%% Books
@book{fds320415,
Author = {Getz, J and Goresky, M},
Title = {Hilbert modular forms with coefficients in intersection
homology and quadratic base change},
Volume = {298},
Pages = {1-256},
Booktitle = {Progress in Mathematics},
Publisher = {Springer Basel},
Year = {2012},
Month = {January},
ISBN = {9783034803502},
url = {http://dx.doi.org/10.1007/978-3-0348-0351-9},
Abstract = {In the 1970s Hirzebruch and Zagier produced elliptic modular
forms with coefficients in the homology of a Hilbert modular
surface. They then computed the Fourier coefficients of
these forms in terms of period integrals and L-functions. In
this book the authors take an alternate approach to these
theorems and generalize them to the setting of Hilbert
modular varieties of arbitrary dimension. The approach is
conceptual and uses tools that were not available to
Hirzebruch and Zagier, including intersection homology
theory, properties of modular cycles, and base change.
Automorphic vector bundles, Hecke operators and Fourier
coefficients of modular forms are presented both in the
classical and adèlic settings. The book should provide a
foundation for approaching similar questions for other
locally symmetric spaces.},
Doi = {10.1007/978-3-0348-0351-9},
Key = {fds320415}
}
@book{fds328604,
Author = {Getz, J and Goresky, M},
Title = {Introduction},
Volume = {298},
Pages = {1-19},
Publisher = {Springer Basel},
Year = {2012},
Month = {January},
ISBN = {9783034803502},
url = {http://dx.doi.org/10.1007/978-3-0348-0351-9_1},
Abstract = {In their seminal paper [Hirz] on the intersection theory of
Hilbert modular surfaces, F. Hirzebruch and D. Zagier
mentioned that the motivation for their work was to explain
an observation of J.-P. Serre.},
Doi = {10.1007/978-3-0348-0351-9_1},
Key = {fds328604}
}
%% Papers Published
@article{fds373482,
Author = {Getz, J and Hsu, C-H and Leslie, S},
Title = {Harmonic analysis on certain spherical varieties},
Journal = {Journal of the European Mathematical Society},
Publisher = {EMS Press},
Year = {2023},
Month = {October},
url = {http://dx.doi.org/10.4171/JEMS/1381},
Doi = {10.4171/JEMS/1381},
Key = {fds373482}
}
@article{fds348383,
Author = {Getz, JR and Liu, B},
Title = {A refined Poisson summation formula for certain
Braverman-Kazhdan spaces},
Journal = {Science China Mathematics},
Volume = {64},
Number = {6},
Pages = {1127-1156},
Year = {2021},
Month = {June},
url = {http://dx.doi.org/10.1007/s11425-018-1616-0},
Abstract = {Braverman and Kazhdan (2000) introduced influential
conjectures aimed at generalizing the Fourier transform and
the Poisson summation formula. Their conjectures should
imply that quite general Langlands L-functions have
meromorphic continuations and functional equations as
predicted by Langlands’ functoriality conjecture. As an
evidence for their conjectures, Braverman and Kazhdan (2002)
considered a setting related to the so-called doubling
method in a later paper and proved the corresponding Poisson
summation formula under restrictive assumptions on the
functions involved. The connection between the two papers is
made explicit in the work of Li (2018). In this paper, we
consider a special case of the setting in Braverman and
Kazhdan’s later paper and prove a refined Poisson
summation formula that eliminates the restrictive
assumptions of that paper. Along the way we provide analytic
control on the Schwartz space we construct; this analytic
control was conjectured to hold (in a slightly different
setting) in the work of Braverman and Kazhdan
(2002).},
Doi = {10.1007/s11425-018-1616-0},
Key = {fds348383}
}
@article{fds353052,
Author = {Getz, JR},
Title = {A summation formula for the Rankin-Selberg monoid and a
nonabelian trace formula},
Journal = {American Journal of Mathematics},
Volume = {142},
Number = {5},
Pages = {1371-1407},
Year = {2020},
Month = {October},
url = {http://dx.doi.org/10.1353/ajm.2020.0035},
Abstract = {Let F be a number field and let AF be its ring of adeles.
