arXiv subjects

Kartik Prasanna

Publications and source records attributed to Kartik Prasanna.

6 recordsLinked to original sources

arXiv2026

Cycles for Rankin-Selberg $L$-functions, I: automorphic periods

In this paper, we establish an explicit formula for automorphic periods which will be used in a sequel to study special values of $p$-adic Rankin-Selberg $L$-functions. Our motivation is to extend the Bertolini-Darmon-Prasanna formula to the case where the archimedean local sign is opposite to that in the original setting. To this end, we prove a formula for the $\mathrm{GU}(1,1)$-period of a theta lift from $\mathrm{GSO}(2)$ to $\mathrm{GSp}_4$, which is adapted to $p$-adic interpolation. We also introduce a $p$-depletion Hecke operator for this theta lift, which will play a crucial role in relating the automorphic period to the image of the $p$-adic Abel-Jacobi map.

math.NT↗

arXiv2025

Motivic action for Siegel modular forms

We study the coherent cohomology of automorphic sheaves corresponding to Siegel modular forms $f$ of low weight on ${\rm GSp}(4)$ Shimura varieties. Inspired by the work of Prasanna--Venkatesh on singular cohomology of locally symmetric spaces, we propose a conjecture that explains all the contributions of a Hecke eigensystem to coherent cohomology in terms of the action of a motivic cohomology group. Under some technical conditions, we prove that our conjecture is equivalent to Beilinson's conjecture for the adjoint $L$-function of $f$. We also prove some unconditional results in special cases. For a lift $f$ of a Hilbert modular form $f_0$ to ${\rm GSp}(4)$, we produce elements in the motivic cohomology group for which the conjecture holds, using the results of Ramakrishnan on the Asai $L$-function of $f_0$. For a lift $f$ of a Bianchi modular form $f_0$ to ${\rm GSp}(4)$, we show that our conjecture for $f$ is equivalent to the conjecture of Prasanna-Venkatesh for $f_0$, thus establishing a connection between the motivic action conjectures for locally symmetric spaces of non-hermitian type and those for coherent cohomology of Shimura varieties.

math.NT↗

arXiv2025

Representations of $\mathrm{GL}_2$ over $\mathbb{Z}/p^n\mathbb{Z}$ and supercongruences for hypergeometric polynomials

For an odd prime $p$, we realize the trivial representation of $\mathrm{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$ on the free $\mathbb{Z}/p^n \mathbb{Z}$-module of rank one as a subquotient of a direct sum of symmetric power representations (twisted by appropriate powers of the determinant) of rank strictly greater than one. The proof eventually reduces to establishing some novel supercongruences for hypergeometric polynomials.

math.RT↗

arXiv2023

Hodge classes and the Jacquet-Langlands correspondence

We prove that the Jacquet-Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz's conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in $\ell$-adic cohomology is Galois invariant for all $\ell$.

math.NT↗

arXiv2017

Automorphic cohomology, motivic cohomology, and the adjoint $L$-function

We propose a relationship between the cohomology of arithmetic groups, and the motivic cohomology of certain (Langlands-)attached motives. The motivic cohomology group in question is that related, by Beilinson's conjecture, to the adjoint $L$-function at $s=1$. We present evidence for the conjecture using the theory of periods of automorphic forms, and using analytic torsion.

math.NT↗

arXiv2016

Periods of quaternionic Shimura varieties. I

We study "quadratic periods" on quaternionic Shimura varieties and formulate an integral refinement of Shimura's conjecture regarding Petersson inner products of automorphic forms that are related by the Jacquet-Langlands correspondence. The main result is that this integral refinement is implied by another conjecture (Conjecture D below) regarding integrality of theta lifts between certain quaternionic unitary groups.

math.NT↗