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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.
19
votes
1
answer
2k
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 elli …
- 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
17
votes
Accepted
What is the Perrin-Riou logarithm (or regulator)?
I am sure I've already written an expository account of this somewhere, but I looked over the lecture and seminar notes on my webpage and couldn't find it, so I'll write one here instead.
Suppose we s …
- 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
14
votes
Accepted
BSD conjecture for rank 1 elliptic curves
The following theorem is due to Chris Skinner, in this 2014 paper.
Let E/Q be an elliptic curve such that rank E(Q) = 1 and the
Tate-Shafarevich group Sha(E / Q) is finite, and some other techni …
- 37k
14
votes
1
answer
982
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 …
- 37k
13
votes
What kind of non-cuspidal automorphic representation are not isobaric sums?
EDIT. A colleague wrote to me to point out that my original answer to this question was actually completely wrong: I had confused "isobaric" representations with "pure" representations (which are not …
- 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
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
10
votes
Symmetric powers of Ramanujan tau-function
It is indeed true that substantially more is known for holomorphic cusp forms than for general automorphic representations, as a consequence of modularity lifting theorems.
The strongest result so f …
- 37k
10
votes
Accepted
Main conjecture for elliptic curves
The main conjecture is a theorem if the image of the mod $p$ Galois representation of E is the whole of $GL_2(\mathbf{F}_p)$. The full statement of the conjecture, which implies what you wrote about l …
- 37k
9
votes
Regulator of abelian extensions of Q
All of your questions have comprehensive and beautiful answers: providing them is exactly what the subject of Iwasawa theory was developed to do. You should perhaps read either Washington's Cyclotomic …
- 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
9
votes
$p$-adic analogue of modular forms, upper half-plane, and $L$-functions
The subjects of "p-adic L-functions" and "p-adic modular forms" are so closely intertwined that it's virtually impossible to talk about either one without immediately running into the other. Applicati …
- 37k