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Michael Harris

Publications and source records attributed to Michael Harris.

At least 37 records · Page 2Linked to original sources
arXiv2021

A local Langlands parameterization for generic supercuspidal representations of $p$-adic $G_2$

We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / \QQ_p$ we will construct a bijection \[ \CL_g : \CA^0_g(G_2,K) \rightarrow \CG^0(G_2,K) \] from the set of generic supercuspidal representations of $G_2(K)$ to the set of irreducible continuous homomorphisms $ρ: W_K \to G_2(\CC)$ with $W_K$ the Weil group of $K$. The construction of the map is simply a matter of assembling arguments that are already in the literature, together with a previously unpublished theorem of G. Savin on exceptional theta correspondences, included as an appendix. The proof that the map is a bijection is arithmetic in nature, and specifically uses automorphy lifting theorems. These can be applied thanks to a recent result of Hundley and Liu on automorphic descent from $GL(7)$ to $G_2$.

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arXiv2020

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 $L(1/2,BC(π) \times BC(π'))$, where $π$ and $π'$ 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 $π$ is a holomorphic representation and $π'$ varies in an ordinary Hida family (of antiholomorphic forms). The construction of the measure attached to $π$ uses recent work of Eischen, Fintzen, Mantovan, and Varma.

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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↗
arXiv2020

p-adic L-functions for unitary groups

This paper completes the construction of $p$-adic $L$-functions for unitary groups. More precisely, in 2006, the last three named authors proposed an approach to constructing such $p$-adic $L$-functions (Part I). Building on more recent results, including the first named author's construction of Eisenstein measures and $p$-adic differential operators, Part II of the present paper provides the calculations of local $ζ$-integrals occurring in the Euler product (including at $p$). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method.

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arXiv2019

Chern classes of automorphic vector bundles, II

We prove that the $\ell$-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over $\bar{ \mathbb{Q}}_p$, descend to classes in the $\ell$-adic cohomology of the minimal compactifications. These are invariant under the Galois group of the $p$-adic field above which the variety and the bundle are defined.

math.AG↗
arXiv2019

$\hat{G}$-local systems on smooth projective curves are potentially automorphic

Let $X$ be a smooth, projective, geometrically connected curve over a finite field $\mathbb{F}_q$, and let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Its dual group $\hat{G}$ is a split reductive group over $\mathbb{Z}$. Conjecturally, any $l$-adic $\hat{G}$-local system on $X$ (equivalently, any conjugacy class of continuous homomorphisms $π_1(X) \to \hat{G}(\bar{\mathbb{Q}}_l)$) should be associated to an everywhere unramified automorphic representation of the group $G$. We show that for any homomorphism $π_1(X) \to \hat{G}(\bar{\mathbb{Q}}_l)$ of Zariski dense image, there exists a finite Galois cover $Y \to X$ over which the associated local system becomes automorphic.

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arXiv2019

Incorrigible Representations

As a consequence of his numerical local Langlands correspondence for $GL(n)$, Henniart deduced the following theorem: If $F$ is a nonarchimedean local field and if $π$ is an irreducible admissible representation of $GL(n,F)$, then, after a finite sequence of cyclic base changes, the image of $π$ contains a vector fixed under an Iwahori subgroup. This result was indispensable in all proofs of the local Langlands correspondence. Scholze later gave a different proof, based on the analysis of nearby cycles in the cohomology of the Lubin-Tate tower. Let $G$ be a reductive group over $F$. Assuming a theory of stable cyclic base change exists for $G$, we define an incorrigible supercuspidal representation $π$ of $G(F)$ to be one with the property that, after any sequence of cyclic base changes, the image of $π$ contains a supercuspidal member. If F is of positive characteristic then we define $π$ to be pure if the Langlands parameter attached to $π$ by Genestier and Lafforgue is pure in an appropriate sense. We conjecture that no pure supercuspidal representation is incorrigible. We prove this conjecture for $GL(n)$ and for classical groups, using properties of standard $L$-functions; and we show how this gives rise to a proof of Henniart's theorem and the local Langlands correspondence for $GL(n)$ based on V. Lafforgue's Langlands parametrization, and thus independent of point-counting on Shimura or Drinfel'd modular varieties.

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arXiv2017

Derived Hecke algebra for weight one forms

We study the action of the derived Hecke algebra on the space of weight one forms. By analogy with the topological case, we formulate a conjecture relating this to a certain Stark unit. We verify the truth of the conjecture numerically, for the weight one forms of level $23$ and $31$, and many derived Hecke operators at primes less than $200$. Our computation depends in an essential way on Merel's evaluation of the pairing between the Shimura and cuspidal subgroups of $J_0(q)$.

