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A $p$ -adic interpolation of the Cogdell lift

This paper establishes a $p$ -adic interpolation of the classical Cogdell lift by $p$ -adically interpolating the adjoint Kudla lift, constructing higher-weight cycle analogues on Picard modular surfaces, and utilizing Loeffler's formalism to produce $p$ -adic analytic cohomology classes whose generating series form a Hida family interpolating these lifts across weight and level variables.

Original authors: Francesco Maria Iudica

Published 2026-09-02✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Francesco Maria Iudica

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of number theory, mathematicians often study shapes that exist not in our physical world, but in the realm of complex numbers. These shapes, known as Shimura varieties, act as a bridge between two seemingly unrelated worlds: the geometry of these high-dimensional surfaces and the arithmetic of numbers. On one side of this bridge sit special geometric objects called cycles, which are like loops or surfaces drawn on the larger shape. On the other side sit modular forms, which are highly structured functions that encode deep information about numbers. For decades, mathematicians have sought ways to translate information from the geometric side to the arithmetic side, creating a "lift" that turns a shape into a function. One such successful translation, known as the Cogdell lift, was discovered in the 1980s. It showed that by counting how these special loops intersect with one another on a specific type of surface called a Picard modular surface, one could generate the coefficients of a modular form. This was a profound discovery, linking the physical act of counting intersections to the abstract world of number patterns. However, a major challenge remained: this lift was fixed to a specific weight, a measure of the function's complexity. Mathematicians wanted to know if this connection could be made flexible, allowing the weight to vary continuously, and if it could be studied using p-adic numbers, a different kind of number system that reveals hidden patterns in arithmetic that standard numbers often miss.

This paper tackles that challenge by constructing a flexible, p-adic version of the Cogdell lift. The author, Francesco Maria Iudica, achieves this by first revisiting an older, related lift known as the adjoint Kudla lift. While the original lift takes a modular form and creates a geometric shape, the adjoint version does the reverse: it takes a geometric shape and produces a modular form. The paper demonstrates that this reverse process can be interpolated, meaning it can be extended to work across a continuous family of weights and levels, rather than just at isolated points. The author builds a mathematical machine that takes a family of modular forms and outputs a corresponding family of geometric-to-arithmetic translations. This machine is constructed using a sophisticated p-adic L-function, a tool that allows the mathematician to track how these values change as the weight shifts. The result is a smooth, analytic connection that proves the relationship between geometry and arithmetic holds true not just for a single case, but for an entire spectrum of variations.

In the second part of the work, the author expands the scope of these geometric objects. Instead of looking at simple loops, the paper constructs more complex, higher-weight cycles. These are built by taking the original Picard surfaces and attaching them to a family of abelian varieties, which are higher-dimensional generalizations of elliptic curves. By creating these new, more intricate shapes, the author shows that the generating series of their intersections still produces modular forms, but now with higher weights. This is a significant generalization, proving that the deep link between geometry and arithmetic is robust enough to handle these more complicated structures. The paper then ties these two strands together. By using a method developed by another mathematician, the author packages these higher-weight cycles into a single, massive p-adic family. This family is designed so that if you "zoom in" on any specific point within it, you recover the original Cogdell lift or its higher-weight analogues.

The final and most powerful result is the construction of a single, overarching object that interpolates all these lifts simultaneously. The author creates a formal power series, a type of infinite polynomial, whose coefficients are built from the intersection numbers of these special cycles. The paper proves that when you evaluate this series at any specific arithmetic point, the result is a modular form of the expected weight and level. This means that the entire family of lifts, from the original 1980s discovery to the new higher-weight versions, is unified under one p-adic umbrella. The work confirms that the arithmetic information hidden in the geometry of these surfaces is not static; it flows continuously across different weights and levels, governed by a single, coherent structure. This provides a new, flexible tool for understanding the deep connections between the shape of numbers and the patterns they form, offering a way to study these relationships in a dynamic, rather than static, manner. The findings are rigorous and proven within the framework of the paper, establishing a firm foundation for future exploration in this corner of number theory.

Technical Summary: A p-Adic Interpolation of the Cogdell Lift

Problem Statement
The paper addresses the arithmetic interpolation of the classical Cogdell lift, a correspondence introduced by Cogdell ([Cog85]) that maps special cycles on Picard modular surfaces (associated with the unitary group $GU(2,1)$ ) to elliptic modular forms of weight 3. While the classical lift is well-understood, constructing a $p$ -adic analytic family that interpolates these lifts across varying weights and levels remains a significant challenge. The author aims to achieve a $p$ -adic interpolation of the adjoint Kudla lift (which coincides with the Cogdell lift) and to extend this framework to higher weights using Kuga-Sato varieties.

