Daniel Loughran
|
Registered User
|
I am a postdoc at the University of Bristol. I do arithmetic geometry and analytic number theory.
|
|
May 8 |
awarded | ● Citizen Patrol |
|
Apr 21 |
comment |
non-singular cubics are not rational @ Jérémy: I think Ashwath is pointing out a minor typo, and suggesting that you replace "surface" with "threefold" in 4). |
|
Apr 10 |
revised |
What does the tensor product of two central simple algebras correspond to geometrically? edited body |
|
Apr 9 |
comment |
What does the tensor product of two central simple algebras correspond to geometrically? @Michael and Will: Thanks for your comments. This shows that there is always a morphism $B(A_1) \times B(A_2) \to B(A_1 \otimes_k A_2)$. It seems quite possible that $B(A_1 \otimes_k A_2)$ is in some respects determined by this morphism, i.e. this morphism should satisfy some kind of universal property. Perhaps $B(A_1 \otimes_k A_2)$ is the Brauer-Severi variety of smallest dimension that $B(A_1) \times B(A_2)$ embeds into? |
|
Apr 8 |
asked | What does the tensor product of two central simple algebras correspond to geometrically? |
|
Apr 2 |
comment |
Meromorphic continuation of a Dirichlet series associated to an irrational number @Pieter: Tschinkel has done lots of work on height zeta functions. In particular I think that his recent work with Chambert-Loir on integral points of bounded height on toric varieties applies to your setting (see his webpage). The analytic behaviour of these zeta functions is closely related to corresponding number of rational/integral points of bounded height. In their paper they use smooth norms like you, but I seem to remember some trick which allows you to pass to non-smooth norms as I mention above, however the precise details of this trick currently elude me... |
|
Apr 2 |
awarded | ● Yearling |
|
Apr 1 |
awarded | ● Good Question |
|
Apr 1 |
comment |
Meromorphic continuation of a Dirichlet series associated to an irrational number Also, do you know about Height zeta functions and Epstein zeta functions? The zeta functions $\zeta_d$ which you are studying are very closely related to these. For example your zeta function with $d(x,y)=\mathrm{max}\{|y|,|x|/\theta\}$ is a height zeta function for integral points in the affine plane. Here meromorphic continuation is known in some region past $s=2$ (though I am not sure if meromorphic continuation is know to the whole complex plane). |
|
Apr 1 |
comment |
Meromorphic continuation of a Dirichlet series associated to an irrational number Here is one idea for studying $\zeta_d$ with $d$ not smooth: Find a sequence of smooth norms $(d_t)$ with $t \in \mathbb{R}$ which converge to the norm $d$ you are interested in. If you have enough control over the convergence you might be able to deduce the meromorphic continuation of $\zeta_d$ from the meromorphic continuation of each $\zeta_{d_t}$. |
|
Mar 22 |
comment |
Coutour Integral of Gamma Functions What does $ii$ mean? And is $j^2=-1$? |
|
Mar 19 |
answered | Producing $(-2)$ curves on a smooth surface |
|
Feb 26 |
awarded | ● Nice Answer |
|
Feb 26 |
revised |
group actions on blow-ups added 3 characters in body |
|
Feb 25 |
revised |
group actions on blow-ups added 83 characters in body |
|
Feb 25 |
answered | group actions on blow-ups |
|
Feb 5 |
comment |
Birational Automorphisms and infinite divisibility Right I see. I got confused with the notation and thought that $\mathbb{Z}[1/2]$ meant $(1/2)\mathbb{Z}$. Thanks for clearing it up. |
|
Feb 5 |
comment |
Birational Automorphisms and infinite divisibility Why is it clear that $\mathbb{Z}[1/2]$ belongs to $ker(\phi)$? I'm thinking about something like an elliptic surface which could have a copy of $\mathbb{Z} \cong \mathbb{Z}[1/2]$ in its automorphism group, given by translation by a non-torsion section. It is not clear to me that this copy of $\mathbb{Z}$ acts trivially on the cohomology of the surface. |
|
Jan 31 |
comment |
galois cohomology over finite field I think you might be able to deal with such a cohomology group using global tate duality. Try for example the book "Cohomology of number fields" by Neukirch, Schmidt and Winberg. Note that despite the title, they do indeed study all global fields |
|
Jan 17 |
answered | What are the general techniques for proving a variety is not toric? |
|
Jan 16 |
comment |
Functorial properties of blow-up Perhaps you know this already, but the universal property of blow-ups should give you what you need in the case that $\phi$ is algebraic. See e.g. Hartshorne Corollary II.7.15. |
|
Jan 14 |
comment |
Brauer group elements associated to conic bundles @MBeasy: Thanks for the references. I have had a look and I feel like I'm getting closer to being able to answer my questions. One thing which is still confusing me is how does one construct an order in the associated central simple algebra $A$ from the conic bundle? The key thing about this order is that it should be "defined everywhere" (i.e. not just on some open subset like the associated Brauer group element $A$ above). If anyone could help me with this I would be much obliged. |
|
Jan 11 |
revised |
Brauer group elements associated to conic bundles deleted 69 characters in body |
|
Jan 10 |
comment |
Brauer group elements associated to conic bundles @Jason: Thanks for the hint. I have come across a paper by Artin-Mumford which seems to suggest that relatively minimal conic bundles should correspond to maximal orders in the associated quaternion algebra. Do you have any ideas for my other questions? |
|
Jan 9 |
asked | Brauer group elements associated to conic bundles |
|
Dec 22 |
comment |
How we obtain information about a variety from an algebraic group acting on it It's not clear to me exactly what you are after. Do you have a specific variety and group action in mind that you want to know more about? |
|
Dec 16 |
comment |
How does the line bundles look like on a proper model (or Néron model) of an abelian variety? If $\mathcal{A}$ denotes a model for $A$ over some appropriate base, there is a homomorphism $\mathrm{Pic}(\mathcal{A}) \to \mathrm{Pic}(A)$ given by restricting line bundles to the generic fibre. Is this what you are interested in? |
|
Dec 15 |
answered | Cohomology of vector bundles via Intersection Theory |
|
Dec 13 |
awarded | ● Nice Question |
|
Dec 5 |
comment |
Average orders of multiplicative functions @Alexey: Thanks for the answer. I will try to find this book in our library and get back to you. Is it possible for you to give a simple example in the mean time? |
|
Dec 5 |
comment |
Average orders of multiplicative functions @Greg: Do you know an example of such a set? |
|
Dec 4 |
asked | Average orders of multiplicative functions |
