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.
435
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Kronecker limit formula, modular curves, and the class number problem
Let
$$Q(x,y)=ax^2+bxy+cy^2$$
be a positive definite quadratic form with $a>0$ and $D=b^2-4ac<0$. Let
$$\zeta_Q(s)=\sideset{}{'}\sum_{m,n}Q(m,n)^{-s},$$
the accent indicating that $(0,0)$ is ...
- 2,335
12
votes
3
answers
2k
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Are L-functions uniquely determined by their values at negative integers?
Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a_1, a_2, a_3, \cdots$ such that
the corresponding L-function
$$L_{\{a_n\}...
- 1,410
7
votes
1
answer
658
views
Change of variables for $p$-adic integral
Say $p$ is an odd prime. Suppose I have a measure $\mu$ on $\mathbf{Z}_p$. As in II.4.3 in Colmez - Fonctions d'une variable $p$-adique, I can restrict $\mu$ to $1+p\mathbf Z_p$, and there is a ...
- 399
3
votes
2
answers
307
views
Explicit formula: explicit work with general smoothing?
The following is a literature question, in the sense that I already know how to do what I am asking about, and in fact have already done it; now I'd like to write a brief historical overview as an ...
- 19.4k
3
votes
0
answers
159
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Similarity between two $L$-functions (Hasse-Weil $L$-function of twisted ellptic curve and Dirichlet $L$-function)
Let $E$ be an elliptic curve over $\mathbb Q$ with conductor $N$ and $E_d$ be its twisted curve by $d$, where $d$ is a fundamental discriminant with $(d,N)=1$. Let $\chi_d$ be a Dirichlet character ...
- 653
3
votes
1
answer
313
views
Complex L-functions for Hermitian modular forms?
Fix an imaginary quadratic field $K$, and let $\mathcal{O}_K$ be its ring of integers. A Hermitian modular form of genus 1 (i.e., an automorphic form on $GU(1,1)$) of weight $(k_1,k_2)$ on a ...
- 909
5
votes
0
answers
140
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Is there a converse to Vatsal's theorem on congruence of p-adic L-functions?
Let $f=\sum_n a_n(f) q^n$ and $g=\sum_n a_n(g) q^n$ be normalized (cuspidal) newforms whose Fourier coefficients are contained in the p-adic field K for which the uniformizer of $\mathcal{O}_K$ is ...
user130124
2
votes
0
answers
179
views
Adelic Mellin transform with nontrivial character
This is a short question with a lot of setup. I apologize in advance.
In Dan Bump's "Automorphic Forms and Representations," he constructs the L-function of a modular form via an "adelic Mellin ...
- 909
2
votes
0
answers
144
views
Well-known estimate for $L(s,\chi)$ for $\sigma=\text{Re}s\geq 1/2$
This is a very short question.
Let $s=\sigma+it$ be a complex number with $\sigma \geq 1/2$.
In the paper 'Jutila, Matti. "On the Mean Value of $L(1/2, \chi)$ FW Real Characters." Analysis 1.2 (...
- 653
13
votes
1
answer
742
views
Least quadratic residue under GRH: an explicit bound
Let $m$ be a positive integer and $\chi$ a primitive character mod $m$. Let $x$ be such that $\chi(p)\ne 1$ for all primes $p<x$. Assume GRH. How can one bound $x$ in terms of $m$ ? I do not need ...
- 1,130
4
votes
2
answers
461
views
Real non trivial zeros of Dirichlet L-functions
When dealing with the prime number theorem in arithmetic progressions, one cannot exclude the possible presence of a real zero close to $1$ for at most one real character mod $q$. On the other hand, ...
- 1,174
4
votes
0
answers
249
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Second derivative at 1 of L function of elliptic curve
Let $E$ be an elliptic curve over $\mathbb Q$ of conductor $N$ and rank $0$. It follows from the functional equation that
$$L'(E,1)=(\log(2\pi/\sqrt{N})+\gamma)L(E,1)$$
where $\gamma$ is Euler's ...
- 11.8k
5
votes
2
answers
416
views
Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms
Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients.
Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know the ...
