### $p$-adic $L$-functions for modular forms

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Over a non-archimedean local field the absolute value, raised to any positive power $\alpha \>0$, is a negative definite function and generates (the analogue of) the symmetric stable process. For $\alpha \in (0,1)$, this process is transient with potential operator given by M. Riesz’ kernel. We develop this potential theory purely analytically and in an explicit manner, obtaining special features afforded by the non-archimedean setting ; e.g. Harnack’s inequality becomes an equality.

We develop the basic theory of the Weyl symbolic calculus of pseudodifferential operators over the $p$-adic numbers. We apply this theory to the study of elliptic operators over the $p$-adic numbers and determine their asymptotic spectral behavior.

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