arXiv subjects

Michael Harris

Publications and source records attributed to Michael Harris.

At least 19 recordsLinked to original sources

arXiv2026

Laser Remelting for Reduced Porosity on Additively Manufactured Aluminium Mirrors

Additively manufactured (AM) AlSi10Mg mirrors are fabricated through laser powder bed fusion (LPBF), allowing the use of complex geometries such as lattices and organic structures that enable high mass reduction while maintaining mechanical stiffness. Micron-sized pores that cause optical scatter may form during LPBF as a consequence of deviations from the optimal processing window, particularly from laser energy input and scan strategy. This work proposes a laser remelting strategy aimed at reducing porosity; standard LPBF build steps automatically alternate with laser remelting passes, where previously deposited material is remelted during fabrication. Laser remelting is evaluated through fabricating 10 mm proof-of-concept cubes. Following single point diamond turning (SPDT), optical measurements characterised surface roughness and identified surface artefacts. The best-performing AlSi10Mg remelted cube exhibited no pores within sampled regions and achieved 6.4 nm average surface roughness, comparable to a conventionally manufactured RSA 6061 control cube (5.8 nm). Driven by these results, AM 52 mm diameter secondary sandwich mirrors were manufactured using LPBF and laser remelting. These incorporate an optimised diamond TPMS lattice to achieve a 50% mass reduction while accommodating design for AM considerations. Unlike the cube study, the optical surface of the remelted mirror after SPDT exhibited residual porosity and 11.8 nm average surface roughness. These results show that while the proof-of-concept confirmed the viability of laser remelting in reducing porosity within simple geometries, optimisation of the LPBF and SPDT processes are required to translate the benefits of laser remelting to lightweight AlSi10Mg AM mirrors.

physics.optics↗

arXiv2025

$p$-adic $L$-functions for $\mathrm U(2,1)\times\mathrm U(1,1)$

We construct the five-variable $p$-adic $L$-function attached to Hida families on $\mathrm U(2,1)\times\mathrm U(1,1)$, interpolating the square-root of Rankin-Selberg $L$-values in the \emph{shifted piano} range. Our construction relies on a new theta operator and its $p$-adic variation which plays a role analogous to the classical Ramanujan-Serre theta operator in Hida's $p$-adic Rankin-Selberg method. The interpolation formula, including the modified Euler factors at $p$ and at the real place, is consistent with the conjectural shape of $p$-adic $L$-functions predicted by Coates and Perrin-Riou.

math.NT↗

arXiv2025

Translation functors, branching problems, and applications to the restriction of coherent cohomology of Shimura varieties

We study properties of the restriction of discrete series representations of $G=U(p,q)$ to $G'= U(p-1,q)$ and the corresponding symmetry breaking operators in $\operatorname{Hom}_{G'}(\pi|_{G'}, \pi')$. This leads to the introduction of elementary and coherent pairs of discrete series representations and their classification. Translations of symmetry breaking operators are defined via tensor products with finite-dimensional representations, which leads to the study of the coherent cohomology of discrete series representations under restriction and translations. This is applied to the study of cup products of coherent cohomology of associated Shimura varieties, and to the arithmetic of central values of certain Rankin--Selberg $L$-functions of $GL(n+1)\times GL(n)$.

math.RT↗

arXiv2025

Factorization of periods, construction of automorphic motives and Deligne's conjecture over CM-fields

The present paper is devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. As an application of our main result, we establish Deligne's conjecture for a class of CM-automorphic motives, which we construct in this paper. Our proof uses the results of our recent joint work with Raghuram in combination with the Ichino--Ikeda--Neal-Harris (IINH) formula for unitary groups -- which is now a theorem -- and an analysis of cup products of coherent cohomological automorphic forms on Shimura varieties to establish relations between certain automorphic periods and critical values of Rankin-Selberg and Asai $L$-functions of $\GL(n)\times\GL(m)$ over CM fields. By reinterpreting these critical values in terms of automorphic periods of holomorphic automorphic forms on unitary groups, we show that the automorphic periods of holomorphic forms can be factored as products of coherent cohomological forms, compatibly with a motivic factorization predicted by the Tate conjecture. All of these results are stated under a certain regularity condition and an hypothesis of rationality on archimedean zeta-integrals.