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arXiv2017

Chern classes of automorphic vector bundles

We prove that Chern classes in continuous $\ell$-adic cohomology of automorphic bundles associated to representations of $G$ on a projective Shimura variety with data $(G,X)$ are trivial rationally. It is a consequence of Beilinson's conjectures which predict that the Chern classes in the Chow groups vanish rationally.

math.AG↗
arXiv2016

Period relations and special values of Rankin-Selberg $L$-functions

This is a survey of recent work on values of Rankin-Selberg $L$-functions of pairs of cohomological automorphic representations that are {\it critical} in Deligne's sense. The base field is assumed to be a CM field. Deligne's conjecture is stated in the language of motives over $\QQ$, and express the critical values, up to rational factors, as determinants of certain periods of algebraic differentials on a projective algebraic variety over homology classes. The results that can be proved by automorphic methods express certain critical values as (twisted) period integrals of automorphic forms. Using Langlands functoriality between cohomological automorphic representations of unitary groups, which can be identified with the de Rham cohomology of Shimura varieties, and cohomological automorphic representations of $GL(n)$, the automorphic periods can be interpreted as motivic periods. We report on recent results of the two authors, of the first-named author with Grobner, and of Guerberoff.

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arXiv2014

Whittaker rational structures and special values of the Asai $L$-function

Let $F$ be a totally real number field and $E/F$ a totally imaginary quadratic extension of $F$. Let $Π$ be a cohomological, conjugate self-dual cuspidal automorphic representation of $GL_n(\mathbb A_E)$. Under a certain non-vanishing condition we relate the residue and the value of the Asai $L$-functions at $s=1$ with rational structures obtained from the cohomologies in top and bottom degrees via the Whittaker coefficient map. This generalizes a result in Eric Urban's thesis when $n = 2$, as well as a result of the first two named authors, both in the case $F = \mathbb Q$.

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arXiv2014

On the Rigid Cohomology of Certain Shimura Varieties

We construct the compatible system of $l$-adic representations associated to a regular algebraic cuspidal automorphic representation of $GL_n$ over a CM (or totally real) field and check local-global compatibility for the $l$-adic representation away from $l$ and finite number of rational primes above which the CM field or the automorphic representation ramify. The main innovation is that we impose no self-duality hypothesis on the automorphic representation.

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arXiv2014

Whittaker periods, motivic periods, and special values of tensor product L-functions

Let $\mathcal K$ be an imaginary quadratic field. Let $Π$ and $Π'$ be irreducible generic cohomological automorphic representation of $GL(n)/{\mathcal K}$ and $GL(n-1)/{\mathcal K}$, respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, the other is given in terms of the Whittaker model. The ratio between these rational structures is called a {\it Whittaker period}. An argument presented by Mahnkopf and Raghuram shows that, at least if $Π$ is cuspidal and the weights of $Π$ and $Π'$ are in a standard relative position, the critical values of the Rankin-Selberg product $L(s,Π\times Π')$ are essentially algebraic multiples of the product of the Whittaker periods of $Π$ and $Π'$. We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal $Π$ can be given a motivic interpretation, and can also be related to a critical value of the adjoint $L$-function of related automorphic representations of unitary groups. The resulting expressions for critical values of the Rankin-Selberg and adjoint $L$-functions are compatible with Deligne's conjecture.

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arXiv2012

Testing rationality of coherent cohomology of Shimura varieties

Let $G' \subset G$ be an inclusion of reductive groups whose real points have a non-trivial discrete series. Combining ergodic methods of Burger-Sarnak and the author with a positivity argument due to Li and the classification of minimal $K$-types of discrete series, due to Salamanca-Riba, we show that, if $π$ is a cuspidal automorphic representation of $G$ whose archimedean component is a sufficiently general discrete series, then there is a cuspidal automorphic representation of $G'$, of (explicitly determined) discrete series type at infinity, that pairs non-trivially with $π$. When $G$ and $G'$ are inner forms of U(n) and $U(n-1)$, respectively, this result is used to define rationality criteria for sufficiently general coherent cohomological forms on $G$.

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arXiv2006

The Advanced Compton Telescope Mission

The Advanced Compton Telescope (ACT), the next major step in gamma-ray astronomy, will probe the fires where chemical elements are formed by enabling high-resolution spectroscopy of nuclear emission from supernova explosions. During the past two years, our collaboration has been undertaking a NASA mission concept study for ACT. This study was designed to (1) transform the key scientific objectives into specific instrument requirements, (2) to identify the most promising technologies to meet those requirements, and (3) to design a viable mission concept for this instrument. We present the results of this study, including scientific goals and expected performance, mission design, and technology recommendations.

astro-ph↗
arXiv2003

On the local Langlands correspondence

The local Langlands correspondence for GL(n) of a non-Archimedean local field $F$ parametrizes irreducible admissible representations of $GL(n,F)$ in terms of representations of the Weil-Deligne group $WD_F$ of $F$. The correspondence, whose existence for $p$-adic fields was proved in joint work of the author with R. Taylor, and then more simply by G. Henniart, is characterized by its preservation of salient properties of the two classes of representations. The article reviews the strategies of the two proofs. Both the author's proof with Taylor and Henniart's proof are global and rely ultimately on an understanding of the $\ell$-adic cohomology of a family of Shimura varieties closely related to GL(n). The author's proof with Taylor provides models of the correspondence in the cohomology of deformation spaces, introduced by Drinfeld, of certain $p$-divisible groups with level structure.

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