- 153
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votes
0
answers
416
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A mysterious number related to Hasse-Weil L-function of elliptic curve
Let $E/K$ be a non-isotrivial elliptic curve over a function field $K$ of characteristic $p$, with field of constant $F_q$, with semistable reduction. Its Hasse-Weil L-function $L(s)$ is a polynomial ...
4
votes
0
answers
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How many exceptional conductors are there?
We say that a conductor $q$ is exceptional if there is a primitive quadratic character $\chi$ modulo $q$ such that $L(s,\chi)$ has a real zero $\beta$ such that $\beta > 1-c/\log q$ (where $c$ is ...
- 6,608
7
votes
1
answer
519
views
Functional equation for general number fields
When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, ...
- 907
2
votes
0
answers
100
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Function equation over general number fields
Let $\chi$ be a Hecke character on a number field $k$, where could I find a precise reference for the function equation of the $GL(1)$ L-functions
$$L(s, \chi)?$$
I only find references for the case ...
- 907
4
votes
1
answer
666
views
Modular forms and Period Polynomials
1.) What is the importance of special values of L functions in connection to weakly holomorphic modular forms? Why is the study of special values a subject of intense study except the fact it is ...
- 43
4
votes
0
answers
226
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holomorphic continuation of motivic $L$-functions
The question is rather easy to formulate: when is the $L$-function of a pure motive over $\mathbb{Q}$ expected to have a holomorphic (as opposed to simply meromorphic) continuation to the complex ...
- 41
1
vote
0
answers
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Mean value estimates for general number fields
Results are known in many different cases to bound powers of L-functions on average over a wide enough family. I am interested in results for general number fields, not only for the rationals, for ...
- 907
11
votes
2
answers
1k
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Relation between Fourier coefficients and Satake parameters
Let $L(s)$ be an automorphic L-function (attached to a self contragredient automorphic representation on $GL(3)$), according to the following notations for $s$ of sufficiently large real part:
$$L(s) =...
- 5,314
7
votes
1
answer
578
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 $\phi$. ...
- 467
14
votes
1
answer
951
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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 ...
- 11.8k
9
votes
1
answer
395
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 ...
- 1,702
14
votes
1
answer
868
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 ...
- 243
4
votes
1
answer
200
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 $\...
- 3,764
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votes
0
answers
959
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 ...
user127738
0
votes
0
answers
124
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 $N$, $\chi$...
- 303
6
votes
1
answer
466
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 ...
- 3,317
10
votes
0
answers
362
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 ...
- 6,110
1
vote
0
answers
149
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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
391
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, ...
- 1,191
3
votes
0
answers
125
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
$$L(s,\...
- 6,110
18
votes
1
answer
1k
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Definitions of $\pi_1 \times \pi_2, \pi_1 \boxplus \pi_2, \pi_1 \boxtimes \pi_2$
Let $\pi_i$ be a smooth, admissible (possibly irreducible) representation of $\operatorname{GL}_{n_i}(k)$ for $k$ a $p$-adic field. I have seen the following representations defined in terms of $\...
- 6,110
4
votes
0
answers
207
views
Generalization of Weil's theorem for L-functions
I have a question reffering to a theorem by Weil, which gives sufficient conditions that a given L-series $$ L(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}$$ which is convergent somewhere comes from a ...
- 163
2
votes
0
answers
94
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$-...
- 6,110
13
votes
3
answers
674
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-...
- 7,646
3
votes
2
answers
197
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 ...
- 5,314
1
vote
1
answer
302
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}(...
- 131
4
votes
1
answer
287
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 ...
- 6,932
14
votes
2
answers
974
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/...
11
votes
1
answer
306
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 ...
- 1,401
6
votes
1
answer
319
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 ...
- 7,646
3
votes
0
answers
138
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 ...
- 31
3
votes
1
answer
254
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 ...
- 5,314
6
votes
2
answers
451
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)$...
- 3,764
6
votes
1
answer
238
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 ...
- 7,646
6
votes
1
answer
241
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 "...
- 3,975
18
votes
0
answers
711
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 $\mathbf Q$ that is unramified away from $p$ (more generally, away from a finite set $S$...
- 27.3k
5
votes
1
answer
2k
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=...
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