Search arXiv⌕ Search

arXiv subjects

Andrew Graham

Publications and source records attributed to Andrew Graham.

14 recordsLinked to original sources
arXiv2026

$p$-adic Maass--Shimura operators on $\mu$-ordinary Igusa varieties

For Shimura varieties of Hodge type, we optimally extend algebraic Maass--Shimura differential operators on $p$-integral nearly holomorphic automorphic forms to differential operators on $\mu$-ordinary Mantovan Igusa varieties. We then show that the rank one operators can be integrated to an action of an explicit formal group. Via $p$-adic Fourier theory, this provides a $p$-adic interpolation by extending the action of a symmetric algebra of differential operators to the algebra of functions on the Tate module of the dual $p$-divisible group. Passing to the generic fiber, we obtain an action of an explicit algebra of $p$-adic locally analytic functions, and we show that the action of the subalgebra of locally constant functions is equivalent to a natural Hecke action and thus preserves classical forms. In the ordinary case, we further show that the locally analytic action extends to nearly overconvergent automorphic forms. Our results extend, clarify, and recover prior constructions.

math.NT↗
arXiv2026

The non-abelian Leopoldt conjecture and equalities of $\mathcal{L}$-invariants

Let $G$ be a reductive group quasi-split at $p$. Using arguments of Hansen--Thorne, we show that under the non-abelian Leopoldt conjecture (NALC), Hansen's $p$-adic overconvergent cohomology eigenvariety for $G$ is \'etale over its image in weight space at any non-critical classical tempered cuspidal point of `cohomological multiplicity one'. This applies to all non-critical classical cuspidal points if $G = \mathrm{Res}_{F/\mathbb{Q}}\mathrm{GL}_n$. We then let $\pi$ be a $p$-ordinary regular algebraic cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ such that $\pi_p$ is Steinberg. Combining the above \'etaleness result for the classical point attached to $\pi$, and a local-global compatibility result from our earlier work, we deduce -- under a tangent vector hypothesis that is true for at least half the simple roots -- the equality of Fontaine--Mazur and automorphic $\mathcal{L}$-invariants for $\pi$. Where this assumption is satisfied, we deduce the NALC implies a conjecture of Gehrmann: that automorphic $\mathcal{L}$-invariants are independent of cohomological degree. Our approach is inspired by (and generalises) previous work of Gehrmann--Rosso. When $\pi = \operatorname{Sym}^{n-1} \pi_f$ is the symmetric power lift of a modular form, we verify all assumptions other than the NALC, and deduce a functoriality result for the automorphic $\mathcal{L}$-invariants.

math.NT↗
arXiv2025

Local-global compatibility and the exceptional zero conjecture for GL(3)

We prove exceptional zero conjectures for $p$-ordinary regular algebraic cuspidal automorphic representations of $\mathrm{GL}_3(\mathbb{A})$ which are Steinberg at $p$. We make no self-duality assumptions. The paper has two parts. In Part 1, we use $p$-arithmetic cohomology to unconditionally prove an automorphic exceptional zero conjecture in this setting, using Gehrmann's automorphic $\mathcal{L}$-invariant. In Part 2 we prove, under mild assumptions that are expected to always hold, the equality of automorphic and Fontaine--Mazur $\mathcal{L}$-invariants, and thus deduce cases of the full Greenberg--Benois exceptional zero conjecture. As one of the key ingredients for this, we establish local-global compatibility at $\ell = p$ for Galois representations attached to $p$-ordinary torsion classes for $\mathrm{GL}_n$, confirming a conjecture of Hansen in this setting. We prove this for all $n$ following the strategy in the "10-author paper", and use the $n=3$ case to deduce the desired equality of $\mathcal{L}$-invariants.

math.NT↗
arXiv2025

Deep Neural Network Driven Simulation Based Inference Method for Pole Position Estimation under Model Misspecification

