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The Riemann Hypothesis for Function Fields by Machiel van Frankenhuijsen

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6

Shift operators

In this chapter, we compute the trace of the shift on the local additive group image This trace gives the Weil term in the explicit formula for the counting function of points on image. Then we give a possible definition and computation of a global trace formula. Even though we suspect that this trace will yield Weil’s explicit formula (up to a function that vanishes in the limit), we are not able to complete the computations and consequently, we do not reach our goal of proving Weil positivity.

We encounter two problems: we do not know the ...

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