math.NT↗

arXiv2025

Transverse single-spin asymmetries in $\gamma$SIDIS as a direct probe of quark-gluon-quark longitudinal momentum structure

Transverse single-spin asymmetries in the semi-inclusive deep-inelastic production of isolated photons ($\gamma$SIDIS), $A_{UT}^{\gamma {\rm SIDIS}}$, provide an unprecedented opportunity to extract the quark-gluon-quark correlators $F_{FT}(x,x')$ and $G_{FT}(x,x')$ point-by-point in their full support $x,x'$. We utilize realistic models for these functions, based on input from the Sivers transverse momentum dependent parton distribution function and imposing constraints from the $d_2$ matrix element calculated in lattice QCD, in order to provide numerical estimates for $A_{UT}^{\gamma {\rm SIDIS}}$ at the Electron-Ion Collider (EIC). We thoroughly explore the EIC phase space in order to isolate in which regions the asymmetry can be sizable, finding it can be as much as $10\%$ or larger for certain kinematics. Given that $F_{FT}(x,x')$ and $G_{FT}(x,x')$ are basically unknown, $A_{UT}^{\gamma {\rm SIDIS}}$ will be an important future measurement to learn about multi-parton correlations in the nucleon.

hep-ph↗

arXiv2025

Inductive construction of supercuspidal $L$-packets

Genestier--Lafforgue and Fargues--Scholze have constructed a semisimple local Langlands paramterization for reductive groups over equicharacteristic local fields. Assuming a version of the stable twisted trace formula for function fields, we prove the surjectivity of this parameterization for split groups in sufficiently large characteristic.

math.NT↗

arXiv2024

Derived structures in the Langlands Correspondence

We survey several recent examples of derived structures emerging in connection with the Langlands correspondence. Cases studies include derived Galois deformation rings, derived Hecke algebras, derived Hitchin stacks, and derived special cycles. We also highlight some open problems that we expect to be important for future progress.

math.NT↗

arXiv2024

Investigating Mass Reduction Capabilities of Additive Manufacturing through the Re-Design of a Space-Based Mirror

Additive manufacture (AM) involves creating a part layer by layer and is a rapidly evolving manufacturing process. It has multiple strengths that apply to space-based optics, such as the ability to consolidate multiple parts into one, reducing the number of interfaces. The process also allows for greater mass reduction, making parts more cost-effective to launch, achieved by optimising the shape for intended use or creating intricate geometries like lattices. However, previous studies have highlighted issues associated with the AM process. For example, when trying to achieve high-precision optical surfaces on AM parts, the latticing on the underside of mirrors can provide insufficient support during machining, resulting in the quilting effect. This paper builds on previous work and explores such challenges further. This will be implemented by investigating ways to apply AM to a deployable mirror from a CubeSat project called A-DOT. The reflective surface has a spherical radius of curvature of 682 mm and approximate external dimensions of 106 mm x 83 mm. The aim is to produce two mirrors that will take full advantage of AM design benefits and account for the challenges in printing and machining a near-net shape. The designs will have reduced mass by using selected internal lattice designs and topology-optimised connection points, resulting in two mirrors with mass reduction targets of 50% and 70%. Once printed in aluminium using laser powder bed fusion, the reflective surface will be created using single point diamond turning. Finally, an evaluation of the dimensional accuracy will be conducted, using interferometry, to quantify the performance of the reflective surface.

