QUADRATIC ENRICHMENT OF THE LOGARITHMIC DERIVATIVE OF THE ZETA FUNCTION

Margaret Bilu; Wei Ho; Padmavathi Srinivasan; Isabel Vogt; Kirsten Wickelgren

doi:10.1090/btran/201

QUADRATIC ENRICHMENT OF THE LOGARITHMIC DERIVATIVE OF THE ZETA FUNCTION

Margaret Bilu, Wei Ho, Padmavathi Srinivasan, Isabel Vogt, Kirsten Wickelgren

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck–Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure.

Original language	English (US)
Pages (from-to)	1183-1225
Number of pages	43
Journal	Transactions of the American Mathematical Society Series B
Volume	11
DOIs	https://doi.org/10.1090/btran/201
State	Published - 2024

All Science Journal Classification (ASJC) codes

Mathematics (miscellaneous)

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10.1090/btran/201

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abstract = "We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck–Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure.",

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AU - Vogt, Isabel

AU - Wickelgren, Kirsten

PY - 2024

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AB - We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck–Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure.

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JO - Transactions of the American Mathematical Society Series B

JF - Transactions of the American Mathematical Society Series B

ER -

QUADRATIC ENRICHMENT OF THE LOGARITHMIC DERIVATIVE OF THE ZETA FUNCTION

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