Let B be a quaternion algebra over F and let ν: B → F be
the reduced norm. Consider the reductive monoid M over F
whose points in an F-algebra R are given by (Formula
Presented). Motivated by an influential conjecture of
Braverman and Kazhdan we prove a summation formula analogous
to the Poisson summation formula for certain spaces of
functions on the monoid. As an application, we define new
zeta integrals for the Rankin-Selberg L-function and prove
their basic properties. We also use the formula to prove a
nonabelian twisted trace formula, that is, a trace formula
whose spectral side is given in terms of automorphic
representations of the unit group of M that are isomorphic
(up to a twist by a character) to their conjugates under a
simple nonabelian Galois group.},
Doi = {10.1353/ajm.2020.0035},
Key = {fds353052}
}
@article{fds341875,
Author = {Getz, JR and Liu, B},
Title = {A summation formula for triples of quadratic
spaces},
Journal = {Advances in Mathematics},
Volume = {347},
Pages = {150-191},
Year = {2019},
Month = {April},
url = {http://dx.doi.org/10.1016/j.aim.2019.02.023},
Abstract = {Let V 1 ,V 2 ,V 3 be a triple of even dimensional vector
spaces over a number field F equipped with nondegenerate
quadratic forms Q 1 ,Q 2 ,Q 3 , respectively. Let
Y⊂∏i=1V i be the closed subscheme consisting of (v 1 ,v
2 ,v 3 ) on which Q 1 (v 1 )=Q 2 (v 2 )=Q 3 (v 3 ).
Motivated by conjectures of Braverman and Kazhdan and
related work of Lafforgue, Ngô and Sakellaridis we prove an
analogue of the Poisson summation formula for certain
functions on this space.},
Doi = {10.1016/j.aim.2019.02.023},
Key = {fds341875}
}
@article{fds340244,
Author = {Getz, JR},
Title = {Secondary terms in asymptotics for the number of zeros of
quadratic forms over number fields},
Journal = {Journal of the London Mathematical Society},
Volume = {98},
Number = {2},
Pages = {275-305},
Publisher = {Wiley},
Year = {2018},
Month = {April},
url = {http://dx.doi.org/10.1112/jlms.12130},
Abstract = {Let Q be a nondegenerate quadratic form on a vector space V
of even dimension n over a number field F. Via the circle
method or automorphic methods one can give good estimates
for smoothed sums over the number of zeros of the quadratic
form whose coordinates are of size at most X (properly
interpreted). For example, when F = Q and dim V >4
Heath-Brown has given an asymptotic of the form: c1Xn−2 +
OQ,ε,f (Xn/2+ε), for any ε > 0. Here c1 ∈ C and f ∈
S(V (R)) is a smoothing function. We refine Heath-Brown's
work to give an asymptotic of the form: c1Xn−2 + c2Xn/2 +
OQ,ε,f (Xn/2+ε−1) over any number field. Here c2 ∈ C.
Interestingly the secondary term c2 is the sum of a rapidly
decreasing function on V (R) over the zeros of Q∨, the
form whose matrix is inverse to the matrix of Q. We also
prove analogous results in the boundary case n=4,
generalizing and refining Heath-Brown's work in the case F =
Q.},
Doi = {10.1112/jlms.12130},
Key = {fds340244}
}
@article{fds320413,
Author = {Getz, JR},
Title = {Nonabelian fourier transforms for spherical
representations},
Journal = {Pacific Journal of Mathematics},
Volume = {294},
Number = {2},
Pages = {351-373},
Publisher = {Mathematical Sciences Publishers},
Year = {2018},
Month = {January},
url = {arXiv:1506.09128},
Abstract = {Braverman and Kazhdan have introduced an influential
conjecture on local functional equations for general
Langlands L-functions. It is related to L. Lafforgue's
equally influential conjectural construction of kernels for
functorial transfers. We formulate and prove a version of
Braverman and Kazhdan's conjecture for spherical
representations over an archimedean field that is suitable
for application to the trace formula. We then give a global
application related to Langlands' beyond endoscopy proposal.
