What does this research mean for the field?

A limit-free form of Dwork's trace formula demonstrates that Gauss sums are normalized traces of words in a monoid generated by integer matrices derived from Dwork's splitting function. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish a direct, limit-free algebraic formulation of Dwork's trace formula that represents Gauss sums as eigenvalues of matrix products factored digit by digit.

September 2, 2026Open Access

Gauss Sums Are Eigenvalues: The Dwork Monoid, the Trace Formula Without Limits, and Stickelberger Digit by Digit

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To establish a direct, limit-free algebraic formulation of Dwork's trace formula that represents Gauss sums as eigenvalues of matrix products factored digit by digit.
Constructed a matrix monoid generated by integer matrices B(δ) derived from the series coefficients of Dwork's splitting function.
Factorized Dwork kernels over finite fields of characteristic p entry-by-entry based on base-p digit expansions of character exponents.
Instantiated and verified the exact algebraic statements and truncation windows for prime values p in {3, 5, 7, 11}.
Proved a limit-free trace formula where the level-k relation (q^k − 1) Tr(A_a^k) = −(−g_a)^k operates as an explicit bijection of finite index sets.
Derived the Fredholm determinant as an explicit q-difference product with an exact Newton ladder valuation formula v_π(e_m) = m sp(a) + f(p − 1) binom(m, 2).
Demonstrated that Stickelberger exponents and spectrum generation by Morita Gamma values can be read digit by digit via word rotation under Frobenius action.

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Implication

Mathematical analysis demonstrates a limit-free Dwork trace formula expressing Gauss sums as matrix traces, revealing digit-by-digit factorization across finite field kernels.

Cite This Study

xuewen guan (2026) studied this question.

synapsesocial.com/papers/6a97e275c562ede874ec69fb https://doi.org/10.5281/zenodo.22199013

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Implication

Mathematical analysis demonstrates a limit-free Dwork trace formula expressing Gauss sums as matrix traces, revealing digit-by-digit factorization across finite field kernels.

Gauss Sums Are Eigenvalues: The Dwork Monoid, the Trace Formula Without Limits, and Stickelberger Digit by Digit

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Discussion

Implication

Cite This Study

Gauss Sums Are Eigenvalues: The Dwork Monoid, the Trace Formula Without Limits, and Stickelberger Digit by Digit

Key Points

Discussion

Implication

Cite This Study

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