%% Papers Published
@article{fds345674,
Author = {Dasgupta, S and Spiess, M},
Title = {On the characteristic polynomial of the gross regulator
matrix},
Journal = {Transactions of the American Mathematical
Society},
Volume = {372},
Number = {2},
Pages = {803827},
Year = {2019},
Month = {January},
url = {http://dx.doi.org/10.1090/tran/7393},
Abstract = {© 2019 American Mathematical Society. We present a
conjectural formula for the principal minors and the
characteristic polynomial of Gross’s regulator matrix
associated to a totally odd character of a totally real
field. The formula is given in terms of the Eisenstein
cocycle, which was defined and studied earlier by the
authors and collaborators. For the determinant of the
regulator matrix, our conjecture follows from recent work of
Kakde, Ventullo, and the first author. For the diagonal
entries, our conjecture overlaps with the conjectural
formula presented in our prior work. The intermediate cases
are new and provide a refinement of the GrossStark
conjecture.},
Doi = {10.1090/tran/7393},
Key = {fds345674}
}
@article{fds339636,
Author = {Dasgupta, S and Kakde, M and Ventullo, K},
Title = {On the GrossStark Conjecture},
Journal = {Annals of Mathematics},
Volume = {188},
Number = {3},
Pages = {833870},
Publisher = {Annals of Mathematics, Princeton U},
Year = {2018},
Month = {November},
url = {http://dx.doi.org/10.4007/annals.2018.188.3.3},
Abstract = {© 2018 Department of Mathematics, Princeton University. In
1980, Gross conjectured a formula for the expected leading
term at s=0 of the DeligneRibet padic Lfunction
associated to a totally even character ϕ of a totally real
field F. The conjecture states that after scaling by
L(ϕω1,0), this value is equal to a padic regulator of
units in the abelian extension of F cut out by ϕω1. In
this paper, we prove Gross's conjecture.},
Doi = {10.4007/annals.2018.188.3.3},
Key = {fds339636}
}
@article{fds338508,
Author = {Dasgupta, S and Voight, J},
Title = {Sylvester’s problem and mock heegner points},
Journal = {Proceedings of the American Mathematical
Society},
Volume = {146},
Number = {8},
Pages = {32573273},
Publisher = {American Mathematical Society (AMS)},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1090/proc/14008},
Abstract = {© 2018 American Mathematical Society. We prove that if p
≡ 4, 7 (mod 9) is prime and 3 is not a cube modulo p, then
both of the equations x3 + y3 = p and x3 + y3 = p2 have a
solution with x, y ∈ ℚ.},
Doi = {10.1090/proc/14008},
Key = {fds338508}
}
@article{fds339289,
Author = {Dasgupta, S and Spieß, M},
Title = {Partial zeta values, Gross's tower of fields conjecture, and
GrossStark units},
Journal = {Journal of the European Mathematical Society},
Volume = {20},
Number = {11},
Pages = {26432683},
Publisher = {European Mathematical Publishing House},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.4171/JEMS/821},
Abstract = {© European Mathematical Society 2018. We prove a conjecture
of Gross regarding the “order of vanishing” of
Stickelberger elements relative to an abelian tower of
fields and give a cohomological construction of the
conjectural GrossStark units. This is achieved by
introducing an integral version of the Eisenstein
cocycle.},
Doi = {10.4171/JEMS/821},
Key = {fds339289}
}
@article{fds338509,
Author = {Dasgupta, S and Spieß, M},
Title = {The Eisenstein cocycle and Gross’s tower of fields
conjecture},
Journal = {Annales Mathématiques Du Québec},
Volume = {40},
Number = {2},
Pages = {355376},
Publisher = {Springer Nature},
Year = {2016},
Month = {August},
url = {http://dx.doi.org/10.1007/s4031601500462},
Abstract = {© 2016, Fondation CarlHerz and Springer International
Publishing Switzerland. This paper is an announcement of the
following result, whose proof will be forthcoming. Let F be
a totally real number field, and let F⊂ K⊂ L be a tower
of fields with L / F a finite abelian extension. Let I
denote the kernel of the natural projection from Z[ Gal (L/
F) ] to Z[ Gal (K/ F) ]. Let Θ ∈ Z[ Gal (L/ F) ] denote
the Stickelberger element encoding the special values at
zero of the partial zeta functions of L / F, taken
