# ITP

## Dr Shin Hattori in Durham University, UK (Part III)

[Introduction]

I have been in Durham for three months. It is getting colder and I am even using an oil heater. It rained almost everyday in the first half of this month. Now we are in the middle of a series of sunny days with slightly cold wind. It is just like how the autumn begins in Japan. As many people told me, there seems to be no pronounced summer in this country. But they said that even so it was unusually cold and rainy in this summer. In fact, one day in this month, several rivers in this area flooded due to heavy rain and the floods caused the cancellation of many railway services. The River Wear in Durham also flooded, and a lot of local people were watching and taking photos of the flooded river curiously from the bridges above the river. An old lady told me that she has lived in the town for many years but has never seen such a scene.

 The flooding Wear river A double rainbow on a summer day

[My Research]

Professor Abrashkin and I kept having weekly seminars also in this month. To attack my problem of obtaining better ramification estimate for a better situation, we made many trials and errors of how to descent down some algebra over the integer ring of a local field to a smaller ring. Thanks to these discussions with him, I found interesting examples showing how this goodness of the situation makes such a descent unexpectedly hopeful. I hope that these examples give me some hint to a solution of the whole problem. We also discussed about an analog of the criterion of Neron-Ogg-Shafarevich for torsion Galois representations. He has many experiences in this topic and kindly let me share a lot of interesting ideas with him. I am going to carry on studying these things ahead.

We were planning for me to give several talks in this country, but as I found I have to go back to Japan at the end of this month, we decided to make only one talk. I gave a talk about my work in Arithmetic Study Group Seminar of Durham univerisity. I was very happy to have a lot of audiences though we were in the summer vacation period, and have a vigorous discussion after the talk, which unfortunately made me forget taking a photo of the talk.

[The town of Newcastle]

On a holiday, I visited Newcastle, the biggest town in this area, which is 15-minutes railway ride from Durham. Newcastle is also an old town dating back to the 2nd century when Romans settled in this place along the River Tyne. The town was named after a castle built in the 11th century to protect it from invasions of Scots. There were wars against Scots in the medieval era and this area was in the front lines of wars. By this reason the town was walled off by stone walls. Even now parts of the castle and the walls remain and these are main sight spots in Newcastle. There is also a very tall monument celebrating a former prime minister the 2nd Earl Grey who was born in this area and is famous for his parliamentary reform and pro-laboror policy in the 18th century. He is also famous as the origin of the name of a famous British citrus-flavored tea, Earl Grey tea. In the Eldon Square at the center of the town, I found a tropaeum of the two World Wars. I am interested in it because we have apparently few such modern statues celebrating triumphs in a war in Japan.

 The Central Station of Newcastle Strange statues at the center of Newcastle The old castle keep. Once this had been used as a prison Old walls and a gate of a China town A tall monument of Earl Grey. On the top of the tower is his statue A tropaeum in the Eldon Square

I also visited the St. Nicholas Cathedral in this town. The pamphlet of the cathedral says that there has been a church for over 900 years, and this cathedral was built in the 14th century. This church is rather smaller than the cathedral of Durham, but even so I was very much impressed when I heard a monk playing sublime musics by the big pipe organ in the cathedral. I was also amused by beautiful stained glasses inside the cathedral.

 The St. Nicholas Cathedral The inside of the cathedral A beautiful stained glass in the cathedral Another stained glass

Newcastle is also famous for its traditional beer, the Newcastle Brown Ale. It is very popular in Durham, and when I went to a pub in Durham on a holiday, I found many people were drinking this brown-colored beer. This beer is a kind of ales, which are brewed differently from lagers that we mainly drink in Japan. Generally ales have less aroma of malts than lagers, but instead come in a variety of flavors. Another difference of this British brown ale from lagers is that they drink it usually at room temperature. A porter of the college told me a connoisseur's way of drinking this beer. He said that you should drink it with a half-pint glass to reduce gases inside it.

 The Newcastle Brown Ale A distant view of the town from the cathedral at dusk in late September, just before my departurey

[Conclusion]

As I mentioned above, I have to go back to Japan urgently because I got a job in a university. It is a pity that I can't stay at Durham for the full length of the arranged period. Though it is shorter than I planned, this stay was a very nice research experience for me. I would like to thank Professor Abrashkin for his hospitality and many invaluable discussions, and the ITP program for giving me such a nice opportunity.