We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To save content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about saving content to .
To save content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
We prove the existence of a vector-valued cusp form for the full modular group for which the nth derivative of its L-function does not vanish under certain conditions. As an application, we generalize our result to Kohnen’s plus space and prove an analogous result for Jacobi forms.
The final chapter treats minimal threefolds. We explain the abundance for threefolds due to Miyaoka and Kawamata depending on the numerical Kodaira dimension. The initial step is to prove the non-vanishing which means the existence of a global section of some pluricanonical divisor. If the irregularity is not zero, then the Albanese map provides enough geometric information. In the case of irregularity zero, Miyaoka applied the generic semi-positivity via positive characteristic. We derive abundance from non-vanishing after replacing the threefold by a special divisorially log terminal pair. Birational minimal models are connected by flops and have the same Betti and Hodge numbers. In dimension three, they have the same analytic singularities. One can expect the finiteness of minimal models ignoring the marking map. This is a part of Kawamata and Morrison's cone conjecture for Calabi-Yau fibrations. We explain Kawamata's work on the conjecture for threefold fibrations with non-trivial base. In dimension three, there exists a uniform number for l such that the l-th pluricanonical map is birational to the Iitaka fibration. We find this number explicitly in the case of general type.
Let $E/\mathbb {Q}$ be a number field of degree $n$. We show that if $\operatorname {Reg}(E)\ll _n |\!\operatorname{Disc}(E)|^{1/4}$ then the fraction of class group characters for which the Hecke $L$-function does not vanish at the central point is $\gg _{n,\varepsilon } |\!\operatorname{Disc}(E)|^{-1/4-\varepsilon }$. The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in $\mathbf {PGL}_n(\mathbb {Z})\backslash \mathbf {PGL}_n(\mathbb {R})$ associated to the maximal order of $E$, and the escape of mass of the torus orbit associated to the trivial ideal class.
Recommend this
Email your librarian or administrator to recommend adding this to your organisation's collection.