Author

# Paul Garrett

Bio: Paul Garrett is an academic researcher from University of Minnesota. The author has contributed to research in topics: Eisenstein series & Automorphic form. The author has an hindex of 7, co-authored 23 publications receiving 526 citations.

##### Papers

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TL;DR: In this article, it was shown that the space of holomorphic cuspforms of such weight and with respect to a principal congruence subyroup is spanned by rational Fourier coefficients (for weights as above).

Abstract: The latter is a sort of 'L-indistinguishability' result concerning the (presumably transcendental) Petersson norms-squared of Hecke eigenfunctions. The general assertion is essential in discussion of special values of L-functions obtained as inner products. Incidental to the proof of this fact, we obtain a very short proof of the important (known) result that the space of holomorphic cuspforms of such weight and with respect to a principal congruence subyroup is spanned by those with rational Fourier coefficients (for weights as above). Rather than starting from the theory of canonical models, we begin with consideration of the arithmetic of the Fourier coefficients of Siegel's Eisenstein series: this is a relatively elementary issue, amenable to direct calculation (although, ironically, the best reference currently available seems to be [H3], wherein canonical models results are invoked). By contrast, previous proofs of results concerning rationality properties of Fourier coefficients have relied essentially upon the theory of canonical models: the paradigms are the two papers [Sh2] and [Sh3] of Shimura, which depend upon the sequence of his papers culminating in [Shl] which developed the necessary theory of canonical models.

48 citations

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TL;DR: In this article, the convexity bound in the t-aspect for L-functions attached to cusp forms f for GL2(k) over arbitrary number fields k was broken.

Abstract: We break the convexity bound in the t-aspect for L-functions attached to cusp forms f for GL2(k) over arbitrary number fields k. The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twists L(s,f⊗χ) by Grossencharacters χ, from our previous paper on integral moments.

26 citations

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TL;DR: In this article, the second integral moments of GL2 automorphic L-functions over an arbitrary number field are presented in a form enabling application of the structure of adele groups and their representation theory.

Abstract: This paper exposes the underlying mechanism for obtaining second integral moments of GL2 automorphic L–functions over an arbitrary number field. Here, moments for GL2 are presented in a form enabling application of the structure of adele groups and their representation theory. To the best of our knowledge, this is the first formulation of integral moments in adele-group-theoretic terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers Q, we recover the classical results. §

20 citations

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01 Sep 2018TL;DR: A self-contained introduction to the theory and application of automorphic forms, using examples to illustrate several critical analytical concepts surrounding and supporting the theory of automomorphic forms is given in this paper.

Abstract: This is Volume 1 of a two-volume book that provides a self-contained introduction to the theory and application of automorphic forms, using examples to illustrate several critical analytical concepts surrounding and supporting the theory of automorphic forms. The two-volume book treats three instances, starting with some small unimodular examples, followed by adelic GL2, and finally GLn. Volume 1 features critical results, which are proven carefully and in detail, including discrete decomposition of cuspforms, meromorphic continuation of Eisenstein series, spectral decomposition of pseudo-Eisenstein series, and automorphic Plancherel theorem. Volume 2 features automorphic Green's functions, metrics and topologies on natural function spaces, unbounded operators, vector-valued integrals, vector-valued holomorphic functions, and asymptotics. With numerous proofs and extensive examples, this classroom-tested introductory text is meant for a second-year or advanced graduate course in automorphic forms, and also as a resource for researchers working in automorphic forms, analytic number theory, and related fields.

13 citations

##### Cited by

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TL;DR: There is, I think, something ethereal about i —the square root of minus one, which seems an odd beast at that time—an intruder hovering on the edge of reality.

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.
Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

33,785 citations

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TL;DR: In this article, the authors introduced and investigated the total graph of R, denoted by T ( Γ ( R ) ), which is the (undirected) graph with all elements of R as vertices.

290 citations

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TL;DR: In this article, the authors solve the subconvexity problem for the L-functions of GL-1 and GL-2 automorphic representations over a fixed number field, uniformly in all aspects.

Abstract: Generalizing and unifying prior results, we solve the subconvexity problem for the L-functions of GL 1 and GL 2 automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino–Ikeda.

286 citations

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TL;DR: In this paper, strongly Gorenstein projective, injective, and flat modules are studied, which they call strongly GORNEINSTEIN projective and injective.

162 citations

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TL;DR: In this article, a commutative ring R and a proper ideal I ⊂ R were constructed and a new ring denoted by R⋈I was studied.

161 citations