relative to sets S and T in the usual way. Let r denote the
number of places in S that split completely in K. We show
that Θ ∈ Ir, unless K is totally real in which case we
obtain Θ ∈ Ir1 and 2 Θ ∈ Ir. This proves a conjecture
of Gross up to the factor of 2 in the case that K is totally
real and # S≠ r. In this article we sketch the proof in
the case that K is totally complex.},
Doi = {10.1007/s4031601500462},
Key = {fds338509}
}
@article{fds338510,
Author = {Dasgupta, S},
Title = {Factorization of padic Rankin Lseries},
Journal = {Inventiones Mathematicae},
Volume = {205},
Number = {1},
Pages = {221268},
Publisher = {Springer Nature},
Year = {2016},
Month = {July},
url = {http://dx.doi.org/10.1007/s0022201506344},
Abstract = {© 2015, SpringerVerlag Berlin Heidelberg. We prove that
the padic Lseries of the tensor square of a pordinary
modular form factors as the product of the symmetric square
padic Lseries of the form and a Kubota–Leopoldt padic
Lseries. This establishes a generalization of a conjecture
of Citro. Greenberg’s exceptional zero conjecture for the
adjoint follows as a corollary of our theorem. Our method of
proof follows that of Gross, who proved a factorization
result for the Katz padic Lseries associated to the
restriction of a Dirichlet character. Whereas Gross’s
method is based on comparing circular units with elliptic
units, our method is based on comparing these same circular
units with a new family of units (called Beilinson–Flach
units) that we construct. The Beilinson–Flach units are
constructed using Bloch’s intersection theory of higher
Chow groups applied to products of modular curves. We relate
these units to special values of classical and padic
Lfunctions using work of Beilinson (as generalized by
Lei–Loeffler–Zerbes) in the archimedean case and
Bertolini–Darmon–Rotger (as generalized by
Kings–Loeffler–Zerbes) in the padic case. Central to
our method are two compatibility theorems regarding
Bloch’s intersection pairing and the classical and padic
Beilinson regulators defined on higher Chow
groups.},
Doi = {10.1007/s0022201506344},
Key = {fds338510}
}
@article{fds338511,
Author = {Bellaïche, J and Dasgupta, S},
Title = {The padic Lfunctions of evil Eisenstein
series},
Journal = {Compositio Mathematica},
Volume = {151},
Number = {6},
Pages = {9991040},
Publisher = {Oxford University Press (OUP)},
Year = {2015},
Month = {January},
url = {http://dx.doi.org/10.1112/S0010437X1400788X},
Abstract = {© Foundation Compositio Mathematica 2015. We compute the
padic Lfunctions of evil Eisenstein series, showing that
they factor as products of two KubotaLeopoldt padic
Lfunctions times a logarithmic term. This proves in
particular a conjecture of Glenn Stevens.},
Doi = {10.1112/S0010437X1400788X},
Key = {fds338511}
}
@article{fds338512,
Author = {Charollois, P and Dasgupta, S and Greenberg, M},
Title = {Integral Eisenstein cocycles on GLn, II:
Shintani's method},
Journal = {Commentarii Mathematici Helvetici},
Volume = {90},
Number = {2},
Pages = {435477},
Publisher = {European Mathematical Publishing House},
Year = {2015},
Month = {January},
url = {http://dx.doi.org/10.4171/CMH/360},
Abstract = {© Swiss Mathematical Society. We define a cocycle on GLn(Q)
using Shintani's method. This construction is closely
related to earlier work of Solomon and Hill, but differs in
that the cocycle property is achieved through the
introduction of an auxiliary perturbation vector Q. As a
corollary of our result we obtain a new proof of a theorem
of Diaz y Diaz and Friedman on signed fundamental domains,
and give a cohomological reformulation of Shintani's proof
of the KlingenSiegel rationality theorem on partial zeta
functions of totally real fields. Next we relate the
Shintani cocycle to the Sczech cocycle by showing that the
two differ by the sum of an explicit coboundary and a simple
"polar" cocycle. This generalizes a result of Sczech and
Solomon in the case n = 2. Finally, we introduce an integral
version of our cocycle by smoothing at an auxiliary prime l.
This integral refinement has strong arithmetic consequences.