Simulation Based Inference (SBI) is shown to yield more accurate resonance parameter estimates than traditional chi-squared minimization in certain cases of model misspecification, demonstrated through a case study of pi-pi scattering and the rho(770) resonance. Models fit to some data sets using chi-squared minimization can predict inaccurate pole positions for the rho(770), while SBI provides more robust predictions across the same models and data. This result is significant both as a proof of concept that SBI can handle model misspecification, and because accurate modeling of pi-pi scattering is essential in the study of many contemporary physical systems (e.g., a1(1260), omega(782)). The method of Simulation Based Inference is shown to lead to a more accurate resonance parameter estimation than traditional chi-squared minimization in certain cases of model misspecification in a case-study of pi-pi scattering and the rho(770)-resonance. Models fit to certain data sets using chi-squared minimization can make inaccurate predictions for the pole position of the rho(770). SBI is shown to make a more robust predictions for the pole positions. This is significant, both as a proof of concept that the SBI method can be used in cases of model misspecification, and because models of pi-pi scattering are a crucial part to many physical systems of contemporary interest (e.g., a1(1260), omega(782)).

hep-ph↗
arXiv2025

$p$-adic Fourier theory in families

We construct Fourier transforms relating functions and distributions on finite height $p$-divisible rigid analytic groups and objects in a dual category of $\mathbb{Z}_p$-local systems with analyticity conditions. Our Fourier transforms are formulated as isomorphisms of solid Hopf algebras over arbitrary small v-stacks, and generalize earlier constructions of Amice and Schneider--Teitelbaum. We also construct compatible integral Fourier transforms for $p$-divisible groups and their dual Tate modules. As an application, we use the Weierstrass $\wp$-function to construct a global Eisenstein measure over the $p$-adic modular curve, extending previous constructions of Katz over the ordinary locus and at CM points, and show its generic fiber, the global Eisenstein distribution, gives rise to new families of quaternionic modular forms that overconverge from profinite sets in the rigid analytic supersingular locus.

math.NT↗
arXiv2024

Nearly higher Coleman theory and p-adic L-functions for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ and $\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$

We construct four-variable $p$-adic $L$-functions for the spin Galois representation of a Siegel modular form of genus 2 twisted by the Galois representation of a cuspidal modular form as the modular forms vary in Coleman families. The main ingredient is the construction of a space of nearly overconvergent modular forms in the coherent cohomology of the Siegel threefold, extending the spaces of overconvergent modular forms appearing in higher Coleman theory. In addition to this, we construct $p$-adic distributions interpolating the Gan-Gross-Prasad automorphic periods for $(\mathrm{GSpin}(5), \mathrm{GSpin}(4))$ which, conditional on the local and global Gan-Gross-Prasad conjectures for this pair of groups, provides a construction of "square-root" $p$-adic $L$-functions for $\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$ as the automorphic forms vary in Coleman families.

math.NT↗
arXiv2024

A Test of the Thermodynamics of Evolution

Recent research has extended methods from the fields of thermodynamics and statistical mechanics into other disciplines. Most notably, one recent work creates a unified theoretical framework to understand evolutionary biology, machine learning, and thermodynamics. We present simulations of biological evolution used to test this framework. The test simulates organisms whose behavior is determined by specific parameters that play the role of genes. These genes are passed on to new simulated organisms with the capacity to mutate, allowing adaption of the organisms to the environment. With this simulation, we are able to test the the framework in question. The results of our simulation are consistent with the work being tested, providing evidence for it.

q-bio.PE↗
arXiv2024

Unitary Friedberg--Jacquet periods and anticyclotomic p-adic L-functions

We extend the construction of the $p$-adic $L$-function interpolating unitary Friedberg--Jacquet periods in previous work of the author to include the $p$-adic variation of Maass--Shimura differential operators. In particular, we develop a theory of nearly overconvergent automorphic forms in higher degrees of coherent cohomology for unitary Shimura varieties generalising previous work for modular curves. The construction of this $p$-adic $L$-function can be viewed as a higher-dimensional generalisation of the work of Bertolini--Darmon--Prasanna and Castella--Hsieh, and the inclusion of this extra variable arising from the $p$-adic iteration of differential operators will play a key role in relating values of this $p$-adic $L$-function to $p$-adic regulators of special cycles on unitary Shimura varieties.

