Shintani's Invariant in Real Quadratic Fields

• Shintani’s Invariant is a family of special real numbers defined via partial zeta functions and the double sine function for ray classes in real quadratic fields.
• It underpins a conjectural approach to explicit class field theory by generating abelian extensions, linking analytic, geometric, and q-series reformulations.
• Recent developments using q-Pochhammer symbols and cyclic quantum dilogarithm offer modular and arithmetic refinements, enhancing computational and theoretical insights.

Shintani’s invariant most commonly denotes a family of special real numbers attached to ray classes of real quadratic fields, defined from partial zeta functions and originally expressed through the double sine function. In that sense, these invariants are conjectured to generate abelian extensions of real quadratic fields and therefore belong to the conjectural real-quadratic analogue of explicit class field theory and Hilbert’s 12th problem (Yalkinoglu, 2024). In adjacent literatures, the same phrase is also used more broadly for residues or special values of Shintani-type zeta functions, especially those attached to binary cubic forms, so the term is context-sensitive and requires disambiguation (Hough et al., 2022).

1. Definition in the real quadratic setting

Let $K=\mathbb{Q}(\sqrt{d})$ be a real quadratic field, let $\mathcal{O}_K$ be its ring of integers, and let $\varepsilon$ be its fundamental unit. For a ray class $\mathfrak{C}$ with conductor $\mathfrak{f}$, Shintani defined the partial zeta function

$$\zeta(s,\mathfrak{C})=\sum_{\mathfrak{a}\in \mathfrak{C},\ \mathfrak{a}\ \text{integral}} \mathfrak{N}(\mathfrak{a})^{-s},$$

and the corresponding invariant

$$X(\mathfrak{C})=X_1(\mathfrak{C})X_2(\mathfrak{C}), \qquad X_1(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C})), \qquad X_2(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C}')).$$

Here $\mathfrak{C}'$ is the auxiliary ray class appearing in Shintani’s construction (Yalkinoglu, 2024).

For principal ideals, the literature also writes $X(\mathfrak{f})$ for the invariant attached to the identity class, with $\mathfrak{f}=(u+v\sqrt d)$ in the notation of the recent reformulations. In that setting one again has a factorization

$\mathcal{O}_K$0

now written in terms adapted to explicit product formulas (Yalkinoglu, 24 Aug 2025).

The central arithmetic claim attached to these numbers is conjectural but very specific: Shintani’s invariants are conjectured to generate abelian extensions of real quadratic fields. This is the precise sense in which they are said to provide a conjectural solution to Hilbert’s 12th problem for real quadratic fields (Yalkinoglu, 2024).

2. Classical analytic construction and geometric background

Shintani originally expressed the invariant by means of the double sine function. In the formulation recorded in the recent literature,

$\mathcal{O}_K$1

and for the trivial ray class one has a finite product of double sine values of the form

$\mathcal{O}_K$2

with analogous formulas for the second factor $\mathcal{O}_K$3 (Yalkinoglu, 2024).

This analytic construction sits inside a broader geometric method. Shintani’s method decomposes the positive cone, or equivalently a torus modulo a lattice, into simplicial cones and uses that decomposition to compute Hecke $\mathcal{O}_K$4-functions at integer points; in particular, one version proves the analytic class number formula for a totally real field directly at $\mathcal{O}_K$5 (Colmez, 2024). The same geometric infrastructure underlies the calculation of abelian $\mathcal{O}_K$6-functions by cone-wise summation.

A practical refinement is the theory of signed fundamental domains for the action of the totally positive units $\mathcal{O}_K$7 on $\mathcal{O}_K$8. In that framework one has

$\mathcal{O}_K$9

where the $\varepsilon$0 are $\varepsilon$1-rational cones and $\varepsilon$2 are explicit orientation weights (Diaz et al., 2013). These signed domains are stated to be as useful as Shintani’s true ones for calculating abelian $\varepsilon$3-functions, but they have the computational advantage that they can be constructed from any set of fundamental units, whereas in practice there is no algorithm producing Shintani’s $\varepsilon$4-rational cones (Diaz et al., 2013).

A cohomological reformulation of this method is given by the Shintani cocycle on $\varepsilon$5, where the cocycle property is achieved by introducing an auxiliary perturbation vector $\varepsilon$6. That formalism yields a new proof of the theorem of Diaz y Diaz and Friedman on signed fundamental domains and a cohomological reformulation of Shintani’s proof of the Klingen–Siegel rationality theorem on partial zeta functions of totally real fields (Charollois et al., 2014).

3. Reformulation via $\varepsilon$7-Pochhammer symbols

A major recent development is the replacement of the double sine function by $\varepsilon$8-series expressions. Yamamoto had observed that the double sine can be written in terms of $\varepsilon$9-Pochhammer symbols, and this observation is pushed substantially further in the recent note on Shintani’s invariants (Yalkinoglu, 2024).

The relevant $\mathfrak{C}$0-Pochhammer function is

$\mathfrak{C}$1

equivalently the two-variable notation

$\mathfrak{C}$2

In this language, the double sine function is represented as a limit of a ratio of such products, and the invariant is recast as a limit along a discrete modular geodesic (Yalkinoglu, 2024).

One explicit form is

$\mathfrak{C}$3

where the sequence $\mathfrak{C}$4 traces a discrete modular geodesic in the upper half-plane and is constructed from powers of a matrix $\mathfrak{C}$5 determined by the fundamental unit $\mathfrak{C}$6 (Yalkinoglu, 24 Aug 2025).

The same note also gives a product formula using only one base $\mathfrak{C}$7, together with a further trigonometric-hyperbolic product expression involving sine and hyperbolic sine functions (Yalkinoglu, 2024). The significance of these formulas is twofold. First, they replace the double sine function by classical $\mathfrak{C}$8-analogues. Second, they expose explicit modular-geometric data—continued fractions, modular geodesics, and Chebyshev polynomials—that were more implicit in the original analytic presentation (Yalkinoglu, 2024).

4. Cyclic quantum dilogarithm and algebraic approximation

A further reformulation expresses Shintani’s invariant in terms of the cyclic quantum dilogarithm. In the 2025 announcement by Yalkinoglu, the cyclic quantum dilogarithm is defined by

$\mathfrak{C}$9

for $\mathfrak{f}$0, $\mathfrak{f}$1, $\mathfrak{f}$2, and $\mathfrak{f}$3 (Yalkinoglu, 24 Aug 2025).

The new limit formulas are

$\mathfrak{f}$4

where

$\mathfrak{f}$5

and $\mathfrak{f}$6 are Chebyshev polynomials of the first kind (Yalkinoglu, 24 Aug 2025). In the special case $\mathfrak{f}$7, the formula simplifies to

$\mathfrak{f}$8

The conceptual point of this reformulation is that each finite step in the limit lies in a Kummer extension of a cyclotomic field, since the bases are roots of unity and the exponents are rational (Yalkinoglu, 24 Aug 2025). This moves the presentation from transcendental functions such as the double sine and $\mathfrak{f}$9-Pochhammer symbols toward finite products encoding cyclotomic data. The paper explicitly presents this as a new arithmetic perspective on Shintani’s construction and as a possible bridge to explicit class field theory, modular and quantum topology, and Manin’s “real multiplication” program (Yalkinoglu, 24 Aug 2025). Since the paper is an announcement, the detailed proofs are deferred to a forthcoming full account (Yalkinoglu, 24 Aug 2025).

5. Relation to Shintani zeta functions and binary cubic forms

A common source of confusion is the distinction between Shintani’s invariant in the real quadratic setting and the Shintani zeta function attached to binary cubic forms. The latter is the Dirichlet series

$$\zeta(s,\mathfrak{C})=\sum_{\mathfrak{a}\in \mathfrak{C},\ \mathfrak{a}\ \text{integral}} \mathfrak{N}(\mathfrak{a})^{-s},$$0

which counts $$\zeta(s,\mathfrak{C})=\sum_{\mathfrak{a}\in \mathfrak{C},\ \mathfrak{a}\ \text{integral}} \mathfrak{N}(\mathfrak{a})^{-s},$$1-equivalence classes of integral binary cubic forms by discriminant (Thorne, 2011). In the binary cubic literature, “Shintani’s invariant” is used in a looser sense for the residue at the principal pole or a special value of this zeta function, relating to counting cubic rings and number fields (Hough et al., 2022).

The analytic behavior of the binary-cubic zeta function is markedly different from that of standard automorphic $$\zeta(s,\mathfrak{C})=\sum_{\mathfrak{a}\in \mathfrak{C},\ \mathfrak{a}\ \text{integral}} \mathfrak{N}(\mathfrak{a})^{-s},$$2-functions. Frank Thorne proved that the Shintani zeta function for binary cubic forms cannot be written as a finite sum of Euler products, even though it admits representations as infinite sums of Euler products (Thorne, 2011). This negative result answers a question of Wright and shows that the arithmetic of cubic field discriminants is not captured by any finite list of multiplicative building blocks.

Recent work has nevertheless established strong analytic bounds. For the completed linear combinations

$$\zeta(s,\mathfrak{C})=\sum_{\mathfrak{a}\in \mathfrak{C},\ \mathfrak{a}\ \text{integral}} \mathfrak{N}(\mathfrak{a})^{-s},$$3

the completed functions are degree $$\zeta(s,\mathfrak{C})=\sum_{\mathfrak{a}\in \mathfrak{C},\ \mathfrak{a}\ \text{integral}} \mathfrak{N}(\mathfrak{a})^{-s},$$4 with analytic conductor $$\zeta(s,\mathfrak{C})=\sum_{\mathfrak{a}\in \mathfrak{C},\ \mathfrak{a}\ \text{integral}} \mathfrak{N}(\mathfrak{a})^{-s},$$5 in the $$\zeta(s,\mathfrak{C})=\sum_{\mathfrak{a}\in \mathfrak{C},\ \mathfrak{a}\ \text{integral}} \mathfrak{N}(\mathfrak{a})^{-s},$$6-aspect, convexity gives $$\zeta(s,\mathfrak{C})=\sum_{\mathfrak{a}\in \mathfrak{C},\ \mathfrak{a}\ \text{integral}} \mathfrak{N}(\mathfrak{a})^{-s},$$7, and the main theorem proves the subconvexity estimate

$$\zeta(s,\mathfrak{C})=\sum_{\mathfrak{a}\in \mathfrak{C},\ \mathfrak{a}\ \text{integral}} \mathfrak{N}(\mathfrak{a})^{-s},$$8

(Hough et al., 2021). The twisted Maass-form version satisfies

$$\zeta(s,\mathfrak{C})=\sum_{\mathfrak{a}\in \mathfrak{C},\ \mathfrak{a}\ \text{integral}} \mathfrak{N}(\mathfrak{a})^{-s},$$9

despite the absence of a known functional equation in the twisted case (Hough et al., 2022). The papers emphasize that this is a non-Eulerian setting: the untwisted zeta function has functional equations, whereas the twisted version is handled by an approximate functional equation without a full functional equation (Hough et al., 2022).

These results do not redefine the original real-quadratic invariant, but they show how the phrase “Shintani’s invariant” migrates into arithmetic statistics, where it refers to special values or residues of zeta functions enumerating integral orbits in prehomogeneous vector spaces (Hough et al., 2021).

6. Extensions, analogues, and adjacent constructions

The classical Shintani lift is another major branch of the subject. It associates to a modular form of even integral weight $$X(\mathfrak{C})=X_1(\mathfrak{C})X_2(\mathfrak{C}), \qquad X_1(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C})), \qquad X_2(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C}')).$$0 a modular form of half-integral weight $$X(\mathfrak{C})=X_1(\mathfrak{C})X_2(\mathfrak{C}), \qquad X_1(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C})), \qquad X_2(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C}')).$$1, with Fourier coefficients given by period integrals over geodesic cycles attached to binary quadratic forms. Bringmann, Guerzhoy, and Kane extended this lift from cusp forms to integral-weight weakly holomorphic modular forms and, in the process, obtained new cycle-integral formulas for the coefficients of the classical Shintani lift and new formulas for $$X(\mathfrak{C})=X_1(\mathfrak{C})X_2(\mathfrak{C}), \qquad X_1(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C})), \qquad X_2(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C}')).$$2-values of Hecke eigenforms (Bringmann et al., 2014). Their construction also introduces a Hecke-equivariant “fractional derivative” between half-integral-weight harmonic weak Maass forms and half-integral-weight weakly holomorphic modular forms (Bringmann et al., 2014).

A $$X(\mathfrak{C})=X_1(\mathfrak{C})X_2(\mathfrak{C}), \qquad X_1(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C})), \qquad X_2(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C}')).$$3-adic analogue appears in the rigid-analytic setting. The rigid Shintani lift attaches to a cuspidal rigid cocycle $$X(\mathfrak{C})=X_1(\mathfrak{C})X_2(\mathfrak{C}), \qquad X_1(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C})), \qquad X_2(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C}')).$$4 a modular form

$$X(\mathfrak{C})=X_1(\mathfrak{C})X_2(\mathfrak{C}), \qquad X_1(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C})), \qquad X_2(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C}')).$$5

whose Fourier coefficients are expressed in terms of residues of $$X(\mathfrak{C})=X_1(\mathfrak{C})X_2(\mathfrak{C}), \qquad X_1(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C})), \qquad X_2(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C}')).$$6 and pairings with quadratic forms. The paper explicitly states that for certain cocycles $$X(\mathfrak{C})=X_1(\mathfrak{C})X_2(\mathfrak{C}), \qquad X_1(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C})), \qquad X_2(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C}')).$$7, these residues correspond to periods of modular forms associated to Zagier’s forms and provide $$X(\mathfrak{C})=X_1(\mathfrak{C})X_2(\mathfrak{C}), \qquad X_1(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C})), \qquad X_2(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C}')).$$8-adic analogues to Shintani’s invariants associated with real quadratic fields (Negrini, 2024). This situates Shintani-type invariants within the nascent $$X(\mathfrak{C})=X_1(\mathfrak{C})X_2(\mathfrak{C}), \qquad X_1(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C})), \qquad X_2(\mathfrak{C})=\exp(\zeta'(0,\mathfrak{C}')).$$9-adic Kudla program.

In the cohomological theory of zeta values, Shintani cocycles on $\mathfrak{C}'$0 yield signed cone decompositions, integral Eisenstein cocycles, and $\mathfrak{C}'$1-adic $\mathfrak{C}'$2-functions of totally real fields. After smoothing at an auxiliary prime $\mathfrak{C}'$3, certain specializations of the smoothed class yield the $\mathfrak{C}'$4-adic $\mathfrak{C}'$5-functions of totally real fields, and the resulting cohomological construction is combined with a theorem of Spiess to obtain the lower bound on the order of vanishing predicted by Gross’s conjecture (Charollois et al., 2014). This is not the same object as the real-quadratic invariant $\mathfrak{C}'$6, but it belongs to the same methodological lineage.

Finally, the word “Shintani” also enters representation theory through Shintani descent and Shintani twisting for finite groups of Lie type. There the relevant statement is that Shintani twisting preserves the span of geometric Lusztig series, and in type $\mathfrak{C}'$7 with $\mathfrak{C}'$8 prime it is shown to commute with Jordan decomposition after a suitable choice of parametrization (Digne et al., 2020). This usage is terminologically related but conceptually separate from the real-quadratic invariant.

Taken together, these developments show that “Shintani’s invariant” names a specific real-quadratic class invariant in its strictest sense, while also designating a broader family of special-value phenomena generated by Shintani’s methods: cone decompositions, zeta functions of prehomogeneous vector spaces, theta-type lifts, and $\mathfrak{C}'$9-adic or representation-theoretic analogues.