We showed in previous work that certain specializations of
the smoothed class yield the padic Lfunctions of totally
real fields. Furthermore, combining our cohomological
construction with a theorem of Spiess, one deduces that that
the order of vanishing of these padic Lfunctions is at
least as large as the expected one.},
Doi = {10.4171/CMH/360},
Key = {fds338512}
}
@article{fds338513,
Author = {Bertolini, M and Castella, F and Darmon, H and Dasgupta, S and Prasanna,
K and Rotger, V},
Title = {Padic Lfunctions and Euler systems: A tale in two
trilogies},
Pages = {52101},
Booktitle = {Automorphic Forms and Galois Representations:
volume1},
Year = {2014},
Month = {January},
ISBN = {9781107691926},
url = {http://dx.doi.org/10.1007/9781107446335.004},
Abstract = {© Cambridge University Press 2014. This chapter surveys six
different special value formulae for padic Lfunctions,
stressing their common features and their eventual
arithmetic applications via Kolyvagin’s theory of “Euler
systems”, in the spirit of CoatesWiles and
KatoPerrinRiou.},
Doi = {10.1007/9781107446335.004},
Key = {fds338513}
}
@article{fds338514,
Author = {Dasgupta, S},
Title = {A conjectural product formula for BrumerStark units over
real quadratic fields},
Journal = {Journal of Number Theory},
Volume = {133},
Number = {3},
Pages = {915925},
Publisher = {Elsevier BV},
Year = {2013},
Month = {March},
url = {http://dx.doi.org/10.1016/j.jnt.2012.02.013},
Abstract = {Following methods of Hayes, we state a conjectural product
formula for ratios of BrumerStark units over real quadratic
fields. © 2012 Elsevier Inc.},
Doi = {10.1016/j.jnt.2012.02.013},
Key = {fds338514}
}
@article{fds338515,
Author = {Dasgupta, S and Greenberg, M},
Title = {ℒinvariants and Shimura curves},
Journal = {Algebra & Number Theory},
Volume = {6},
Number = {3},
Pages = {455485},
Publisher = {Mathematical Sciences Publishers},
Year = {2012},
Month = {July},
url = {http://dx.doi.org/10.2140/ant.2012.6.455},
Abstract = {In earlier work, the second named author described how to
extract Darmonstyle ℒinvariants from modular forms on
Shimura curves that are special at p. In this paper, we show
that these ℒinvariants are preserved by the
JacquetLanglands correspondence. As a consequence, we prove
the second named author's period conjecture in the case
where the base field is ℚ. As a further application of our
methods, we use integrals of Hida families to describe
StarkHeegner points in terms of a certain AbelJacobi map.
©2012 by Mathematical Sciences Publishers.},
Doi = {10.2140/ant.2012.6.455},
Key = {fds338515}
}
@article{fds338516,
Author = {Dasgupta, S and Darmon, H and Pollack, R},
Title = {Hilbert modular forms and the GrossStark
conjecture},
Journal = {Annals of Mathematics},
Volume = {174},
Number = {1},
Pages = {439484},
Publisher = {Annals of Mathematics, Princeton U},
Year = {2011},
Month = {July},
url = {http://dx.doi.org/10.4007/annals.2011.174.1.12},
Abstract = {Let F be a totally real field and χ an abelian totally odd
character of F. In 1988, Gross stated a padic analogue of
Stark's conjecture that relates the value of the derivative
of the padic Lfunction associated to χ and the padic
logarithm of a punit in the extension of F cut out by χ.