It is motivated by Ngô's suggestion that one combine
nonabelian Fourier transforms with the trace formula in
order to prove the functional equations of Langlands
L-functions in general.},
Doi = {10.2140/pjm.2018.294.351},
Key = {fds320413}
}
@article{fds320411,
Author = {Getz, JR},
Title = {A four-variable automorphic kernel function},
Journal = {Research in Mathematical Sciences},
Volume = {3},
Number = {1},
Publisher = {Springer Nature},
Year = {2016},
Month = {December},
url = {http://dx.doi.org/10.1186/s40687-016-0069-6},
Abstract = {Let F be a number field, let AF be its ring of adeles, and
let g1, g2, h1, h2∈ GL 2(AF). We provide an absolutely
convergent geometric expression for ∑πKπ(g1,g2)Kπ∨(h1,h2)Ress=1LS(s,π×π∨),where
the sum is over isomorphism classes of cuspidal automorphic
representations π of GL 2(AF). Here Kπ is the typical
kernel function representing the action of a test function
on the space of π.},
Doi = {10.1186/s40687-016-0069-6},
Key = {fds320411}
}
@article{fds320412,
Author = {Getz, JR and Herman, PE},
Title = {A nonabelian trace formula},
Journal = {Research in Mathematical Sciences},
Volume = {2},
Number = {1},
Publisher = {Springer Nature},
Year = {2015},
Month = {December},
url = {http://dx.doi.org/10.1186/s40687-015-0025-x},
Abstract = {Let E/F be an everywhere unramified extension of number
fields with Gal(E/F) simple and nonabelian. In a recent
paper, the first named author suggested an approach to
nonsolvable base change and descent of automorphic
representations of GL2 along such an extension. Motivated by
this, we prove a trace formula whose spectral side is a
weighted sum over cuspidal automorphic representations of
GL2 (AE) that are isomorphic to their Gal(E/F)-conjugates.
The basic method, which is of interest in itself, is to use
functions in a space isolated by Finis, Lapid, and Müller
to build more variables into the trace formula.},
Doi = {10.1186/s40687-015-0025-x},
Key = {fds320412}
}
@article{fds292889,
Author = {Getz, JR and Hahn, H},
Title = {A general simple relative trace formula},
Journal = {Pacific Journal of Mathematics},
Volume = {277},
Number = {1},
Pages = {99-118},
Publisher = {Mathematical Sciences Publishers},
Year = {2015},
Month = {January},
ISSN = {0030-8730},
url = {http://dx.doi.org/10.2140/pjm.2015.277.99},
Abstract = {In this paper we prove a relative trace formula for all
pairs of connected algebraic groups H ≤ G × G, with G a
reductive group and H the direct product of a reductive
group and a unipotent group, given that the test function
satisfies simplifying hypotheses. As an application, we
prove a relative analogue of the Weyl law, giving an
asymptotic formula for the number of eigenfunctions of the
Laplacian on a locally symmetric space associated to G
weighted by their L<sup>2</sup>-restriction norm over a
locally symmetric subspace associated to H<inf>0</inf> ≤
G.},
Doi = {10.2140/pjm.2015.277.99},
Key = {fds292889}
}
@article{fds292890,
Author = {Getz, JR and Klassen, J},
Title = {Isolating rankin-selberg lifts},
Journal = {Proceedings of the American Mathematical
Society},
Volume = {143},
Number = {8},
Pages = {3319-3329},
Year = {2015},
Month = {January},
ISSN = {0002-9939},
url = {http://dx.doi.org/10.1090/proc/12389},
Abstract = {Let F be a number field and let π be a cuspidal unitary
automorphic representation of GL<inf>mn</inf>(A<inf>F</inf>)
where m and n are integers greater than one. We propose a
conjecturally necessary condition for π to be a
Rankin-Selberg transfer of an automorphic representation of
GL<inf>m</inf> × GL<inf>n</inf>(A<inf>F</inf>). As evidence
for the conjecture we prove the corresponding statement
about automorphic L-parameters.},
Doi = {10.1090/proc/12389},
Key = {fds292890}
}
@article{fds292892,
Author = {Getz, JR and Hahn, H},
Title = {Algebraic cycles and tate classes on hilbert modular
varieties},
Journal = {International Journal of Number Theory},
Volume = {10},
Number = {1},
Pages = {161-176},
Year = {2014},
Month = {February},
ISSN = {1793-0421},
url = {http://dx.doi.org/10.1142/S1793042113500875},
Abstract = {Let E/ be a totally real number field that is Galois over ,
and let be a cuspidal, nondihedral automorphic
representation of GL2(E) that is in the lowest weight
discrete series at every real place of E. The representation
cuts out a motive Mét(π∞) from the ℓ-adic middle
degree intersection cohomology of an appropriate Hilbert
modular variety. If ℓ is sufficiently large in a sense
