Exponential Sums and L-functions in Algebraic Geometry

Exponential sums appear in the study of algebraic varieties over finite fields through sums of additive or multiplicative characters evaluated at polynomial functions. L-functions encapsulate generating series whose coefficients derive from these sums, encoding arithmetic and geometric information about the underlying variety. Within algebraic geometry, cohomological methods interpret exponential sums as traces of Frobenius on suitable cohomology groups, linking de Rham, rigid and p-adic theories. This approach yields deep results on distribution of points on varieties, monodromy of families, and p-adic properties such as the slopes of Newton polygons. Recent advances have refined comparison theorems between cohomology theories, leading to precise control over the analytic behaviour of L-functions and enabling explicit computations of their zeros and poles. The interplay between geometric structures—such as Dwork crystals and Monsky–Washnitzer complexes—and arithmetic invariants illuminates both classical objects like character sums and modern developments in p-adic Hodge theory. This synergy has broad applications to coding theory, cryptography, the study of motives and the Langlands programme.

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Exponential Sums and L-functions in Algebraic Geometry publication trend

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Technical terms

Exponential sum: A sum over finite field points of additive or multiplicative characters applied to polynomial values.

L-function: A generating series whose coefficients encode exponential sums, capturing arithmetic properties of a variety.

De Rham cohomology: A cohomology theory computing differential forms on algebraic varieties over fields of characteristic zero.

Dwork cohomology: A p-adic cohomology theory using Dwork’s differential operator to study exponential sums and L-functions.

Rigid cohomology: A p-adic cohomology theory for algebraic varieties over finite fields, extending Monsky–Washnitzer methods.

Newton polygon: A convex polygon associated with the p-adic valuations of coefficients of a power series, indicating slopes of Frobenius eigenvalues.

References

1. Exponentially twisted de Rham cohomology and rigid cohomology. Mathematische Annalen (2023).
2. ON A COMPARISON BETWEEN DWORK AND RIGID COHOMOLOGIES OF PROJECTIVE COMPLEMENTS. Nagoya Mathematical Journal (2023).
3. On the L L -function of multiplicative character sums. Transactions of the American Mathematical Society (2012).