Methodology
The paper employs a combination of geometric cycle constructions, theta correspondence theory, and Hida theory for $p$ -adic modular forms.

Adjoint Kudla Lift and Theta Correspondence: The author utilizes the theta kernel associated with the dual reductive pair $(GU(1,1), GU(2,1))$ . By analyzing the adjoint Kudla lift $L^\dagger$ , the paper establishes its equivalence to the Cogdell lift via the Kudla-Millson and Tong-Wang theories. The composition $L^\dagger L$ is studied using the Rallis inner product formula, linking the lift to special values of $L$ -functions.
Higher Weight Cycles: To generalize the lift to higher weights, the author constructs algebraic cycles $C^k_n$ on Kuga-Sato varieties attached to Picard modular surfaces. These cycles are defined geometrically as projections of fundamental classes of sub-varieties, ensuring they are algebraic cycles with constant coefficients. The modularity of their generating series is proven by comparing them to cycles with coefficients defined by Little ([L19]) and applying the formalism of Funke-Millson ([FM11]).
$p$ -adic Interpolation via Hida Theory: The paper constructs $\Lambda$ -adic families of modular forms. It defines analytic weight spaces and utilizes the theory of ordinary $\Lambda$ -adic forms. A key technical tool is the construction of a $p$ -adic $L$ -function (following [EHLS20]) to interpolate the inner products required for the lift.
Loeffler's Machinery: For the final construction of a "big" family, the author applies the formalism developed by Loeffler ([L21]). This involves constructing "big" cohomology classes in Iwasawa cohomology using mirabolic subgroups and specific level structures ( $K_r, V_r$ ). These classes are paired via a twisted "big" Poincaré pairing to generate a formal $q$ -expansion.

Key Contributions and Results

Main Theorem A (Adjoint Kudla Interpolation): The paper proves the existence of an analytic endomorphism $\Psi$ on the space of ordinary $\Lambda$ -adic forms $S_{ord}(\omega_{K/\mathbb{Q}})$ . For any arithmetic weight $\kappa$ and Hida family $F$ , the specialization of $\Psi(F)$ at $\kappa$ (up to a period factor) recovers the classical adjoint Kudla lift $L^\dagger L(F(\kappa))$ . This provides an arithmetic $p$ -adic interpolation of the Cogdell lift compatible with previous interpolations of the Kudla lift.
Main Theorem B (Higher Weight Modularity): The author constructs higher weight cycles $C^k_n$ in the cohomology of Kuga-Sato varieties $A^k$ . It is shown that the generating series $\sum C^k_n q^n$ is modular of weight $2k+3$ . Specifically, the $q$ -expansion of the pairing of these cycles with compactly supported cohomology classes yields elliptic modular forms of weight $2k+3$ .
Main Theorem C (Big Cogdell Theorem): By constructing a cuspidal "big" cohomology class $\zeta_\infty$ orthogonal to the Chern class, the paper defines a $\Lambda$ -adic $q$ -expansion $\Phi_\zeta(z)$ . The specializations of this series at arithmetic points of weight $k$ and level $r$ are proven to be modular forms in $S_{2k+3}(\Gamma_1(p^r D))$ . This series interpolates both the classical Cogdell lifts and their higher weight analogues.

Significance and Claims
The paper claims to provide a unified framework for the $p$ -adic variation of the Cogdell lift. Its significance lies in:

Arithmetic Interpolation: It successfully interpolates the Cogdell lift, which maps geometric intersection numbers to Fourier coefficients, into a $p$ -adic analytic family varying in both weight and level.
Geometric Construction: Unlike previous approaches that might rely on coefficient systems, the higher weight cycles are constructed as genuine algebraic cycles in the cohomology of Kuga-Sato varieties, offering a geometric interpretation of the higher weight adjoint Kudla lift.
Generalization: The results establish a "Big Cogdell Theorem," packaging the intersection multiplicities of special cycles into a single $\Lambda$ -adic object. The author notes that the methods used are potentially applicable to unitary Shimura varieties of signature $(m, 1)$ , though this is left for future work.

The paper relies heavily on the compatibility of the geometric cycle constructions with the analytic properties of Hida families and the algebraic structure of the Rallis inner product formula, bridging the gap between the geometry of Shimura varieties and the arithmetic of $p$ -adic modular forms.

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