Cris Poor
Math Department, Fordham University
Publications
 Thesis: CrossRatio Identities for Theta Functions on Jacobi Varieties

Princeton University, (124), 1988.

My thesis derived the crossratio identities for Jacobian theta function
in general and for hyperelliptic theta functions in particular. For
the hyperelliptic case, this work was directly continued in article 3.
The cycliccovering case was further developed in [7] by H. Farkas.
 Fay's Trisecant Formula and CrossRatios

Proc. AMS, vol. 114. no. 3, (667671), 1992.

This proof of Fay's trisecant formula is based on the Riemann surface
function theory developed by my thesis advisor, R. C. Gunning. I
regret not explicitly including a similar proof of the multisecant identity.
One can find it hidden on page 848 of article 3. An exposition
of R. Gunning's crossratio function is in the final section added to
the second edition of Riemann Surfaces by H. Farkas and I. Kra, the
excellent text book whose first edition I used as a graduate student.
 The Hyperelliptic Locus (PDF)

Duke Math. J., vol. 76. no. 3, (809884), 1994.

Here it is proved that irreducible hyperelliptic Jacobians are characterized
among all principally polarized abelian varieties by their vanishing
nullwerte, an unresolved problem for over a century. In 1984,
D. Mumford [22] proved that hyperelliptic Jacobians are characterized
among all principally polarized abelian varieties by the vanishing and
nonvanishing of their nullwerte. Neither result seems to be a corollary
of the other. This article is a continuation of article 1. H. J. Weber
[28] used this result to locate hyperelliptic Jacobians related to modular
curves. I have unpublished results that hyperelliptic Jacobians are
characterized by their vanishing nullwerte alone in genera four and five.
 Relations on the Period Mapping giving Extensions of Mixed
Hodge Structures on Compact Riemann Surfaces (PDF)

Geometriae Dedicata, vol. 59, (243291), 1996. (with D. Yuen)

R. Gunning [10] gave a novel proof of the symmetry of the Riemann
period matrix by studying the periods of iterated integrals of holomorphic
differentials, work continued by his student E. Jablow [18]. R. Hain
[11] and his student M. Pulte [26] studied the mixed Hodge structure
on compact Riemann surfaces via periods of homotopy functionals consisting
of twice iterated integrals of holomorphic and antiholomorphic
differentials. We attempted to unify these two approaches. We proved
that the holomorphic periods alone generically determine the mixed
Hodge structure. We found all higher order symmetries of the period
map arising from homotopy functionals among triple iterated integrals.
An intrinsic formulation of these symmetries is that that period map
from Teichmuller space factors through a third exterior power bundle
over Α_{g}. This subject as been furthered [19] in the thesis of R.
Kaenders.
 Schottky's Form and the Hyperelliptic Locus (PDF)

Proc. AMS, vol. 124, no. 7, (19871991), 1996.

J. I. Igusa proved [17] that Schottky's form in genus four was given by
the difference of the theta series for the two classes of even unimodular
lattices in dimension sixteen. Schottky's form vanishes on the Jacobian
locus in genus four. Here it is proven that this difference of theta series
vanishes on the hyperelliptic locus in every genus. S. Grushevsky and
R. Salvati Manni recently proved [9] that this difference does not vanish
on the Jacobian locus in genus five but rather cuts out the trigonal
divisor on M5.
 Dimensions of spaces of Siegel modular forms of low weight in
degree four (PDF)

Bull. Austral. Math, Soc., vol. 54, (309315), 1996. (with D. Yuen)

We found the dimensions of genus four Siegel cusp forms in weights 6,
8 and 12. The initial classification of the theta series of the Niemeier
lattices [5][6]
by V. A. Erokhin was the main tool. The cases of weight 6
and 8 are corollaries of a theorem [27] by R. Salvati Manni but we were
unaware of this at the time. This note was immediately responded to
by W. Duke and O. Imamoglu in an article [3] using explicit formulae.
Our results were also used by E. Freitag and M. Oura in [8].
 Scaling theory and solutions for steadystate coagulation and set
tling of fractal aggregates in aquatic systems.

Colloids and Surfaces A, vol. 107, (155174), 1996. (S. Grant, S. Relle)

The Smoluchowski coagulation equation, in integrodifferential form,
is used to model steadystate coagulation in combination with gravitational
settling. Our model fits the particle volume concentrations of
Lake Zurich, Switzerland pretty well except for the large aggregates
which are likely being removed from the water column by other means.
 Estimates for Dimensions of Spaces of Siegel Modular Cusp Forms (PDF)

Abh. Math. Sem. Univ. Hamburg, vol. 66, (337354), 1996. (D. Yuen)

This paper grew from a desire to explicitly compute with theta series
in higher genera. Siegel gave a constructive proof of the finite dimensionality
of vector spaces of Siegel modular forms. He showed that the
vector spaces were determined by Fourier coecients with indices of
bounded trace. M. Eichler has a sharper vanishing theorem [4] that
uses the Minimum function m(v) = min x ∊ Z_{g}∖ {0}x′vx
and relies on Hermite's
constant from the geometry of numbers. We interpolated the
proofs of Siegel and Eichler to get a reasonably sharp estimation theorem
for any convex function. As an application, we reproved Witt's
conjecture [29] by computing just one Fourier coefficient. In this sense,
this article is a sequel to [29], where Witt lamented the ungeheuere
Rechnungen. This work is continued in article 11.
 Particle coagulation and the memory of initial conditions.

