Faisceaux pervers (Asterisque 100 1982) by Beilinson A., Bernstein J., Deligne P.

By Beilinson A., Bernstein J., Deligne P.

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Example text

Now drop the second term, and the proof is complete. In a similar way, it is useful to embed an L-function into a family. First let us cut the sum (6) into h pieces according to the ideal class of the ideal a. For simplicity we will only work with the principal class. We consider now the second moment ω∈Ω |L(π ⊗ ω, 1/2 + i t )|2 , where Ω is a family of characters containing χ, for example the family of all characters of (O /q)× . These characters are in general not Hecke characters, because they may not be trivial on units, and so strictly speaking the expression L(π ⊗ ω, 1/2 + i t ) does not make sense as a value of an automorphic L-function.

1) Of course, since both statements are true, they are in particular equivalent. But even without knowing the truth of either of these statements one can deduce one from the other. (2) Here it is unknown if either of these statement holds for some δ > 0. Valentin Blomer & Gergely Harcos: L-functions, automorphic forms, and arithmetic 13 3) Let χ be a primitive Dirichlet character to some large modulus q. This gives rise to a Dirichlet L-function L(s, χ) := χ(n) , s n=1 n ∞ which again can be continued to an entire function.

The notation (n) refers to the G action. The map can be computed. It sends χu i v n−i for a locally constant function χ to χζi . 51) 1 (n+1) f (z)(u − zv)n d z, is the exact sequence in which the first arrow takes f to n! 52) 0 ← C an [(−n − 2)∞](−n − 2) ← C an [n∞]/P n (n) ← C pol ,n [n∞]/P n (n) ← 0 n n+1 d where the first (backward) arrow is (−1) . Note that this operator kills the n! dζ principal (polynomial) part at ∞ and increases the order of the zero at infinity to n + 2. Summary. 53) 0 → 0 → 1 d n!

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