In this paper we prove Gross's conjecture when F is a real
quadratic field and χ is a narrow ring class character. The
main result also applies to general totally real fields for
which Leopoldt's conjecture holds, assuming that either
there are at least two primes above p in F, or that a
certain condition relating the Linvariants of χ and χ1
holds. This condition on Linvariants is always satisfied
when χ is quadratic.},
Doi = {10.4007/annals.2011.174.1.12},
Key = {fds338516}
}
@article{fds338517,
Author = {Dasgupta, S and Miller, A},
Title = {A Shintanitype formula for GrossStark units over function
fields},
Journal = {Journal of Mathematical Sciences},
Volume = {16},
Number = {3},
Pages = {415440},
Year = {2009},
Month = {December},
Abstract = {Let F be a totally real number field of degree n, and let H
be a finite abelian extension of F. Let p denote a prime
ideal of F that splits completely in H. Following Brumer and
Stark, Tate conjectured the existence of a punit u in H
whose padic absolute values are related in a precise way to
the partial zetafunctions of the extension H/F. Gross later
refined this conjecture by proposing a formula for the
padic norm of the element u. Recently, using methods of
Shintani, the first author refined the conjecture further by
proposing an exact formula for u in the padic completion of
H. In this article we state and prove a function field
analogue of this Shintanitype formula. The role of the
totally real field F is played by the function field of a
curve over a finite field in which n places have been
removed. These places represent the "real places" of F. Our
method of proof follows that of Hayes, who proved Gross's
conjecture for function fields using the theory of Drinfeld
modules and their associated exponential
functions.},
Key = {fds338517}
}
@article{fds338518,
Author = {Dasgupta, S},
Title = {Shintani zeta functions and grossstark units for totally
real fields},
Journal = {Duke Mathematical Journal},
Volume = {143},
Number = {2},
Pages = {225279},
Publisher = {Duke University Press},
Year = {2008},
Month = {June},
url = {http://dx.doi.org/10.1215/001270942008019},
Abstract = {Let F be a totally real number field, and let p be a finite
prime of F such that p splits completely in the finite
abelian extension H of F. Tate has proposed a conjecture
[22, Conjecture 5.4] stating the existence of a punit u in
H with absolute values at the places above p specified in
terms of the values at zero of the partial zeta functions
associated to H/F. This conjecture is an analogue of Stark's
conjecture, which Tate called the BrumerStark conjecture.
Gross [12, Conjecture 7.6] proposed a refinement of the
BrumerStark conjecture that gives a conjectural formula for
the image of u in Fpx/Ê, where FP denotes the completion of
F at p and Ê denotes the topological closure of the group
of totally positive units E of F. We present a further
refinement of Gross's conjecture by proposing a conjectural
formula for the exact value of u in Fpx.},
Doi = {10.1215/001270942008019},
Key = {fds338518}
}
@article{fds338519,
Author = {Dasgupta, S},
Title = {Computations of elliptic units for real quadratic
fields},
Journal = {Canadian Journal of Mathematics},
Volume = {59},
Number = {3},
Pages = {553574},
Publisher = {Canadian Mathematical Society},
Year = {2007},
Month = {January},
url = {http://dx.doi.org/10.4153/CJM20070230},
Abstract = {Let K be a real quadratic field, and p a rational prime
which is inert in K. Let a be a modular unit on Γ0(N). In
an earlier joint article with Henri Darmon, we presented the
definition of an element u(α, τ) ε Kpx attached to a and
each τ ε K. We conjectured that the padic number u(α,
τ) lies in a specific ring class extension of K depending
on τ, and proposed a "Shimura reciprocity law" describing
the permutation action of Galois on the set of u(α, τ),
This article provides computational evidence for these
conjectures. We present an efficient algorithm for computing
u(α, τ), and implement this algorithm with the modular
unit α(z) = Δ(z) 2 Δ(4z)3,Δ(2z)3. Using p = 3, 5, 7, and
11, and all real quadratic fields K with discriminant D <
500 such that 2 splits in K and K contains no unit of
negative norm, we obtain results supporting our conjectures.
One of the theoretical results in this paper is that a
certain measure used to define w(α, τ) is shown to be
Zvalued rather than only Zp ∩ Qvalued; this is an
improvement over our previous result and allows for a
precise definition of u(α, τ), instead of only up to a
root of unity. © Canadian Mathematical Society
2007.},
Doi = {10.4153/CJM20070230},
Key = {fds338519}
}
@article{fds338520,
Author = {Darmon, H and Dasgupta, S},
Title = {Elliptic units for real quadratic fields},
Journal = {Annals of Mathematics},
Volume = {163},
Number = {1},
Pages = {301346},
Publisher = {Annals of Mathematics, Princeton U},
Year = {2006},
Month = {July},
url = {http://dx.doi.org/10.4007/annals.2006.163.301},
Doi = {10.4007/annals.2006.163.301},
Key = {fds338520}
}
@article{fds338521,
Author = {Dasgupta, S},
Title = {StarkHeegner points on modular Jacobians},
Journal = {Annales Scientifiques De L’École Normale
Supérieure},
Volume = {38},
Number = {3},
Pages = {427469},
Publisher = {Societe Mathematique de France},
Year = {2005},
Month = {May},
url = {http://dx.doi.org/10.1016/j.ansens.2005.03.002},
Abstract = {We present a construction which lifts Darmon's StarkHeegner
points from elliptic curves to certain modular Jacobians.
