The undetermined algebraic factors in Weil's period formula for Fermat-type curves: the exact multipliers, their number fields, denominators, and p-adic properties

Weil's 1976 paper Sur les periodes des integrales abeliennes expresses the periods of the Fermat-type curves $C_a: y^l=x^a(1-x)$, and the special values of the Jacobi-sum Hecke L-series attached to their Jacobians, in terms of products of Euler beta and gamma values - in both cases only up to undetermined algebraic factors - and asks two things: to identify the number fields, denominators, and p-adic continuity properties of these factors, and to extend the method beyond the special index sets $T_a$ that the curves $C_a$ supply (the quadratic residues mod 11 already form no $T_a$, and primes such as 23 point to intermediate Jacobians of higher-dimensional varieties). Later work resolved the structural extension up to a rational factor within the Jacobi-sum (abelian-field) range and for the prime discriminants $p\equiv 3 \pmod 4$: Gross (1978) treats all prime discriminants $p\equiv 3 \pmod 4$ geometrically, and Anderson (1986) proves Lichtenbaum's gamma hypothesis for every abelian field. What remains open is the exact arithmetic of the undetermined factor - its number field, integrality, and p-adic interpolation - which later authors tie to integral Beilinson and Tamagawa-number refinements.

Background

A period of a smooth projective curve or abelian variety over $\overline{\mathbb{Q}}$ is the integral of an algebraic differential form over a topological cycle; periods are the coordinates through which de Rham cohomology and singular homology are compared. The Jacobian of a curve is the abelian variety carrying its holomorphic differentials. An abelian variety has complex multiplication (CM) when its endomorphism ring is an order in a number field of degree twice its dimension; the L-function of such a variety factors into Hecke L-series, where a Hecke character (Grossencharakter) is a continuous complex-valued character on the ideals of a number field. The Hecke characters attached to Fermat-type covers of the line are generated by Jacobi sums, finite-field analogues of beta integrals studied by Weil in 1952.

The prototype result is the Chowla-Selberg formula (1967): the periods of an elliptic curve with CM by an imaginary quadratic order equal, up to algebraic factors, explicit products of values of Euler's gamma function $\Gamma$ at rational arguments; their proof was analytic, via the Kronecker limit formula, which computes the constant term at $s=1$ in the expansion of the Epstein zeta function (the zeta function attached to a positive definite quadratic form).

Weil's 1976 paper Sur les periodes des integrales abeliennes provides a geometric route. For the Fermat-type curves $C_a: y^{l}=x^{a}(1-x)$ with $l$ prime and $a$ coprime to $l$, write $\zeta$ for a primitive $l$-th root of unity and $\langle t\rangle$ for the fractional part of $t$. A natural basis of differentials on $C_a$ has periods equal, up to roots of unity and algebraic numbers, to beta-function values $B(\langle at\rangle,\langle t\rangle)$ with $B(x,y)=\Gamma(x)\Gamma(y)/\Gamma(x+y)$, for $t=n/l$, $1\le n\le l-1$; the index set governing which pairs occur is $T_a=\{n\in\{1,\dots,l-1\} : \langle an/l\rangle+\langle n/l\rangle<1\}$. Via a theorem of Shimura relating periods of CM abelian varieties to special values of automorphic forms, the computation transfers to L-values: the special values at $s=1$ of the Jacobi-sum Hecke L-series attached to the Jacobian of $C_a$ equal, up to algebraic factors, the same gamma products, such as $\Gamma(\langle at\rangle)\Gamma(\langle t\rangle)\Gamma(\langle -(a+1)t\rangle)$. Weil's paper records two issues it does not resolve. First, the algebraic factors relating the L-values and the gamma products are unidentified: their number fields, denominators, and p-adic continuity properties are undetermined. Second, the method is tied to the index sets $T_a$ that the curves $C_a$ supply: the quadratic residues modulo 11, for example, form no $T_a$ (while for $l=7$, $T_2$ is exactly the set of quadratic residues), so the method cannot recover Chowla-Selberg for the elliptic curves with CM by $\mathbb{Q}(\sqrt{-11})$; and for primes such as 23 one is led to intermediate Jacobians - abelian varieties obtained from the middle cohomology of higher-dimensional varieties by integrating middle-dimensional forms over middle-dimensional cycles - where the needed period information was unavailable in 1976.

