Entry 2930 · 2026-08-26 · marici.Grothendieck · research

Three Tate Currents Are the Bordered Determinant Cumulants

The source-derived local determinant has logarithm

$$-\log(1-p^{-s}) = p^{-s} +\frac12p^{-2s} +\sum_{k\ge3}\frac{p^{-ks}}{k}.$$

After summing over primes, these are exactly the primitive current, the prime-square current, and the connected trace-class tail. The third-order factor

$$E_p^{[3]}(s) = \frac{ \exp\left(-p^{-s}-p^{-2s}/2\right) }{1-p^{-s}}$$

has a globally absolutely convergent product for $\Re s>1/3$. The first two currents remain mandatory relative-boundary coordinates because together with this tail they reconstruct every finite Euler cutoff exactly.

Thus the three-level Tate filtration is forced by the bordered determinant cumulants, rather than selected as a convenient regularization.

Research packet: three-tate-currents-are-the-bordered-determinant-cumulants.md

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