2 transition Eis.pdf] [2] This is an example of conversion to automorphic forms on reductive or s... more 2 transition Eis.pdf] [2] This is an example of conversion to automorphic forms on reductive or semi-simple real Lie groups, of which SL 2 (R) is a small example.
After a too-brief introduction to adeles and ideles, we sketch proof of analytic continuation and... more After a too-brief introduction to adeles and ideles, we sketch proof of analytic continuation and functional equation of Riemann’s zeta, in the modern form due independently to Iwasawa and Tate about 1950. The sketch is repeated for Dedekind zeta functions of number fields, noting some additional complications. The sketch is repeated again for Hecke’s (größencharakter) L-functions, noting further complications.
We obtain second integral moments of automorphic L–functions on adele groups GL2 over arbitrary n... more We obtain second integral moments of automorphic L–functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires complete reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(Q) or GL2(Q(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression 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 on moments.
indispensable stuff with few prerequisites • Recollection of some definitions • Irreducibility of... more indispensable stuff with few prerequisites • Recollection of some definitions • Irreducibility of unramified principal series: GL(n) • Irreducibility of unramified principal series: Sp(n) • Relation to spherical representations • Intertwinings of unramified principal series: GL(n) An important but seldom-heard remark is that there is essentially just one irreducible representation of a p-adic group. Thus, rather than opening a Pandora's box of disastrous complication, introduction of representation-theoretic methods in study of automorphic forms and L-functions is exactly the right thing: to the extent that there can be simplicity here, it is in the context of representation-theory. By the claim that there is 'just one' representation I mean that 'almost all the time' in the study of automorphic forms one considers only spherical representations, which miraculously are (essentially) unramified principle series representations, which by definition are the simplest induced representations, and which are described via structure constants depending upon a finite list of complex parameters. The point is that in terms of these parameters it makes sense to talk about 'generic' unramified principal series representations, so there is just one. (Making the idea of 'generic' precise is part of the point of [Garrett 1994]. One should be aware that in representation theory proper the phrase 'generic representation' sometimes has a different meaning: it sometimes means that the representation 'has a Whittaker model', which is to say that it imbeds in a certain induced representation whose definition is descended from study of Fourier coefficients of automorphic forms. The above use of the word 'generic' is, instead, in the spirit of algebraic geometry.) Further, not only do all the unramified principal series fit together into one family, but the fact that unramified principal series are induced makes available many standard techniques, especially Frobenius Reciprocity and further methods involving orbit filtrations and other 'physical' reasoning.
Number Theory, Trace Formulas and Discrete Groups, 1989
Publisher Summary This chapter illustrates the integral representations of Eisenstein series and ... more Publisher Summary This chapter illustrates the integral representations of Eisenstein series and the L-functions. The idea of obtaining integral formulas by integrating against some sort of Eisenstein series (or restriction of Eisenstein series) is not new. The “Rankin-Selberg method” is one idea that has interesting analytic and arithmetic consequences. A different use of Eisenstein series has been considered, obtaining not only integral representations of some L-functions, but also the integral representations of complicated Eisenstein series in terms of simpler ones. For the case of holomorphic cuspforms, the local integrals at infinite primes are expressible in terms of gamma functions. It seems to be an open problem to determine the precise nature of these Archimedean integrals in general.
... above, (Tf)(g) = ∫ G f(gh) η(h) dh = πpq · (E-rational number) · f(g) Page 6. 130 PaulGarrett... more ... above, (Tf)(g) = ∫ G f(gh) η(h) dh = πpq · (E-rational number) · f(g) Page 6. 130 PaulGarrett Proof. For computational purposes, we would like f to be in L2(G), allowing use of the Harish-Chandra decomposition as for η. However ...
We consider integrals of cuspforms f on reductive groups G defined over numberfields k against re... more We consider integrals of cuspforms f on reductive groups G defined over numberfields k against restrictions ι * E of Eisenstein series E on "larger" reductive groupsG over k via imbeddings ι : G →G. We give hypotheses sufficient to assure that such global integrals have Euler products. At good primes, the local factors are shown to be rational functions in the corresponding parameters q −s from the Eisenstein series and in the Satake parameters q −si coming from the spherical representations locally generated by the cuspform. The denominators of the Euler factors at good primes are estimated in terms of "anomalous" intertwining operators, computable via orbit filtrations on test functions. The standard intertwining operators (attached to elements of the (spherical) Weyl group) among these unramified principal series yield symmetries among the anomalous intertwining operators, thereby both sharpening the orbit-filtration estimate on the denominator and implying corresponding symmetry in it. Finally, we note a very simple dimension-counting heuristic for fulfillment of our hypotheses, thereby giving a simple test to exclude configurations ι : G →G, E, as candidates for Euler product factorization. Some simple examples illustrate the application of these ideas.