J. Phys. A., vol. 31, (92419254), 1998. (A. Boehm, S. Grant)

The Smoluchowski coagulation equations are a countably infinite set
of coupled nonlinear ordinary differential equations. When there is
uniform attraction among particles of different size, the large particle
normalized distributions converge to a fixed function of the scaling parameter,
independent of the initial conditions. The scaling parameter
is a ratio of particle size to time. It had been widely assumed that the
long time particle distributions were independent of the initial conditions
as well but we showed that this is not the case. The small particle
distributions retain a memory of the initial conditions indefinitely. The
magnitude of the effect depends upon the proximity to the unit circle
of the the complex roots of the generating function given by the initial
conditions.
 Dimensions of Spaces of Siegel Modular Forms and ThetaSeries
with Pluriharmonics.

Far East J. Math. Sci., vol. 1. no. 6, (849863), 1999. (D. Yuen)

We classified spaces of Siegel modular forms spanned by theta series
with pluriharmonic coefficients in some cases in genus four. The main
value of the article lies in the computational techniques. In particular,
we symmetrized the pluriharmonic polynomials under the automorphism
group of the lattice and used the automorphisms to reduce the
complexity of the computations.
 Linear dependence among Siegel modular forms.(PDF)

Math. Ann., vol. 318. (205234), 2000. (with D. Yuen)

The intrinsic "vanishing order" of a Fourier series f is a convex set
ν(f): the convex semihull of the support of f. We prove the Semihull
Theorem: For Siegel modular cusp forms f of weight k, if Y^{k/2}f(Z)
attains its maximum at X_{0} + iY_{0},
then
k/(4π)Y_{0}^{1}
∊ ν(f). From the Semihull
Theorem, we may recover the estimates for convex functions first
proven in article 8. The theorems proven here are used in articles 13,
14, 15, 16, 17, 18, 19 and 22.
We introduce an important convex function, the dyadic trace, and
use it to give improved determining sets of Fourier coefficients. For
example, in weight 12 and genus 4, a determining set consists of the 23
classes whose dyadic trace is less than or equal to 4; whereas the trace
estimate of Siegel requires over 100; 000 classes. The dyadic trace has
been defined and applied in the case of Hermitian modular forms in
the thesis [20] of M. Klein.
 Kinetic Theories for the Coagulation and Sedimentation of Paticles.

(S. Grant, J. Kim)
J. Colloid and Interface Science, vol. 238. (238250), 2001.

The Smoluchowski coagulation equations are modified to include sedimentation.
We assume that there is uniform attraction among particles
of different size and that sedimentation increases linearly with
particle size. A solution is obtained for the corresponding nonlinear
partial differential equation and we compare the solutions with the
predictions of two common approximations: the similarity theory and
the quasisteadystate hypothesis. The latter does not fare well but the
similarity theory predicts critical exponents that are within about 20%
of those of our solution. The boundary of the coagulation dominated
zone is found by analyzing a nonlinear ordinary differential equation.
 The Dyadic Trace and Odd Weight Computations For Siegel
Modular Cusp Forms.

Bull. Austral. Math, Soc., vol. 63, (269271), 2001. (with D. Yuen)

We illustrate the results of article 11 in the case of odd weights. A
Fourier coecient is no longer a class function but its vanishing is.
 Restriction of Siegel modular forms to modular curves (PDF)

Bull. Austral. Math, Soc., vol. 65, (239252), 2002. (with D. Yuen)

The Restriction Technique is introduced to produce linear relations
among the Fourier coefficients of Siegel modular forms. Siegel forms are
restricted to modular curves and the known linear relations among elliptic
modular forms are pulled back. After article 11 provided tractable
determining sets of Fourier coefficients, a method was needed to find the
linear relations among these Fourier coecients. This is an old problem,
equivalent to finding the linear relations among certain Poincare
series. As an application, we computed the dimension of S^{10}_{4},
a space just beyond the reach of the explicit formula method [3]. If you wish
to learn the Restriction Technique, this is the article to read. Further
applications of the Restriction Technique are given in articles 18, 19
and 20.
 Slopes of Integral lattices (PDF)

J. of Number Theory, vol. 100. (363380), 2003. (with D. Yuen)

J. I. Igusa found critical slopes for the hyperelliptic locus [16]. J. Harris
and I. Morrison found critical slopes for the trigonal locus and gave
an elegant conjecture for the Jacobian locus [12]. We use the dyadic
trace to find critical slopes for the modular curves from article 14. Our
modular curves arise from lattices and the slope of a lattice is a new
integral invariant. The dyadic traces and slopes of all root lattices are
computed. The main result is that a Siegel modular cusp form vanishes
on a modular curve if the slope of the cusp form is less than the slope
of the lattice.
 The Extreme Core.