that depends on π we compute the dimension of the space of
Tate classes in M ét(π∞). Moreover if the space of Tate
classes on this motive over all finite abelian extensions
k/E is at most of rank one as a Hecke module, we prove that
the space of Tate classes in M ét(π∞) is spanned by
algebraic cycles. © 2014 World Scientific Publishing
Company.},
Doi = {10.1142/S1793042113500875},
Key = {fds292892}
}
@article{fds320416,
Author = {Getz, JR and Wambach, E},
Title = {Twisted relative trace formulae with a view towards unitary
groups},
Journal = {American Journal of Mathematics},
Volume = {136},
Number = {1},
Pages = {1-57},
Publisher = {Johns Hopkins University Press: American Journal of
Mathematics},
Year = {2014},
Month = {January},
url = {http://dx.doi.org/10.1353/ajm.2014.0002},
Abstract = {We introduce a twisted relative trace formula which
simultaneously generalizes the twisted trace formula of
Langlands et. al. (in the quadratic case) and the relative
trace formula of Jacquet and Lai [JL]. Certain matching
statements relating this twisted relative trace formula to a
relative trace formula are also proven (including the
relevant fundamental lemma in the "biquadratic case"). Using
recent work of Jacquet, Lapid and their collaborators [J1]
and the Rankin-Selberg integral representation of the Asai
L-function (obtained by Flicker using the theory of Jacquet,
Piatetskii-Shapiro, and Shalika [Fl2]), we give the
following application: Let E/F be a totally real quadratic
extension, let U ' be a quasi-split unitary group with
respect to a CM extension M/F, and let U := U'_E . Under
suitable local hypotheses, we show that a cuspidal
cohomological automorphic representation of U whose Asai
L-function has a pole at the edge of the critical strip is
nearly equivalent to a cuspidal cohomological automorphic
representation 0 of U that is U '-distinguished in the sense
that there is a form in the space of 0 admitting a nonzero
period over U . This provides cohomologically nontrivial
cycles of middle dimension on unitary Shimura varieties
analogous to those on Hilbert modular surfaces studied by
Harder, Langlands, and Rapoport [HLR].},
Doi = {10.1353/ajm.2014.0002},
Key = {fds320416}
}
@article{fds328597,
Author = {Getz, J and Goresky, M},
Title = {Eisenstein series with coefficients in intersection
homology},
Volume = {298},
Pages = {179-182},
Booktitle = {Progress in Mathematics},
Publisher = {Springer Basel},
Year = {2012},
Month = {January},
ISBN = {9783034803502},
url = {http://dx.doi.org/10.1007/978-3-0348-0351-9_11},
Abstract = {Thus far we have ignored classes in (Formula presented.) and
their Poincaré duals in intersection homology. We now take
up the study of these classes.},
Doi = {10.1007/978-3-0348-0351-9_11},
Key = {fds328597}
}
@article{fds328598,
Author = {Getz, J and Goresky, M},
Title = {Generalities on Hilbert modular forms and
varieties},
Volume = {298},
Pages = {57-89},
Booktitle = {Progress in Mathematics},
Year = {2012},
Month = {January},
url = {http://dx.doi.org/10.1007/978-3-0348-0351-9_5},
Abstract = {In this chapter we collect some known facts on Hilbert
modular forms and varieties, mostly for the purpose of
fixing our notation. These concepts will be used in Chapter
7, where we will recall the description of the intersection
cohomology of Hilbert modular varieties in terms of Hilbert
modular forms.},
Doi = {10.1007/978-3-0348-0351-9_5},
Key = {fds328598}
}
@article{fds328599,
Author = {Getz, J and Goresky, M},
Title = {The full version of theorem 1.3},
Volume = {298},
Pages = {167-177},
Booktitle = {Progress in Mathematics},
Publisher = {Springer Basel},
Year = {2012},
Month = {January},
ISBN = {9783034803502},
url = {http://dx.doi.org/10.1007/978-3-0348-0351-9_10},
Abstract = {We use the same setup as in the previous chapter: L/E is a
quadratic extension of totally real fields Gal(L/E) = 〈1,
ς〉 Gal(L/E)^ = 〈1, η〉, c ⊂ OL is an ideal, and
n:=[L:Q], d:=dL/E, D := DL/E cE := c∩OE.},
Doi = {10.1007/978-3-0348-0351-9_10},
Key = {fds328599}
}
@article{fds328600,
Author = {Getz, J and Goresky, M},
Title = {Automorphic vector bundles and local systems},
Volume = {298},
Pages = {91-110},
Booktitle = {Progress in Mathematics},
Publisher = {Springer Basel},
Year = {2012},
Month = {January},
ISBN = {9783034803502},
url = {http://dx.doi.org/10.1007/978-3-0348-0351-9_6},
Abstract = {In this section we begin with the general theory of local
systems, automorphic vector bundles, and automorphy factors.