astro-ph.IM↗

arXiv2023

On the generalized Ramanujan and Arthur conjectures over function fields

Let $G$ be a simple group over a global function field $K$, and let $\pi$ be a cuspidal automorphic representation of $G$. Suppose $K$ has two places $u$ and $v$ (satisfying a mild restriction on the residue field cardinality), at which the group $G$ is quasi-split, such that $\pi_u$ is tempered and $\pi_v$ is unramified and generic. We prove that $\pi_w$ is tempered at all unramified places $K_w$ at which $G$ is unramified quasi-split. More generally, the set of unitary spherical representations is partitioned according to nilpotent conjugacy classes in the Lie algebra of $G$. We show that if $\pi_v$ is in the set corresponding to the nilpotent class $N$, and if $\pi_u$ satisfies an analogous hypothesis, then $\pi_w$ belongs to the same class $N$, where $w$ is as above. These results are consistent with conjectures of Shahidi and Arthur. The proofs use the Galois parametrization of cuspidal representations due to V. Lafforgue to relate the local Satake parameters of $\pi$ to Deligne's theory of Frobenius weights. The main observation is that, in view of the classification of unitary spherical representations, due to Barbasch and the first-named author, the theory of weights excludes almost all complementary series as possible local components of $\pi$. This in turn determines the local Frobenius weights at all unramified places. In order to apply this observation in practice we need a result of the second-named author with Gan and Sawin on the weights of discrete series representations.

math.NT↗

arXiv2023

Derived class field theory

We sketch the construction of a derived enhancement of the reciprocity isomorphism of class field theory. Details will appear in a forthcoming joint paper of the authors with A. Raksit.

math.NT↗

arXiv2022

The derived Hecke algebra for dihedral weight one forms

We study the action of the derived Hecke algebra in the setting of dihedral weight one forms, and prove a conjecture of the second- and fourth- named authors relating this action to certain Stark units associated to the symmetric square L-function. The proof exploits the theta correspondence between various Hecke modules as well as ideas of Merel and Lecouturier on higher Eisenstein elements.

math.NT↗

arXiv2022

Cyclic base change of cuspidal automorphic representations over function fields

Let $G$ be a split semi-simple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on modularity lifting theorems, together with a Smith theory argument to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of representations called toral supercuspidal representations.

math.NT↗

arXiv2022

Local Langlands correspondences

The first part of this article is a review of the properties expected of any local Langlands correspondence that aims to be considered "canonical," and of known results that establish some or all of these properties for specific groups. In the absence of compatibility with a global correspondence it is not known in general that this list of desirable properties suffices to characterize the correspondence. The remainder of the article outlines elements of a strategy to prove that the $L$-packets attached to irreducible local parameters are finite and non-empty, in particular, for the specific parametrization constructed by Genestier and Lafforgue when $F$ is of positive characteristic.

math.RT↗

arXiv2022

On the generalized Ramanujan conjecture over function fields

Let $G$ be a simple group over a global function field $K$, and let $\pi$ be a cuspidal automorphic representation of $G$. Suppose $K$ has two places $u$ and $v$ (satisfying a mild restriction on the residue field cardinality), at which the group $G$ is quasi-split, such that $\pi_u$ is tempered and $\pi_v$ is unramified and generic. We prove that $\pi$ is tempered at all unramified places $K_w$ at which $G$ is unramified quasi-split. The proof uses the Galois parametrization of cuspidal representations due to V. Lafforgue to relate the local Satake parameters of $\pi$ to Deligne's theory of Frobenius weights. The main observation is that, in view of the classification of generic unitary spherical representations, due to Barbasch and the first-named author, the theory of weights excludes generic complementary series as possible local components of $\pi$. This in turn determines the local Frobenius weights at all unramified places. In order to apply this observation in practice we need a result of the second-named author with Gan and Sawin on the weights of discrete series representations.

