|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'504'928
Articles rated: 2609
26 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Generalized Heegner cycles at Eisenstein primes and the Katz $p$-adic $L$-function | | Daniel Kriz
; | | Date: |
16 Dec 2015 | | Abstract: | In this paper, we consider normalized newforms $fin
S_k(Gamma_0(N),varepsilon_f)$ whose non-constant term Fourier coefficients
are congruent to those of an Eisenstein series modulo some prime ideal above a
rational prime $p$. In this situation, we establish a congruence between the
anticyclotomic $p$-adic $L$-function of Bertolini-Darmon-Prasanna and the Katz
two-variable $p$-adic $L$-function. From this, we derive congruences between
images under the $p$-adic Abel-Jacobi map of certain generalized Heegner cycles
attached to $f$ and special values of the Katz $p$-adic $L$-function.
In particular, our results apply to newforms associated with elliptic curves
$E/mathbb{Q}$ whose mod $p$ Galois representations $E[p]$ are reducible at a
good prime $p$. As a consequence, we show the following: if $K$ is an imaginary
quadratic field satisfying the Heegner hypothesis with respect to $E$ and in
which $p$ splits, and if the bad primes of $E$ satisfy certain congruence
conditions mod $p$ and $p$ does not divide certain Bernoulli numbers, then the
Heegner point $P_{E}(K)$ is non-torsion, in particular implying that
$ ext{rank}_{mathbb{Z}}E(K) = 1$. From this, we show that when $E$ is
semistable with reducible mod $3$ Galois representation, then a positive
proportion of real quadratic twists of $E$ have rank 1 and a positive
proportion of imaginary quadratic twists of $E$ have rank 0. | | Source: | arXiv, 1512.5032 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER