Gross–Kudla Period Formula

• The Gross–Kudla period formula is a key result that links central values of triple product L-functions to finite arithmetic sums over quaternionic ideal classes.
• It converts complex analytic averages into explicit combinatorial formulas using techniques like the Jacquet–Langlands correspondence and Parseval's identity.
• The formula has significant arithmetic implications, including non-vanishing results, algebraicity of L-values, and computational accessibility for modular form invariants.

The period formula of Gross and Kudla is a central result in the arithmetic theory of automorphic forms and their $L$-functions, linking the central values or derivatives of special $L$-functions to explicit period integrals or sums over arithmetic data—often special cycles or automorphic forms on associated Shimura varieties and quaternion algebras. This formula provides a bridge between analytic and algebraic invariants via explicit combinatorial or integral expressions and is foundational in several branches of the arithmetic theory of modular forms, Shimura varieties, and the arithmetic Gan–Gross–Prasad program.

1. Foundational Statement: Central Value of Triple Product $L$-Functions

The archetypal Gross–Kudla period formula, as systematically used in the context of triple product $L$-functions for weight 2 newforms $f, g, h$ of prime level $N$, translates the central value $L(2, f \otimes g \otimes h)$ into a finite, explicit sum over representatives of quaternionic left ideal classes attached to the quaternion algebra $D$ ramified at $N$ and $\infty$:

$$\frac{4 \pi N \, L(2, f \otimes g \otimes h)}{(f,f)(g,g)(h,h)} = \sum_{i=1}^{n} w_i^2\, f'(I_i) g'(I_i) h'(I_i)$$

where $I_1, \ldots, I_n$ are left $R$-ideal class representatives for a maximal order $R$ in $D$, $w_i = \# R_{i}^{\times} / 2$ with $R_i$ the right order of $I_i$, and $f'$ (resp. $g', h'$) are quaternionic lifts of the modular forms via the Jacquet–Langlands correspondence, normalized such that their weighted inner products are $1$.

This relates the analytic central $L$-value—otherwise extremely difficult to calculate directly—to finite arithmetic data associated with quaternionic algebra structures and modular forms.

2. Reduction of Analytic Averages to Arithmetic Sums

A major application of the period formula is the reduction of analytic problems involving averages of central $L$-values to explicit weighted sums in finite-dimensional quaternionic module spaces. For example, when averaging $L(2, f\otimes g\otimes h)$ over $f$ and $g$ with $h$ fixed, the period formula yields

$$\sum_{f,g \in F_2(N)} \frac{L(2, f \otimes g \otimes h)}{(f,f)(g,g)} = \frac{(h,h)}{4\pi N} \sum_{i=1}^n w_i^2 h'(I_i)^2$$

Parseval's identity and orthogonality in these spaces allow such sums to be evaluated explicitly, leading to closed formulas involving other arithmetic invariants (often lower-degree $L$-values).

3. Explicit Formulas and Their Arithmetic Consequences

Through the period formula mechanism and further combinatorial/ADELC analysis, explicit arithmetic expressions for averages of triple product $L$-values are obtained. For prime $N$ and any $h \in F_2(N)$, Corollary 4.3 in (Feigon et al., 2010) gives:

$$\frac{4 \pi N}{(h,h)} \sum_{f,g \in F_2(N)} \frac{L(2, f \otimes g \otimes h)}{(f,f)(g,g)} = \begin{cases} \frac{1}{24}(N-1)(h,h) & N \equiv 1 \pmod{12} \ 6\sqrt{3} L(1,h)L(1,h \otimes \chi_{-3}) & N \equiv 5 \pmod{12} \ 4 L(1,h)L(1,h \otimes \chi_{-4}) & N \equiv 7 \pmod{12} \ 6\sqrt{3}L(1,h)L(1,h \otimes \chi_{-3}) + 4L(1,h)L(1,h \otimes \chi_{-4}) & N \equiv 11 \pmod{12} \end{cases}$$

where the right-hand side features standard and quadratic twist $L$-values at $s=1$. The structure of the ideal classes and weights makes these precise identifications possible.

