
141 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 ...
1
vote
0answers
115 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
1answer
214 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, ...
3
votes
0answers
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
votes
1answer
285 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 $\...
3
votes
0answers
160 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
0answers
64 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 $L$-...
12
votes
3answers
453 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
votes
2answers
138 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
1answer
223 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}(...
4
votes
1answer
196 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
votes
2answers
657 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
votes
1answer
224 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
1answer
233 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
0answers
117 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
1answer
122 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
1answer
218 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
votes
1answer
160 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
1answer
203 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 "...
10
votes
0answers
303 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
1answer
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
votes
0answers
91 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
votes
0answers
341 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
1answer
271 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
2answers
194 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
0answers
106 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
1answer
189 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
1answer
179 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
1answer
297 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
1answer
757 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
0answers
86 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
1answer
149 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
1answer
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
0answers
212 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
1answer
411 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, ...