After describing the finite-dimensional representation
theory of GL2 we determine the explicit equations relating
modular forms and differential forms.},
Doi = {10.1007/978-3-0348-0351-9_6},
Key = {fds328600}
}
@article{fds328601,
Author = {Getz, J and Goresky, M},
Title = {Review of chains and cochains},
Volume = {298},
Pages = {21-28},
Booktitle = {Progress in Mathematics},
Publisher = {Springer Basel},
Year = {2012},
Month = {January},
ISBN = {9783034803502},
url = {http://dx.doi.org/10.1007/978-3-0348-0351-9_2},
Abstract = {Recall (e.g.,[Hud ]) that a closed convex linear cell is the
convex hull of finitely many points in Euclidean space. A
convex linear cell complex K is a finite collection of
closed convex linear cells in some ℝN such that if σ ∈
K then every face of σ is in K, and if σ, τ ∈ K then
the intersection σ ∩ τ is in K.},
Doi = {10.1007/978-3-0348-0351-9_2},
Key = {fds328601}
}
@article{fds328602,
Author = {Getz, J and Goresky, M},
Title = {The automorphic description of intersection
cohomology},
Volume = {298},
Pages = {111-134},
Booktitle = {Progress in Mathematics},
Publisher = {Springer Basel},
Year = {2012},
Month = {January},
ISBN = {9783034803502},
url = {http://dx.doi.org/10.1007/978-3-0348-0351-9_7},
Abstract = {In this chapter we use Proposition 6.6 to construct a map
Hilbert modular forms ⟶ intersection cohomology which
takes weight, nebentypus ⟶ local coefficient system Hecke
operator ⟶ action of Hecke correspondence Petersson
product ⟶ intersection product.},
Doi = {10.1007/978-3-0348-0351-9_7},
Key = {fds328602}
}
@article{fds328603,
Author = {Getz, J and Goresky, M},
Title = {Explicit construction of cycles},
Volume = {298},
Pages = {151-166},
Booktitle = {Progress in Mathematics},
Publisher = {Springer Basel},
Year = {2012},
Month = {January},
ISBN = {9783034803502},
url = {http://dx.doi.org/10.1007/978-3-0348-0351-9_9},
Abstract = {In this chapter we consider a quadratic extension L/E of
totally real fields. The inclusion E → L gives rise to
Hilbert modular subvarieties, known as Hirzebruch-Zagier
cycles, Z ⊂ Y with dim(Y ) = 2 dim(Z).},
Doi = {10.1007/978-3-0348-0351-9_9},
Key = {fds328603}
}
@article{fds328605,
Author = {Getz, J and Goresky, M},
Title = {Review of intersection homology and cohomology},
Volume = {298},
Pages = {29-39},
Booktitle = {Progress in Mathematics},
Publisher = {Springer Basel},
Year = {2012},
Month = {January},
ISBN = {9783034803502},
url = {http://dx.doi.org/10.1007/978-3-0348-0351-9_3},
Abstract = {In this chapter we recall the relation between intersection
homology,constructed using (p, i)-allowable chains as in
[Gre4],an d intersection cohomology, constructed via sheaf
theory.},
Doi = {10.1007/978-3-0348-0351-9_3},
Key = {fds328605}
}
@article{fds328606,
Author = {Getz, J and Goresky, M},
Title = {Review of arithmetic quotients},
Volume = {298},
Pages = {41-55},
Booktitle = {Progress in Mathematics},
Publisher = {Springer Basel},
Year = {2012},
Month = {January},
ISBN = {9783034803502},
url = {http://dx.doi.org/10.1007/978-3-0348-0351-9_4},
Abstract = {Possible references for the geometry described in this
section include [BoS,Sa2, Gre6,Gr e3,Gr e2].},
Doi = {10.1007/978-3-0348-0351-9_4},
Key = {fds328606}
}
@article{fds328607,
Author = {Getz, J and Goresky, M},
Title = {Hilbert modular forms with coefficients in a Hecke
module},
Volume = {298},
Pages = {135-150},
Booktitle = {Progress in Mathematics},
Publisher = {Springer Basel},
Year = {2012},
Month = {January},
ISBN = {9783034803502},
