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Results tagged with l-functions
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user 2481
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.
4
votes
Poles of the L-series of the elliptic curve which has CM
It is a general fact that if $E$ is any elliptic curve over $\mathbf{Q}$ (CM or otherwise) its Hasse-Weil $L$-series has no poles, at least if you multiply by the appropriate $\Gamma$ factor. (This fo …
- 37k
10
votes
Accepted
Blueprint of L-functions and need for introducing them ( Hasse-Weil L-functions )
There is an excellent reason why the exponential term and the division by $n$ are there, although they look a bit mysterious at first.
Firstly, a correction to your formula: it should be $|C(\mathbb …
- 37k
10
votes
Accepted
Is there a known construction of Cuspidal representations of GL(3) isomorphic to their own t...
Let $E / F$ be the cyclic cubic extension corresponding to $\chi$ by class field theory. Let $\sigma$ be a generator of $\operatorname{Gal}(E / F)$, and let $\psi$ be a character of $E^\times \backsla …
- 37k
7
votes
Accepted
On computing the periods for $L$-function of a primitive form for $\Gamma_0(N)$ and of weigh...
I think your question is actually not well-posed, because it is not clear if such $\Omega_{\pm}$ actually exist, and if they do exist they are very far from being uniquely determined.
Firstly, it is …
- 37k
5
votes
0
answers
176
views
Non-vanishing of L-values for twists of Hilbert modular forms
Let $F$ be a real quadratic field, and let $f$ be a Hilbert modular form over $F$ of parallel weight 2.
It's known, by a theorem of Rohrlich, that there exist infinitely many Hecke characters of $F$ …
- 37k
9
votes
0
answers
748
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, unrami …
- 37k
18
votes
1
answer
2k
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 rea …
- 37k
3
votes
1
answer
614
views
Which values of symmetric square $L$-functions are critical?
I've recently been learning about the special values of symmetric square $L$-functions of modular forms.
If $f$ is a cuspidal modular eigenform (of some weight $k \ge 2$) then its symmetric square $L …
- 37k
5
votes
Accepted
The effect of base change on the L-function of GL(2)?
If $\pi = \pi(\chi_1, \chi_2)$ is an irreducible principal series, then $BC(\pi)$ is the irreducible principal series attached to the unramified characters $\chi_i \circ N_{E/F}$ of $E^\times$. So if …
- 37k
4
votes
Accepted
Order of vanishing of $L$-function and mixed Hodge-structures
There is a good reason why this particular form of the Beilinson conjecture cannot possibly be valid for the particular $i$ and $n$ you mention.
The real $\mathbb{R}$-MHS $H^1(X(\mathbb{C}), \mathbb{R …
- 37k
9
votes
Accepted
$p$-adic L function of an odd Dirichlet character
Theorem. Let $p > 2$ be prime, and let $\chi$ be a Dirichlet character (non-trivial, and of prime-to-$p$ conductor, for
simplicity). Choose $t \in \mathbb{Z} / (p-1)\mathbb{Z}$ such that
$(-1)^t = \ch …
- 37k
4
votes
What is the conductor of an automorphic representation for $\Gamma_0(q)$ in $GSp(4)$?
To make the question well-posed, I'm going to suppose that we fix $\pi$ and take $q$ to be the smallest integer such that $\pi$ has nonzero invariants under $\Gamma_0(q)$. Then $q$ gives you some info …
- 37k
15
votes
1
answer
1k
views
Jacquet's approach to Rankin--Selberg L-functions
In his book "Automorphic Forms on GL(2), II", Springer Lecture Notes vol. 278, Jacquet defines the Rankin--Selberg L-function of $\pi_1 \times \pi_2$, where $\pi_i$ are automorphic representations of …
- 37k
6
votes
Accepted
degree of an isobaric sum
No, it has nothing to do with the dimension of the representation space. (Automorphic representations of $GL_n$ are always infinite-dimensional for $n > 1$.) The isobaric sum operation changes the gro …
- 37k
10
votes
p-adic L-functions of modular forms: why the condition $v_p(\alpha)<k-1$?
Let me answer your questions in the opposite order.
(2) This question is vacuous: it cannot happen that both roots have valuation $> k-1$, because the product of the roots is $p^{k-1}$. So either bot …
- 37k