Tools and techniques for rational points on curves

We give algorithms to compute Coleman integrals on superelliptic curves over unramified extensions of the p-adics, and apply these via Chabauty methods to determine the set of rational points on such curves. We also determine the solution to an explicit instance of the Shafarevich conjecture by finding all elliptic curves with good reduction outside of the first 6 primes, subject to a heuristic. We use a combination of non-abelian Chabauty and the Mordell--Weil sieve to determine the rational points on several quotient modular curves, and therefore classify pairs of elliptic curves over the rationals with 67-, 73-, and 107-isogenies. We give methods to explicitly compute Coleman integrals on modular curves using a canonical lift of Frobenius and canonical local coordinates in each residue disk, and discuss the problem of computing the Weil pairing in finite rings

Formalized Class Group Computations and Integral Points on Mordell Elliptic Curves

Diophantine equations are a popular and active area of research in number theory. In this paper we consider Mordell equations, which are of the form $y^2=x^3+d$, where $d$ is a (given) nonzero integer number and all solutions in integers $x$ and $y$ have to be determined. One non-elementary approach for this problem is the resolution via descent and class groups. Along these lines we formalized in Lean 3 the resolution of Mordell equations for several instances of $d<0$. In order to achieve this, we needed to formalize several other theories from number theory that are interesting on their own as well, such as ideal norms, quadratic fields and rings, and explicit computations of the class number. Moreover we introduced new computational tactics in order to carry out efficiently computations in quadratic rings and beyond.Comment: 14 pages. Submitted to CPP '23. Source code available at https://github.com/lean-forward/class-group-and-mordell-equatio

Fault and magmatic interaction within Iceland's western rift over the last 9kyr

We present high-resolution 'Chirp' sub-bottom profiler data from Thingvallavatn, a lake in Iceland's western rift zone. These data are combined with stratigraphic constraints from sediment cores to show that movement on normal faults since 9 ka are temporally correlated with magmatic events, indicating that movements were controlled by episodic dyke intrusion. Sediment depo-centres and the focus of subsidence migrated westwards over 3-4 kyr towards the locus of subsequent brittle failure. We interpret this subsidence as related to dyke intrusion a few km along strike, originating from the Hengill volcanic system, which occurred prior to major dyking, faulting and subsidence within the lake at 1.9 ka

New Combinations: Changing Technologies and Infrastructures and the Business Organizations That Will Deal with Them.

This tale is somewhat unique. It is probably one of the few panels, or the only panel in which there is a certain level of agreement. In fact, every one in this room probably has the basic agreement with this panel, that changes in regulation and changes in policy that we are struggling with are driven by changes in technology. This panel has been put together to look at changes in technology. It has three overall goals. The first is, as you heard one of the panels this morning talk about, to talk about technology itself to try to gain some type of understanding. As regulators or practitioners of law in the area, it is hard to be effective without some type of understanding of just what the technology is that we are supposed to be regulating or trying to give people advice concerning. The second area that this panel will hopefully inform us on is how technology continues to change. Many times we look back and see how technological changes have occurred, and how they have forced us to change. However, there is little consideration for the fact that these technologies continue to change and there will be other challenges or maybe even solutions to current problems that will result from that change in technology. And the final area we hope to provide information on is, as technology changes and as it causes regulators to address new issues, how the utilities, are themselves very different, and have been changed and shaped by this changing technology. This group, I think, can demonstrate very well how the conversion of the different technologies have shaped most of the companies and people who are on the panel

Treatment for hepatorenal syndrome in people with decompensated liver cirrhosis: a network meta-analysis

This is a protocol for a Cochrane Review (Intervention). The objectives are as follows: To compare the benefits and harms of different treatments of hepatorenal syndrome in people with decompensated liver cirrhosis