url = {http://dx.doi.org/10.1007/978-3-0348-0351-9_8},
Abstract = {The goal of this chapter is to prove Theorems 8.4 and 8.5,t
he full versions of Theorems 1.1 and 1.2 given in the
introduction. We consider a quadratic extension of totally
real number fields L/E.},
Doi = {10.1007/978-3-0348-0351-9_8},
Key = {fds328607}
}
@article{fds292899,
Author = {Getz, JR},
Title = {An approach to nonsolvable base change and
descent},
Journal = {Journal of the Ramanujan Mathematical Society},
Volume = {27},
Number = {2},
Pages = {143-211},
Year = {2012},
url = {http://www.math.duke.edu/~jgetz/ApproachJRMS.pdf},
Abstract = {We present a collection of conjectural trace identities and
explain why they are equivalent to base change and descent
of automorphic representations of GL(n) along nonsolvable
extensions (under some simplifying hypotheses). The case n =
2 is treated in more detail and applications towards the
Artin conjecture for icosahedral Galois representations are
given.},
Key = {fds292899}
}
@article{fds292898,
Author = {Getz, J},
Title = {Intersection numbers of Hecke cycles on Hilbert modular
varieties},
Journal = {American Journal of Mathematics},
Volume = {129},
Number = {6},
Pages = {1623-1658},
Publisher = {Johns Hopkins University Press},
Year = {2007},
Month = {December},
ISSN = {0002-9327},
url = {http://dx.doi.org/10.1353/ajm.2007.0041},
Abstract = {Let Script O sign be the ring of integers of a totally real
number field E and set G := ResE/ℚ( GL2). Fix an ideal c
⊂ Script O sign. For each ideal m ⊂ Script O sign let
T(m) denote the mth Hecke operator associated to the
standard compact open subgroup Uo(c) of G(double-struck Af).
Setting X0(c) := G(ℚ)\G(double-struck A)/K∞U0(c), where
K ∞ is a certain subgroup of G(ℝ), we use T(m) to define
a Hecke cycle Z(m) ∈ IH2[E:ℚ](X0(c) x X 0(c)). Here
IH• denotes intersection homology. We use Zucker's
conjecture (proven by Looijenga and independently by Saper
and Stern) to obtain a formula relating the intersection
number Z(m)·Z(n) to the trace of *T(m) ○ T(n) considered
as an endomorphism of the space of Hilbert cusp forms on
U0(c). © 2007 by The Johns Hopkins University
Press.},
Doi = {10.1353/ajm.2007.0041},
Key = {fds292898}
}
@article{fds292895,
Author = {Getz, J},
Title = {A generalization of a theorem of Rankin and Swinnerton-Dyer
on zeros of modular forms},
Journal = {Proceedings of the American Mathematical
Society},
Volume = {132},
Number = {8},
Pages = {2221-2231},
Publisher = {American Mathematical Society (AMS)},
Year = {2004},
Month = {January},
ISSN = {0002-9939},
url = {http://dx.doi.org/10.1090/S0002-9939-04-07478-7},
Abstract = {Rankin and Swinnerton-Dyer (1970) prove that all zeros of
the Eisenstein series Ek in the standard fundamental domain
for Γ lie on A:= {eiθ:π/2 ≤ θ ≤ 2π/3}. In this
paper we generalize their theorem, providing conditions
under which the zeros of other modular forms lie only on the
arc A. Using this result we prove a speculation of Ono,
namely that the zeros of the unique "gap function" in M k,
the modular form with the maximal number of consecutive zero
coefficients in its g-expansion following the constant 1,
has zeros only on A. In addition, we show that the
j-invariant maps these zeros to totally real algebraic
integers of degree bounded by a simple function of weight
k.},
Doi = {10.1090/S0002-9939-04-07478-7},
Key = {fds292895}
}
@article{fds292896,
Author = {Basha, S and Getz, J and Nover, H and Smith, E},
Title = {Systems of orthogonal polynomials arising from the modular
j-function},