math.NT↗

arXiv2021

The Taylor-Wiles method for coherent cohomology, II

We show that the Taylor-Wiles method can be applied to the cohomology of a Shimura variety $S$ of PEL type attached to a unitary similitude group $G$, with coefficients in the coherent sheaf attached to an automorphic vector bundle $\CF$ , when $S$ has a smooth model over a $p$-adic integer ring. This generalizes the main results of the article \cite{H13}, which treated the case when $S$ is compact. As in the previous article, the starting point is a theorem of Lan and Suh that proves the vanishing of torsion in the cohomology under certain conditions on the parameters of the bundle $\CF$ and the prime $p$. Most of the additional difficulty in the non-compact case is related to showing that the contributions of boundary cohomology are all of Eisenstein type. We also need to show that the coverings giving rise to the diamond operators can be extended to \'etale coverings of appropriate toroidal compactifications. The result is applied to show that, when the Taylor-Wiles method applies, the congruence ideal attached to a coherent cohomological realization of an automorphic Galois representation is independent of the signatures of the hermitian form to which $G$ is attached. We also show that the Gorenstein hypothesis used to construct $p$-adic $L$-functions in \cite{EHLS} is valid under rather general hypotheses.

math.NT↗

arXiv2021

Local parameters of supercuspidal representations

For a connected reductive group $G$ over a non-archime\-dean local field $F$ of positive characteristic, Genestier and Lafforgue have attached a semisimple parameter $\CL^{ss}(\pi)$ to each irreducible representation $\pi$. Our first result shows that the Genestier-Lafforgue parameter of a tempered $\pi$ can be uniquely refined to a tempered L-parameter $\CL(\pi)$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of $\CL^{ss}(\pi)$ for unramfied $G$ and supercuspidal $\pi$ constructed by induction from an open compact (modulo center) subgroup. If $L^{ss}(\pi)$ is pure in an appropriate sense, we show that $\CL^{ss}(\pi)$ is ramified (unless $G$ is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show $\mathcal{L}^{ss}(\pi)$ is wildly ramified. The proofs are via global arguments, involving the construction of Poincar\'e series with strict control on ramification when the base curve is $\PP^1$ and a simple application of Deligne's Weil II.

math.RT↗

arXiv2020

Virtues of Priority

The conjecture that every elliptic curve with rational coefficients is a so-called modular curve -- since 2000 a theorem due in large part to Andrew Wiles and, in complete generality, to Breuil-Conrad-Diamond-Taylor -- has been known by various names: Weil Conjecture, Taniyama-Weil Conjecture, Shimura-Taniyama-Weil Conjecture, or Shimura-Taniyama Conjecture, among others. The question of the authorship of this conjecture, one of whose corollaries is Fermat's Last Theorem, has been the subject of a priority dispute that has often been quite bitter, but the principles behind one attribution or another have (almost) never been made explicit. The author proposes a reading inspired in part by the "virtue ethics" of Alasdair MacIntyre, analyzing each of the attributions as the expression of a specific value, or virtue, appreciated by the community of mathematicians.

math.HO↗

arXiv2019

Square root $p$-adic $L$-functions, I: Construction of a one-variable measure

The Ichino-Ikeda conjecture, and its generalization to unitary groups by N. Harris, has given explicit formulas for central critical values of a large class of Rankin-Selberg tensor products. Although the conjecture is not proved in full generality, there has been considerable progress, especially for $L$-values of the form $L(1/2,BC(\pi) \times BC(\pi'))$, where $\pi$ and $\pi'$ are cohomological automorphic representations of unitary groups $U(V)$ and $U(V')$, respectively. Here $V$ and $V'$ are hermitian spaces over a CM field, $V$ of dimension $n$, $V'$ of codimension $1$ in $V$, and $BC$ denotes the twisted base change to $GL(n) \times GL(n-1)$. This paper contains the first steps toward generalizing the construction of my paper with Tilouine on triple product $L$-functions to this situation. We assume $\pi$ is a holomorphic representation and $\pi'$ varies in an ordinary Hida family (of antiholomorphic forms). The construction of the measure attached to $\pi$ uses recent work of Eischen, Fintzen, Mantovan, and Varma.

math.NT↗