A modern up-to-algebraic-number reformulation of the first issue is the Gross-Deligne conjecture (stated by Gross in 1978): for an abelian variety with CM, the determinant of its period matrix in a suitable basis equals, up to an algebraic number, an explicit product of gamma values at rational arguments; equivalently, for a CM motive - a piece of the cohomology of a variety cut out by its CM automorphisms, of cohomological degree called its weight - the periods are pinned by gamma values up to an algebraic number. In weight 1 the statement is essentially proved; in higher weight it is open in general and would follow from the Hodge conjecture (the conjecture that every rational Hodge class on a smooth projective complex variety is the class of an algebraic cycle). The exact rational factor in the known up-to-$\mathbb{Q}^{\times}$ statements is governed by Beilinson's conjecture relating L-values to regulators (integrals of motivic cohomology classes) and by the Tamagawa number conjecture of Bloch and Kato, which predicts the exact rational numbers involved; the p-adic side involves the Gross-Koblitz $p$-adic gamma function, the p-adic analogue of $\Gamma$ whose values at fractions are Gauss sums (exponential sums over finite fields).

Problem Statement

Weil's question, in its original generality: for the Fermat-type curves $C_a: y^l=x^a(1-x)$ and the Jacobi-sum Hecke characters their Jacobians carry, the periods and the corresponding special Hecke L-values are beta/gamma products up to unidentified algebraic factors. Determine those factors exactly: (i) the number field to which each factor belongs; (ii) its denominator, i.e. for which integral structures (choice of integral differential basis and normalization of the Hecke L-series) the factor is integral; (iii) its p-adic continuity and interpolation properties as the data vary p-adically; and (iv) extend the formulae beyond the index sets $T_a$ arising from the curves $C_a$ - to arbitrary Jacobi-sum data (the quadratic residues mod 11, for instance, form no $T_a$) and arbitrary imaginary quadratic fields (all abelian fields, in the sense of the Jacobi-sum Hecke characters Weil considers), and to the higher-dimensional cases, such as primes like 23, that lead to intermediate Jacobians. Status of the window: within the Jacobi-sum/abelian-field scope of Weil's question, part (iv) is settled up to a rational factor - Anderson (1986) proves the gamma hypothesis for every abelian field - and Gross (1978) handles the prime-discriminant $p\equiv 3 \pmod 4$ higher-dimensional cases geometrically; for CM abelian varieties with general (non-abelian) CM fields, only the up-to-algebraic-factor structure is known (Deligne's results as formulated and proved in Anderson (1982); Gross's formulae for $B(p)$ recorded in Gross (2020)). The surviving crux is therefore the exact arithmetic of the undetermined factor in (i)-(iii). As related open context: Otsubo (2015) relates the exact rational factors for Fermat motives to the predictions of the Tamagawa-number (Bloch-Kato) conjecture and leaves general formulae and p-adic regulators open, and Fresán (2017) records the Gross-Deligne conjecture - itself an up-to-algebraic-number statement - as wide open in weight greater than one.

The task is to determine the factors (i)-(iii) for the Fermat-type curves $C_a$ and their Jacobi-sum Hecke L-series and for the extension data of (iv) - arbitrary Jacobi-sum data such as the quadratic residues mod 11, arbitrary imaginary quadratic and abelian fields, and the higher-dimensional cases such as prime 23 - over the full range of Weil's question; the target is the exact arithmetic in general, not only for an explicitly bounded family.

References

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