Proceedings of the American Mathematical Society, 1986
In this note we show that given any real analytic function u u in R n {{\mathbf {R}}^n} , there e... more In this note we show that given any real analytic function u u in R n {{\mathbf {R}}^n} , there exists some p > 1 p > 1 for which | u | \left | u \right | is locally an A p {A_p} -weight of Muckenhoupt.
Journal of the Institute of Mathematics of Jussieu, 2009
We break the convexity bound in the t-aspect for L-functions attached to cuspforms f for GL 2 (k)... more We break the convexity bound in the t-aspect for L-functions attached to cuspforms f for GL 2 (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 [Di-Ga].
Journal of the Institute of Mathematics of Jussieu, 2008
We obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary n... more We obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(ℚ) or GL2(ℚ(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression of integral moments in adele-group 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 on moments.
Ikeda lifts form a distinguished subspace of Siegel modular forms. In this paper we prove several... more Ikeda lifts form a distinguished subspace of Siegel modular forms. In this paper we prove several global and local results concerning this space. We find that degenerate principal series representations (for the Siegel parabolic) of the symplectic group Sp 2n of even degree satisfy a Hecke duality relation which has applications to Ikeda lifts and leads to converse theorems. Moreover we apply certain differential operators to study pullbacks of Ikeda lifts.
Let f and ~o be holomorphic Siegel-Hilbert cuspforms on Sp.(&) of weight (x ..... x) with x > 2n ... more Let f and ~o be holomorphic Siegel-Hilbert cuspforms on Sp.(&) of weight (x ..... x) with x > 2n + 1, and suppose thatfis a Hecke eigenfunction at almost all primes. Let a ~ Aut(C/Q). The main result here is the equivariance property of Petersson inner products
2 transition Eis.pdf] [2] This is an example of conversion to automorphic forms on reductive or s... more 2 transition Eis.pdf] [2] This is an example of conversion to automorphic forms on reductive or semi-simple real Lie groups, of which SL 2 (R) is a small example.
After a too-brief introduction to adeles and ideles, we sketch proof of analytic continuation and... more After a too-brief introduction to adeles and ideles, we sketch proof of analytic continuation and functional equation of Riemann’s zeta, in the modern form due independently to Iwasawa and Tate about 1950. The sketch is repeated for Dedekind zeta functions of number fields, noting some additional complications. The sketch is repeated again for Hecke’s (größencharakter) L-functions, noting further complications.
We obtain second integral moments of automorphic L–functions on adele groups GL2 over arbitrary n... more We obtain second integral moments of automorphic L–functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires complete reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(Q) or GL2(Q(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression 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 on moments.
indispensable stuff with few prerequisites • Recollection of some definitions • Irreducibility of... more indispensable stuff with few prerequisites • Recollection of some definitions • Irreducibility of unramified principal series: GL(n) • Irreducibility of unramified principal series: Sp(n) • Relation to spherical representations • Intertwinings of unramified principal series: GL(n) An important but seldom-heard remark is that there is essentially just one irreducible representation of a p-adic group. Thus, rather than opening a Pandora's box of disastrous complication, introduction of representation-theoretic methods in study of automorphic forms and L-functions is exactly the right thing: to the extent that there can be simplicity here, it is in the context of representation-theory. By the claim that there is 'just one' representation I mean that 'almost all the time' in the study of automorphic forms one considers only spherical representations, which miraculously are (essentially) unramified principle series representations, which by definition are the simplest induced representations, and which are described via structure constants depending upon a finite list of complex parameters. The point is that in terms of these parameters it makes sense to talk about 'generic' unramified principal series representations, so there is just one. (Making the idea of 'generic' precise is part of the point of [Garrett 1994]. One should be aware that in representation theory proper the phrase 'generic representation' sometimes has a different meaning: it sometimes means that the representation 'has a Whittaker model', which is to say that it imbeds in a certain induced representation whose definition is descended from study of Fourier coefficients of automorphic forms. The above use of the word 'generic' is, instead, in the spirit of algebraic geometry.) Further, not only do all the unramified principal series fit together into one family, but the fact that unramified principal series are induced makes available many standard techniques, especially Frobenius Reciprocity and further methods involving orbit filtrations and other 'physical' reasoning.