Abh. Math. Sem. Univ. Hamburg, vol. 75, (5175), 2005. (D. Yuen)

This article is a continuation of article 11 and advances the geometry
of numbers for its applications to Siegel modular forms. The topics
include kernels, cores and noble forms. We show the existence of the
extreme core in each genus. This is a core C_{ext} such that k C_{ext}⊆ ν(f)
for every nontrivial cusp form f of weight k. Thus the existence of the
extreme core is a generalization of the Valence Inequality in genus one.
We estimate the extreme core in all genera and almost specify it in
genus two, leaving the true value of ω_{0} = sup_{f}inf < A_{2} (1/k)ν(f)>
as an open problem. The constants in the Vanishing theorems are improved
and the improvements are used in articles 19 and 22 and in [20].
 The BergéMartinet constant and slopes of Siegel Cusp Forms.

Bull. London Math. Soc., vol. 38. (913924), 2006. (with D. Yuen)

This is a topic in the geometry of numbers. We use the dyadic trace
to determine the value of the BergéMartinet constant for degrees 5; 6
and 7. We prove Conjecture 6.4.16 in J. Martinet's recent book [21]
by finding all dualcritical pairs. As an application to Siegel modular
forms, we use these newly found values of the BergéMartinet constant
to improve upon M. Eichler's lower bound [4] for the optimal slope of
a cusp form in degrees 5; 6 and 7.
 Computations of spaces of Siegel modular cusp forms.

J. Math. Soc. Japan, vol. 59. no. 1, (185222), 2007. (with D. Yuen)

This is a systematic exposition of the Restriction Technique written
with Professor David S. Yuen, a preliminary version was published in
[24]. Advances are made in both theory and computation. Determining
sets of Fourier coefficients from article 11 and the Restriction Technique
from article 14 are combined into a systematic method for computing
individual spaces of Siegel modular cusp forms. For g > 3, most known
cases are recovered and some new cases are computed in genera 4, 5
and 6. We compute Hecke eigenforms in the nontrivial cases.
It is natural to ask whether our method of computing spaces of
Siegel modular cusp forms always works. We were able to prove a
partial result: the linear relations generated by the Restriction Technique
characterize the Fourier expansions of Siegel modular cusp forms
from among all convergent Fourier series. Whether or not the linear relations
generated by the Restriction Technique characterize the Fourier
expansions of Siegel modular cusp forms from among all formal Fourier
series is an open problem.
A similar open problem will appear in article 25. The natural application
of our method to congruence subgroups is in article 19. The
Lfunctions of the new Hecke eigenforms will appear in article 21.
 Dimensions of Cusp Forms for Γ_{0}(p) in Degree Two and Small
Weights.

Abh. Math. Sem. Univ. Hamburg, vol. 77, (5980), 2007. (D. Yuen)

We explain how to use the method of article 18 and the constants
of article 16 for spaces of Siegel modular cusp forms of finite index in
the Siegel modular group. We applied our method to the weight one
spaces S^{1}_{2}(Γ_{0}(p)) and found that they were trivial for primes p ≤ 97.
This was unexpected and brought attention to the weight one case. T.
Ibukiyama and N. Skoruppa proved that S^{1}_{2}(Γ_{0}(N)) always vanishes
[14] and their article appears in the same volume of the Abhandlungen.
In weight two, we examined the cases p ≤ 41 and found only Saito
Kurokawa and Yoshida lifts; unlike the weight one case, this pattern
cannot continue indefinitely. An enumeration of weight two forms will
require connections with rational abelian surfaces, see article 22 below.
In weights three and four, we verified conjectures of K. Hashimoto
[13] in some cases. T. Ibukiyama has since proven [15] these conjectures.
The obstruction to the weight three case was the weight one
case, which vanishes as mentioned above.
 Toward the Siegel Ring in Genus Four.

Inter. J. Number Th. vol. 4. no. 4, (563586), 2008. (M. Oura, D. Yuen)

In [8], E. Freitag and M. Oura found the first secondorder theta
relation in genus four. This article is a direct continuation of that work
but the computations require the Restriction Technique from articles 14
and 18. We classify all relations through degree 32, finding six new
relations. A sequel to this paper, showing that the Th_{2} map is not
surjective in genus four, has already been written by M. Oura and R.
Salvati Manni [23].
 Publications accepted and in print: Lifting Puzzles in Degree Four.

(authors: C. Poor, N, Ryan, D. Yuen)
(accepted to: The Bulletin of the Australian Mathematical Society)

Dr. N. Ryan wrote his thesis on the computation of Satake parameters.
We use his work and the Hecke eigenforms found in article 18 to
compute Euler factors of Lfunctions in genus four. These are the first
examples of their type in genus four. Not all the Satake parameters in
our examples are unimodular; therefore, if the Generalized Ramanujan
Petersson conjecture can be properly reformulated at all, there must
be two new types of lifts yet to be discovered.