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Results tagged with zeta-functions
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user 2481
Zeta functions are typically analogues or generalizations of the Riemann zeta function. Examples include Dedekind zeta functions of number fields, and zeta functions of varieties over finite fields. They are typically initially defined as formal generating functions, but often admit analytic continuations.
7
votes
Hasse-Weil Zeta Functions & Fermats Last Theorem
The only method we know of to prove analytic / meromorphic continuation of zeta-functions of alg. varieties over number fields is to go via some kind of modularity, or potential modularity, statement. …
- 37k
8
votes
Accepted
Gamma Factors for Zeta Functions of Abelian Varieties
This amounts to an Euler characteristic argument. A ratio of products of Gamma functions such as
$$ \frac{\prod_{i=1}^r \Gamma(s - a_i)}{\prod_{j=1}^t \Gamma(s - b_j)} $$
is rational if and only if $ …
- 37k
4
votes
Accepted
Can we find a set of elliptic curves over rationals associated with $f$?.
This has already had some votes to close, but I'll see if I can answer it anyway...
The answer is "no". There are lots of motivic L-functions that are not elliptic curve L-functions, just because the …
- 37k
1
vote
Implementing zeta functions of algebraic varieties in SAGE
Sage has a method zeta_series() for arbitrary varieties over finite fields, but it will only give you the first few terms of the power series, not the rational-function closed form; the latter is only …
- 37k
6
votes
Accepted
Values of Artin L-functions at negative integers
The question of order of vanishing is quite elementary: the L-function of a Hecke character (i.e. 1-dimensional Artin representation) over any number field has an Euler product, which is convergent an …
- 37k
7
votes
is there a p-adic Borel theorem?
Yes, there are $p$-adic analogues of (2). The case where $F$ is abelian over $\mathbf{Q}$ is known: see the paper
Manfred Kolster and Thong Nguyen Quang Do, Syntomic regulators and special values of …
- 37k
14
votes
Accepted
What is the value of $p$-adic $\zeta$-function at positive integer point?
If you use this definition, then $\zeta_p(k)$ is zero at negative even integers $k$, so by a $p$-adic continuity argument, it must also be zero at positive even integers.
What about the odd integers? …
- 37k
4
votes
Accepted
Why are the formulations of Deligne-Ribet/Coates congruences for L-functions equivalent?
I think you are misinterpreting Ribet's claim here.
In Ribet's article he defines $\Delta_c(1 -k, \epsilon)$ for an arbitrary function $\epsilon: G_f \to V$ where $V$ is a $\mathbf{Q}_p$-vector space …
- 37k