Let N be a positive integer and let p be a prime not
dividing N. Our essential idea is to replace the modular
symbol attached to an elliptic curve E of conductor Np with
the universal modular symbol for Γ0(Np). We then construct
a certain torus T over Qp and lattice L ⊂ T, and prove
that the quotient T/L is isogenous to the maximal toric
quotient J0(Np)pnew of the Jacobian of X0(Np). This theorem
generalizes a conjecture of Mazur, Tate, and Teitelbaum on
the padic periods of elliptic curves, which was proven by
Greenberg and Stevens. As a byproduct of our theorem, we
obtain an efficient method of calculating the padic periods
of J0(Np)pnew. © 2005 Elsevier SAS. All rights
reserved.},
Doi = {10.1016/j.ansens.2005.03.002},
Key = {fds338521}
}
@article{fds338523,
Author = {Biss, DK and Dasgupta, S},
Title = {A presentation for the unipotent group over rings with
identity},
Journal = {Journal of Algebra},
Volume = {237},
Number = {2},
Pages = {691707},
Publisher = {Elsevier BV},
Year = {2001},
Month = {March},
url = {http://dx.doi.org/10.1006/jabr.2000.8604},
Abstract = {For a ring R with identity, define Unipn(R) to be the group
of uppertriangular matrices over R all of whose diagonal
entries are 1. For i = 1,2,...,n  1, let Si denote the
matrix whose only nonzero offdiagonal entry is a 1 in the
ith row and (i + 1)st column. Then for any integer m
(including m = 0), it is easy to see that the Si generate
Unipn(Z/mZ). Reiner gave relations among the Si which he
conjectured gave a presentation for Unipn(Z/2Z). This
conjecture was proven by Biss [Comm. Algebra26 (1998),
29712975] and an analogous conjecture was made for
Unipn(Z/mZ) in general. We prove this conjecture, as well as
a generalization of the conjecture to unipotent groups over
arbitrary rings. © 2001 Academic Press.},
Doi = {10.1006/jabr.2000.8604},
Key = {fds338523}
}
@article{fds338522,
Author = {Dasgupta, S and Károlyi, G and Serra, O and Szegedy,
B},
Title = {Transversals of additive Latin squares},
Journal = {Israel Journal of Mathematics},
Volume = {126},
Number = {1},
Pages = {1728},
Publisher = {Springer Nature},
Year = {2001},
Month = {January},
url = {http://dx.doi.org/10.1007/BF02784149},
Abstract = {Let A = {a1,..., ak} and B = {b1,..., bk} be two subsets of
an Abelian group G, k ≤ G. Snevily conjectured that,
when G is of odd order, there is a permutation π ≤ Sk
such that the sums ai + bπ(i), 1 ≤ i ≤ k, are pairwise
different. Alon showed that the conjecture is true for
groups of prime order, even when A is a sequence of k < G
elements, i.e., by allowing repeated elements in A. In this
last sense the result does not hold for other Abelian
groups. With a new kind of application of the polynomial
method in various finite and infinite fields we extend
Alon's result to the groups (Zp)α and Zpα in the case k <
p, and verify Snevily's conjecture for every cyclic group of
odd order.},
Doi = {10.1007/BF02784149},
Key = {fds338522}
}
@article{fds338524,
Author = {Dasgupta, S},
Title = {On the size of minimum super arrovian domains},
Journal = {Siam Journal on Discrete Mathematics},
Volume = {12},
Number = {4},
Pages = {524534},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {1999},
Month = {January},
url = {http://dx.doi.org/10.1137/S0895480198332521},
Abstract = {Arrow's celebrated impossibility theorem states that a
sufficiently diverse domain of voter preference profiles
cannot be mapped into social orders of the alternatives
without violating at least one of three appealing
conditions. Following Fishburn and Kelly, we define a set of
strict preference profiles to be super Arrovian if Arrow's
impossibility theorem holds for this set and each of its
strict preference profile supersets. We write σ(m, n) for
the size of the smallest super Arrovian set for m
alternatives and n voters. We show that σ(m, 2) = [2m/m2]
and σ(3, 3) = 19. We also show that σ(m, n) is bounded by
a constant for fixed n and bounded on both sides by a
constant times 2n for fixed m. In particular, we find that
limn→∞ σ(3, n)/2n = 3. Finally, we answer two questions
posed by Fishburn and Kelly on the structure of minimum and
minimal super Arrovian sets.},
Doi = {10.1137/S0895480198332521},
Key = {fds338524}
}