math.NT↗
arXiv2023

p-adic interpolation of Gauss--Manin connections on nearly overconvergent modular forms and p-adic L-functions

In this paper, we give a new geometric definition of nearly overconvergent modular forms and $p$-adically interpolate the Gauss-Manin connection on this space. This can be seen as an ``overconvergent'' version of the unipotent circle action on the space of $p$-adic modular forms, as constructed by Gouv\^{e}a and Howe. This improves on results of Andreatta--Iovita and has applications to the construction of Rankin--Selberg and triple product $p$-adic $L$-functions.

math.NT↗
arXiv2023

On $p$-refined Friedberg-Jacquet integrals and the classical symplectic locus in the $\mathrm{GL}_{2n}$ eigenvariety

Friedberg--Jacquet proved that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A})$, then $\pi$ is a functorial transfer from $\mathrm{GSpin}_{2n+1}$ if and only if a global zeta integral $Z_H$ over $H = \mathrm{GL}_n \times \mathrm{GL}_n$ is non-vanishing on $\pi$. We conjecture a $p$-refined analogue: that any $P$-parahoric $p$-refinement $\tilde\pi^P$ is a functorial transfer from $\mathrm{GSpin}_{2n+1}$ if and only if a $P$-twisted version of $Z_H$ is non-vanishing on the $\tilde\pi^P$-eigenspace in $\pi$. This twisted $Z_H$ appears in all constructions of $p$-adic $L$-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the $\mathrm{GL}_{2n}$ eigenvariety, and -- by proving upper bounds on the dimensions of such families -- obtain various results towards the conjecture.

math.NT↗
arXiv2022

On the GL(2n) eigenvariety: branching laws, Shalika families and $p$-adic $L$-functions

In this paper, we prove that a $\mathrm{GL}(2n)$-eigenvariety is \'etale over the (pure) weight space at non-critical Shalika points, and construct multi-variable $p$-adic $L$-functions varying over the resulting Shalika components. Our constructions hold in tame level 1 and Iwahori level at $p$, and give $p$-adic variation of $L$-values (of regular algebraic cuspidal automorphic representations of $\mathrm{GL}(2n)$ admitting Shalika models) over the whole pure weight space. In the case of $\mathrm{GL}(4)$, these results have been used by Loeffler and Zerbes to prove cases of the Bloch--Kato conjecture for $\mathrm{GSp}(4)$. Our main innovations are: (a) the introduction and systematic study of `Shalika refinements' of local representations of $\mathrm{GL}(2n)$, and evaluation of their attached local twisted zeta integrals; and (b) the $p$-adic interpolation of representation-theoretic branching laws for $\mathrm{GL}(n) \times \mathrm{GL}(n)$ inside $\mathrm{GL}(2n)$. Using (b), we give a construction of multi-variable $p$-adic functionals on the overconvergent cohomology groups for $\mathrm{GL}(2n)$, interpolating the zeta integrals of (a). We exploit the resulting non-vanishing of these functionals to prove our main arithmetic applications.

math.NT↗
arXiv2019

Bounding Selmer groups for the Rankin--Selberg convolution of Coleman families

Let $f$ and $g$ be two cuspidal modular forms and let $\mathcal{F}$ be a Coleman family passing through $f$, defined over an open affinoid subdomain $V$ of weight space $\mathcal{W}$. Using ideas of Pottharst, under certain hypotheses on $f$ and $g$ we construct a coherent sheaf over $V \times \mathcal{W}$ which interpolates the Bloch-Kato Selmer group of the Rankin-Selberg convolution of two modular forms in the critical range (i.e. the range where the $p$-adic $L$-function $L_p$ interpolates critical values of the global $L$-function). We show that the support of this sheaf is contained in the vanishing locus of $L_p$.

math.NT↗