We walked the trails in the morning. It was a little bit misty. The golden sunshine gently sifted through leaves that were shedding their verdant colour. The topic drifted into our past work. It was mostly hard work, laden with agony, yet sprinkled with moments of joy. We have navigated around many dead ends and the voyage led us to where we were. A note of caution is that dead ends may not remain dead ends forever. One should take note of their whereabouts and visit them in the future when the tide is favourable or when one acquires a better ship.
We were proud to be able to work out something. Still it saddened us greatly when we were told that it was of no use.
About today's lectures...
There was a long expository on construction of p-adic L-functions. We wondered when the day would come, when such concepts became part of the language. We are still learning the topic. It is not yet clear to us why people came up with such definitions/what they were trying to achieve. For the good old L-functions, it started with the Riemann zeta function. Then there are some natural generalisations like putting a Dirichlet character or changing the field. Then the Dirichlet characters can be replaced by automorphic representations. This gives the automorphic L-functions. The euler product point of view leads to the consideration of Frobenius elements at each finite places. This will give one the Artin L-functions. There are many many points of view for L-functions. If one really needs to define the L-function, it takes some time to give a self-contained exposition, but we have grown used to the notion.
We dosed off during the second lecture... Please don't tell the speaker. Anything that produces finiteness is probably good. Again this time, Frobenius plays a role. (lol. Frobenius is a person, but we always write Frobenius for a Frobenius element in a Galois group.) It seems that the world of positive characteristic is more amenable to handling.
Then we had a very big dinner. We did not have wine, as we were still rehabilitating. Then people decided to have ice cream after dinner. We did not go, as we were still rehabilitating. Seriously, eating ice cream after a big dinner!?