Number Theory, Trace Formulas and Discrete Groups, 1989
Publisher Summary This chapter illustrates the integral representations of Eisenstein series and ... more Publisher Summary This chapter illustrates the integral representations of Eisenstein series and the L-functions. The idea of obtaining integral formulas by integrating against some sort of Eisenstein series (or restriction of Eisenstein series) is not new. The “Rankin-Selberg method” is one idea that has interesting analytic and arithmetic consequences. A different use of Eisenstein series has been considered, obtaining not only integral representations of some L-functions, but also the integral representations of complicated Eisenstein series in terms of simpler ones. For the case of holomorphic cuspforms, the local integrals at infinite primes are expressible in terms of gamma functions. It seems to be an open problem to determine the precise nature of these Archimedean integrals in general.
... above, (Tf)(g) = ∫ G f(gh) η(h) dh = πpq · (E-rational number) · f(g) Page 6. 130 PaulGarrett... more ... above, (Tf)(g) = ∫ G f(gh) η(h) dh = πpq · (E-rational number) · f(g) Page 6. 130 PaulGarrett Proof. For computational purposes, we would like f to be in L2(G), allowing use of the Harish-Chandra decomposition as for η. However ...
We consider integrals of cuspforms f on reductive groups G defined over numberfields k against re... more We consider integrals of cuspforms f on reductive groups G defined over numberfields k against restrictions ι * E of Eisenstein series E on "larger" reductive groupsG over k via imbeddings ι : G →G. We give hypotheses sufficient to assure that such global integrals have Euler products. At good primes, the local factors are shown to be rational functions in the corresponding parameters q −s from the Eisenstein series and in the Satake parameters q −si coming from the spherical representations locally generated by the cuspform. The denominators of the Euler factors at good primes are estimated in terms of "anomalous" intertwining operators, computable via orbit filtrations on test functions. The standard intertwining operators (attached to elements of the (spherical) Weyl group) among these unramified principal series yield symmetries among the anomalous intertwining operators, thereby both sharpening the orbit-filtration estimate on the denominator and implying corresponding symmetry in it. Finally, we note a very simple dimension-counting heuristic for fulfillment of our hypotheses, thereby giving a simple test to exclude configurations ι : G →G, E, as candidates for Euler product factorization. Some simple examples illustrate the application of these ideas.
Proceedings of the American Mathematical Society, 1986
In this note we show that given any real analytic function u u in R n {{\mathbf {R}}^n} , there e... more In this note we show that given any real analytic function u u in R n {{\mathbf {R}}^n} , there exists some p > 1 p > 1 for which | u | \left | u \right | is locally an A p {A_p} -weight of Muckenhoupt.
Journal of the Institute of Mathematics of Jussieu, 2009
We break the convexity bound in the t-aspect for L-functions attached to cuspforms f for GL 2 (k)... more We break the convexity bound in the t-aspect for L-functions attached to cuspforms f for GL 2 (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 [Di-Ga].
Journal of the Institute of Mathematics of Jussieu, 2008
We obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary n... more We obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(ℚ) or GL2(ℚ(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression of integral moments in adele-group 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 on moments.
Ikeda lifts form a distinguished subspace of Siegel modular forms. In this paper we prove several... more Ikeda lifts form a distinguished subspace of Siegel modular forms. In this paper we prove several global and local results concerning this space. We find that degenerate principal series representations (for the Siegel parabolic) of the symplectic group Sp 2n of even degree satisfy a Hecke duality relation which has applications to Ikeda lifts and leads to converse theorems. Moreover we apply certain differential operators to study pullbacks of Ikeda lifts.
Let f and ~o be holomorphic Siegel-Hilbert cuspforms on Sp.(&) of weight (x ..... x) with x > 2n ... more Let f and ~o be holomorphic Siegel-Hilbert cuspforms on Sp.(&) of weight (x ..... x) with x > 2n + 1, and suppose thatfis a Hecke eigenfunction at almost all primes. Let a ~ Aut(C/Q). The main result here is the equivariance property of Petersson inner products
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