Journal = {Journal of Mathematical Analysis and Applications},
Volume = {289},
Number = {1},
Pages = {336-354},
Publisher = {Elsevier BV},
Year = {2004},
Month = {January},
ISSN = {0022-247X},
url = {http://dx.doi.org/10.1016/j.jmaa.2003.09.067},
Abstract = {Let G-fraktur signp(x) ε double struck F sign p[x] be the
polynomial whose zeros are the j-invariants of supersingular
elliptic curves over double struck F signp. Generalizing a
construction of Atkin described in a recent paper by Kaneko
and Zagier (Computational Perspectives on Number Theory
(Chicago, IL, 1995), AMS/IP 7 (1998) 97-126), we define an
inner product 〈 , 〉ψ on ℝ[x] for every ψ(x) ε
ℚ[x]. Suppose a system of orthogonal polynomials
{Pn,ψ(x)}n=0∞ with respect to 〈, 〉ψ exists. We prove
that if n is sufficiently large and ψ(x)Pn,ψ(x) is
p-integral, then G-fraktur signp(x)\ψ(x)Pn(x) over double
struck F signp[x]. Further, we obtain an interpretation of
these orthogonal polynomials as a p-adic limit of
polynomials associated to p-adic modular forms. © 2003
Elsevier Inc. All rights reserved.},
Doi = {10.1016/j.jmaa.2003.09.067},
Key = {fds292896}
}
@article{fds218117,
Author = {S. Basha and J.R. Getz and H. Nover and E. Smith},
Title = {Systems of orthogonal polynomials arising from the modular
j-functions},
Journal = {J. Math. Anal. Appl.},
Volume = {289},
Number = {1},
Pages = {336-354},
Year = {2004},
Key = {fds218117}
}
@article{fds292894,
Author = {Getz, J and Mahlburg, K},
Title = {Partition identities and a theorem of Zagier},
Journal = {Journal of Combinatorial Theory. Series A},
Volume = {100},
Number = {1},
Pages = {27-43},
Publisher = {Elsevier BV},
Year = {2002},
Month = {January},
ISSN = {0097-3165},
url = {http://dx.doi.org/10.1006/jcta.2002.3276},
Abstract = {In the study of partition theory and q-series, identities
that relate series to infinite products are of great
interest (such as the famous Rogers-Ramanujan identities).
Using a recent result of Zagier, we obtain an infinite
family of such identities that is indexed by the positive
integers. For example, if m = 1, then we obtain the
classical Eisenstein series identity ∑λ≥1odd
(-1)(λ-1)/2qλ/(1 - q2λ) = q ∏n=1∞ (1 - q8n)4/(1 -
q4n)2. If m = 2 and (./3) denotes the usual Legendre symbol
modulo 3, then we obtain ∑λ≥1 (λ/3)qλ/(1 - q2λ) = q
∏n=1∞ (1 - qn)(1 - q6n)6/(1 - q2n)2(1 - q3n)3. We
describe some of the partition theoretic consequences of
these identities. In particular, we find simple formulas
that solve the well-known problem of counting the number of
representations of an integer as a sum of an arbitrary
number of triangular numbers. © 2002 Elsevier Science
(USA).},
Doi = {10.1006/jcta.2002.3276},
Key = {fds292894}
}
@article{fds292893,
Author = {Getz, J},
Title = {Extension of a theorem of Kiming and Olsson for the
partition function},
Journal = {Ramanujan Journal},
Volume = {5},
Number = {1},
Pages = {47-51},
Year = {2001},
Month = {March},
url = {http://dx.doi.org/10.1023/A:1011441111570},
Abstract = {Some congruence properties of the partition function are
proved.},
Doi = {10.1023/A:1011441111570},
Key = {fds292893}
}
@article{fds218123,
Author = {J.R. Getz},
Title = {On congruence properties of the partition
function},
Journal = {Int. J. Math. Math. Sci.},
Volume = {23},
Number = {7},
Pages = {493-496},
Year = {2000},
Key = {fds218123}
}
%% Papers Submitted
@article{fds225056,
Author = {J.R. Getz},
Title = {Automorphic kernel functions in four variables},
Year = {2014},
Key = {fds225056}
}
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Mathematics Department