Arizona State University
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Title: The logarithm matrix Log(a_p) in terms of p-adic digits
Abstract: For a modular form* f of weight two, one can attach a p-adic L-function, which is `good' if p is an ordinary prime, i.e. the p-th Fourier a_p of f is a p-adic unit. `Good' means `Iwasawa function,' or in even simpler terms `coefficients are bounded.' When p is non-ordinary (i.e. a_p is not a p-adic unit), the p-adic L-functions are `bad' -- they have unbounded growth behavior on their coefficients (and note the plural -- there are now two of them).
However, one can factor out the badness. This was done in R. Pollack's PhD thesis when a_p=0, which gave rise to two functions log+ and log-, the `signed logarithms'. The speaker handled the general non-ordinary case via a 2x2- matrix Log(a_p). This matrix is, when a_p=0, essentially a diagonal matrix in which log+ and log- appear.
But where does Log(a_p) come from? We give a simple description in terms of p-adic digits.
*normalized eigenform