Rankin–Selberg Motives

• Rankin–Selberg motives are objects derived from tensor products of automorphic motives, encapsulating special L-value invariants and period comparisons.
• They are constructed from cuspidal automorphic representations of GL(n) and GL(m), bridging Betti, de Rham, and ℓ-adic cohomological realizations.
• Recent advances highlight their role in Euler systems, p-adic variation, Selmer group analysis, and support for conjectures like Bloch–Kato and Deligne’s.

Rankin–Selberg motives are central objects at the interface of the theory of automorphic forms, algebraic geometry, and arithmetic. These motives arise from the tensor product (convolution) constructions in the context of automorphic representations—especially those associated to $\mathrm{GL}(n)$ and its inner forms—and their associated $L$-functions encode deep arithmetic and geometric information. In particular, understanding Rankin–Selberg motives involves the paper of special values of their $L$-functions, their conjectural relation to motives and periods, their $p$-adic variation, and their role in Selmer groups and Iwasawa theory.

1. Construction of Rankin–Selberg Motives

The construction of Rankin–Selberg motives typically begins with cuspidal automorphic representations $\pi$ (for $\mathrm{GL}_n$) and $\pi'$ (for $\mathrm{GL}_m$) over a field (often a number field $F$, including CM fields). Under suitable self-duality and cohomological hypotheses, one associates to each representation a pure motive $M(\pi)$ and $M(\pi')$, often realized in the interior or coherent cohomology of Shimura varieties associated to (possibly unitary) groups.

The Rankin–Selberg motive is then the (restricted) tensor product: $R_{F/F^+}(M(\pi) \otimes M(\pi'))$ over the reflex field, with Betti, de Rham, and $\ell$-adic realizations constructed from the Hecke-equivariant cohomology modules: $$M_B(\pi_f) = \operatorname{Hom}_{H(\mathbf{A})}(\pi_f, H^{n-1}(Sh(H,Y_V), \tilde{V}_\lambda^\vee)), \ M_{dR}(\pi_f) = H^{n-1}_{dR}(Sh(H,Y_V), \tilde{V}_\lambda^\vee).$$ This construction leverages base change, descent, and the explicit understanding of the minimal $K$-types for discrete series at archimedean places. The resulting motive inherits a rational structure suitable for the comparison of periods and $L$-values (Grobner et al., 2 Sep 2025).

2. Rankin–Selberg $L$-Functions and Critical Values

The $L$-function attached to a Rankin–Selberg motive is defined as an Euler product: $$L(s, M(\pi)\otimes M(\pi')) = \prod_v L_v(s, \pi_v \times \pi'_v)$$ where each $L_v$ is determined by the tensor product of the local Galois (or Weil–Deligne) parameters.

A major focus is the paper of critical values in the sense of Deligne—that is, those $s_0\in \mathbb{Z}$ for which both $L(s, M)$ and its functional equation dual are nonzero and the archimedean $\Gamma$-factors are regular. Deligne's conjecture posits that for a pure motive $M$ and critical $s_0$,

$$L^S(s_0, M) \sim_{\mathbf{E}} (2\pi i)^{a} \cdot c^+(M),$$

where $c^+(M)$ is a period computed from the comparison isomorphism between Betti and de Rham cohomology, and $\mathbf{E}$ is a coefficient field (Harris et al., 2016, Grobner et al., 2 Sep 2025).

In the automorphic context, critical values for Rankin–Selberg $L$-functions can be directly expressed as cup products or period integrals of automorphic forms on locally symmetric spaces or coherent cohomology classes on Shimura varieties. Recent work has established (conditional on regularity and rationality hypotheses) explicit period relations for these values, leading to proofs of special instances of Deligne's conjecture for automorphic motives over CM fields (Grobner et al., 2 Sep 2025).

3. Period Relations and Factorization

A striking aspect of modern work on Rankin–Selberg motives is the factorization of period invariants attached to automorphic forms. On the automorphic side, periods $P^{(i)}(\pi, \iota)$ (for each cohomological degree $i$ and complex embedding $\iota$) correspond to arithmetically normalized Petersson norms of holomorphic forms contributing to the cohomology. One can prove factorization statements

$$P^{(i)}(\pi, \iota) \sim P_0(\pi, \iota)\cdot P_1(\pi, \iota) \cdots P_i(\pi, \iota),$$

which mirror the factorization of motivic periods $Q^{(i)}(M(\pi),\iota)$ as predicted by the Tate conjecture (Grobner et al., 2 Sep 2025, Harris et al., 2016).

A central achievement is the identification, up to known algebraic factors, of these local automorphic periods with motivic periods, and the demonstration that the critical values of $L$-functions are proportional to products of these periods up to explicit powers of $2\pi i$ and arithmetic constants—precisely as Deligne’s conjecture suggests.