These explicit formulas provide:

• Non-vanishing results: Lower bounds for the number of non-vanishing central $L$-values.
• Algebraicity and rationality: Deductions about the algebraic properties of $L$-values derived from arithmetic features of quaternionic representations.
• Computability: For small levels, the formulas are computationally accessible, being entirely expressed in terms of modular form coefficients and counting invariants.

4. Link to Period Integrals, Arithmetic Cycles, and Generalizations

The Gross–Kudla period formula is a forerunner of a broad class of results that relate period integrals of automorphic forms (including those computed by theta correspondences) to $L$-function special values or derivatives. In higher rank and general settings:

• It underpins refined period–$L$-value formulas such as those in the Ichino–Ikeda and refined Gross–Prasad conjectures (Harris, 2012), where period integrals of automorphic forms are related, with explicit global and local factors, to special (often central) values of $L$-functions.
• It is intimately connected to the Kudla program, where intersection numbers or heights of special cycles on Shimura varieties correspond to central derivatives of Eisenstein series or $L$-functions, with Green functions or generating series playing a central role (Bruinier et al., 2013, Berndt et al., 2012).

This concept extends through arithmetic fundamental lemma conjectures and the arithmetic Gan–Gross–Prasad program, indicating the profound structural relationship between periods, automorphic forms, and arithmetic invariants.

5. Quaternionic and Shimura Variety Context: The Role of the Jacquet–Langlands Correspondence

The essential mechanism behind the reduction of triple product $L$-values to quaternionic data is the Jacquet–Langlands correspondence, which identifies spaces of modular forms with certain spaces of functions (or automorphic forms) on quaternion algebras. Via this correspondence:

• Modular forms of given weight and level (for $\mathrm{GL}_2$) correspond to modular forms on $D^\times$.
• The explicit evaluation of $f'(I_i), g'(I_i), h'(I_i)$ at left ideal classes encodes the arithmetic of those automorphic forms.
• The finiteness and explicit structure of quaternionic module spaces allow for concrete combinatorial evaluation of global arithmetic invariants.

This approach transforms analytic hazards (infinite sums, analytic continuation of complicated $L$-series) into finite, arithmetic, and often computationally tractable combinatorics.

6. Broader Mathematical Significance and Methodological Impact

The Gross–Kudla period formula for triple product $L$-functions supplied the template for subsequent developments across several research programs:

• Relative trace formula methodology: While the approach via explicit quaternionic sums is classical (and, for the triple product case, optimal at prime level), the viewpoint is echoed in relative trace formula calculations, which generalize to higher-rank groups and non-quaternionic settings (Zhang, 2017).
• Modularity of generating series: In arithmetic geometry, generating series of special cycles equipped with Green functions, whose coefficients are intersection numbers, are often found to be modular or have modular derivatives expressible in terms of $L$-values, following the "arithmetic theta lift" philosophy (Bruinier et al., 2013, Berndt et al., 2012, Berndt et al., 2012).
• Explicit non-vanishing and distribution results: The formula enables a host of quantitative results on non-vanishing and distribution of critical $L$-values, as well as connections to rationality and Galois properties (Feigon et al., 2010).

7. Summary Table: Key Features of the Gross–Kudla Period Formula

Feature	Mathematical Realization	Reference
Central $L$-values	Evaluated as finite sums over quaternion ideal class representatives	Theorem 3.1 in (Feigon et al., 2010)
Averages of $L$-values	Reduced to finite sums via Parseval in quaternionic module spaces	Section 4 in (Feigon et al., 2010)
Arithmetic consequences	Explicit formulas for averages; non-vanishing; algebraicity	Corollaries 4.3, 1.2, 5.2, 5.3
Input data	Prime level $N$, weight 2 newforms, Jacquet–Langlands correspondences	As above
Generalizations	Higher rank period formulas, arithmetic cycles, trace formula approach	(Harris, 2012, Zhang, 2017, Bruinier et al., 2013)

8. Conclusion

The Gross–Kudla period formula rigorously realizes the connection between special values of automorphic $L$-functions and period integrals or finite arithmetic sums in the context of modular forms and quaternion algebras. It has become a prototype for a wide spectrum of results linking analytic $L$-functions with arithmetic geometry, and its techniques underpin both explicit computational approaches and abstract theoretical frameworks in the modern theory of periods, cycles, and $L$-values.