4. Euler Systems, Syntomic Regulators, and $p$-adic Variation

Rankin–Selberg motives play a central role in the paper of Euler systems and $p$-adic phenomena. For pairs of modular forms or their $p$-adic families, motivic cohomology classes (sometimes known as Rankin–Eisenstein or Rankin–Selberg Euler system classes) are constructed in motivic cohomology groups, typically via geometric cycles (Eisenstein elements or diagonal cycles on Kuga–Sato varieties/Shimura varieties) (Kings et al., 2015, Kings et al., 2014).

These classes admit $p$-adic variation—their images in $p$-adic étale or de Rham cohomology via the p-adic regulator map interpolate $p$-adic $L$-values across Hida or Coleman families, often forming "big" Euler systems. Explicit formulas relate syntomic regulators or Abel–Jacobi images of these classes to derivatives and special values of (complex or $p$-adic) Rankin–Selberg $L$-functions (Kings et al., 2014, Kings et al., 2015).

5. Relations to Selmer Groups and the Beilinson–Bloch–Kato Conjecture

A fundamental application of Rankin–Selberg motives is to the paper of Bloch–Kato Selmer groups and their relation to $L$-values. The Beilinson–Bloch–Kato conjecture predicts that for any motive $M$ and critical value $s_0$,

$$\operatorname{ord}_{s=s_0} L(s, M) = \dim_E H^1_f(G_F, V)$$

where $H^1_f$ is the Bloch–Kato Selmer group for the motive. In the Rankin–Selberg setting, geometric Euler system techniques, level-raising, and cohomology of Shimura varieties have been used to verify specific instances of this prediction (Liu et al., 2019, Liu et al., 21 Sep 2025). For example, nonvanishing of $L(1, \Pi_0\times \Pi_1)$ for pairs of conjugate self-dual automorphic representations implies vanishing of the Bloch–Kato Selmer group of the corresponding motive; if the derivative is nonzero, then the Selmer group is rank one, and the corresponding cycle classes in cohomology can be explicitly constructed.

These results connect deep geometric ingredients (moduli spaces, stratifications, explicit correspondences) to arithmetic invariants and Selmer group ranks, providing geometric realizations of abstract conjectures.

6. Iwasawa Theory and Eigenvarieties

Rankin–Selberg motives also underpin modern forms of the Iwasawa main conjecture, particularly in anticyclotomic settings or over families such as $p$-adic eigenvarieties. Here, one relates the $p$-adic $L$-function attached to the Rankin–Selberg motive to the characteristic ideal of the (big) Iwasawa Selmer group. Recent work has proven significant one-sided divisibility results, showing that when the global root number is $+1$, the $p$-adic $L$-function lies in the characteristic ideal of the Selmer group, and when the root number is $-1$, the square of a characteristic ideal is contained in that of the torsion part of the Selmer group (Liu et al., 2 Jun 2024, Liu, 25 Dec 2024).

Furthermore, the construction of Bessel periods and their interpolation across eigenvarieties provides an analytic side to Iwasawa-type main conjectures: the vanishing divisor of the Bessel period on the eigenvariety matches (up to explicit factors) the characteristic divisor of the Selmer group over the eigenvariety, under some regularity and irreducibility hypotheses.

7. Analytic and Representation-Theoretic Aspects

At the analytic level, advances in the explicit computation and analytic continuation of Rankin–Selberg $L$-functions via integral representations (unipotent averaging, new test vector strategies), trace formulas, summation formulas on monoids, and convolution operations for Bessel transforms have deepened the understanding of their pole structures, functional equations, and special value formulas (Herman, 2010, Booker et al., 2018, Getz, 2014).

The compatibility of local and global Rankin–Selberg integrals (including the identification of local Euler and gamma factors) has been clarified for a wide class of groups and representations (Liu et al., 2021, Asgari et al., 25 Sep 2024). These developments are essential for verifying both arithmetic conjectures and functoriality predictions within the Langlands program.

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The paper of Rankin–Selberg motives thus links the arithmetic of automorphic forms, special values of $L$-functions, period relations, regulator maps, and object-level predictions such as the Bloch–Kato and Iwasawa main conjectures, through a rich interplay of geometry, representation theory, and analytic number theory (Grobner et al., 2 Sep 2025, Harris et al., 2016, Liu et al., 2019, Liu, 25 Dec 2024, Liu et al., 2 Jun 2024, Kings et al., 2015, Kings et al., 2014, Getz, 2014, Asgari et al., 25 Sep 2024, Liu et al., 2021, Herman, 2010, Bouganis et al., 2018, Booker et al., 2018, Wu, 2018, Huang, 2020, Koyama et al., 2020, Kim, 2022, Huang, 2023, Liu et al., 21 Sep 2025). The connection between automorphic periods, motivic periods, and critical $L$-values—especially as elucidated by explicit factorization and period comparison theorems—provides a robust framework for understanding the arithmetic of these objects in both classical and $p$-adic families, and suggests further avenues of research towards categorical and functorial generalizations.