%%
@article{fds368211,
Author = {Dasgupta, S and Kakde, M},
Title = {On the Brumer-Stark conjecture},
Journal = {Annals of Mathematics},
Volume = {197},
Number = {1},
Pages = {289-388},
Publisher = {Annals of Mathematics},
Year = {2023},
Month = {January},
url = {http://dx.doi.org/10.4007/annals.2023.197.1.5},
Abstract = {Let H=F be a finite abelian extension of number fields with
F totally real and H a CM field. Let S and T be disjoint
finite sets of places of F satisfying the standard
conditions. The Brumer-Stark conjecture states that the
Stickelberger element ΦH/FS,T annihilates the T-smoothed
class group ClT(H). We prove this conjecture away from p=2,
that is, after tensoring with Z[1/2]. We prove a stronger
version of this result conjectured by Kurihara that gives a
formula for the 0th Fitting ideal of the minus part of the
Pontryagin dual of [Formula Presented] in terms of
Stickelberger elements. We also show that this stronger
result implies Rubin's higher rank version of the
Brumer-Stark conjecture, again away from 2. Our technique is
a generalization of Ribet's method, building upon on our
earlier work on the Gross-Stark conjecture. Here we work
with group ring valued Hilbert modular forms as introduced
by Wiles. A key aspect of our approach is the construction
of congruences between cusp forms and Eisenstein series that
are stronger than usually expected, arising as shadows of
the trivial zeroes of p-adic L-functions. These stronger
congruences are essential to proving that the cohomology
classes we construct are unramified at p.},
Doi = {10.4007/annals.2023.197.1.5},
Key = {fds368211}
}
@article{fds359781,
Author = {Dasgupta, S and Kakde, M},
Title = {On constant terms of Eisenstein series},
Journal = {Acta Arithmetica},
Volume = {200},
Number = {2},
Pages = {119-147},
Year = {2021},
Month = {January},
url = {http://dx.doi.org/10.4064/aa200621-24-2},
Doi = {10.4064/aa200621-24-2},
Key = {fds359781}
}
@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 = {803-827},
Year = {2019},
Month = {January},
url = {http://dx.doi.org/10.1090/tran/7393},
Abstract = {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 Gross-Stark
conjecture.},
Doi = {10.1090/tran/7393},
Key = {fds345674}
}
@article{fds339636,
Author = {Dasgupta, S and Kakde, M and Ventullo, K},
Title = {On the Gross-Stark Conjecture},
Journal = {Annals of Mathematics},
Volume = {188},
Number = {3},
Pages = {833-870},
Publisher = {Annals of Mathematics, Princeton U},
Year = {2018},
Month = {November},
url = {http://dx.doi.org/10.4007/annals.2018.188.3.3},
Abstract = {In 1980, Gross conjectured a formula for the expected
leading term at s=0 of the Deligne-Ribet p-adic L-function
associated to a totally even character ϕ of a totally real
field F. The conjecture states that after scaling by
L(ϕω-1,0), this value is equal to a p-adic regulator of
units in the abelian extension of F cut out by ϕω-1. 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 = {3257-3273},
Publisher = {American Mathematical Society (AMS)},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1090/proc/14008},
Abstract = {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
Gross-Stark units},
Journal = {Journal of the European Mathematical Society},
Volume = {20},
Number = {11},
Pages = {2643-2683},
Publisher = {European Mathematical Publishing House},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.4171/JEMS/821},
Abstract = {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 Gross-Stark 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 Mathematiques du Quebec},
Volume = {40},
Number = {2},
Pages = {355-376},
Publisher = {Springer Nature},
Year = {2016},
Month = {August},
url = {http://dx.doi.org/10.1007/s40316-015-0046-2},
Abstract = {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 ∈ Ir-1 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/s40316-015-0046-2},
Key = {fds338509}
}
@article{fds338510,
Author = {Dasgupta, S},
Title = {Factorization of p-adic Rankin L-series},
Journal = {Inventiones Mathematicae},
Volume = {205},
Number = {1},
Pages = {221-268},
Publisher = {Springer Nature},
Year = {2016},
Month = {July},
url = {http://dx.doi.org/10.1007/s00222-015-0634-4},
Abstract = {We prove that the p-adic L-series of the tensor square of a
p-ordinary modular form factors as the product of the
symmetric square p-adic L-series of the form and a
Kubota–Leopoldt p-adic L-series. 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
p-adic L-series 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 p-adic L-functions 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 p-adic
case. Central to our method are two compatibility theorems
regarding Bloch’s intersection pairing and the classical
and p-adic Beilinson regulators defined on higher Chow
groups.},
Doi = {10.1007/s00222-015-0634-4},
Key = {fds338510}
}
@article{fds338511,
Author = {Bellaïche, J and Dasgupta, S},
Title = {The p-adic L-functions of evil Eisenstein
series},
Journal = {Compositio Mathematica},
Volume = {151},
Number = {6},
Pages = {999-1040},
Publisher = {Oxford University Press (OUP)},
Year = {2015},
Month = {June},
url = {http://dx.doi.org/10.1112/S0010437X1400788X},
Abstract = {We compute the p-adic L-functions of evil Eisenstein series,
showing that they factor as products of two Kubota-Leopoldt
p-adic L-functions 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 = {435-477},
Publisher = {European Mathematical Publishing House},
Year = {2015},
Month = {January},
url = {http://dx.doi.org/10.4171/CMH/360},
Abstract = {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 Klingen-Siegel
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 p-adic L-functions of totally real fields.
Furthermore, combining our cohomological construction with a
theorem of Spiess, one deduces that that the order of
vanishing of these p-adic L-functions 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 = {P-adic L-functions and Euler systems: A tale in two
trilogies},
Pages = {52-101},
Booktitle = {Automorphic Forms and Galois Representations:
volume1},
Year = {2014},
Month = {January},
ISBN = {9781107691926},
url = {http://dx.doi.org/10.1007/9781107446335.004},
Abstract = {This chapter surveys six different special value formulae
for p-adic L-functions, stressing their common features and
their eventual arithmetic applications via Kolyvagin’s
theory of “Euler systems”, in the spirit of Coates-Wiles
and Kato-Perrin-Riou.},
Doi = {10.1007/9781107446335.004},
Key = {fds338513}
}
@article{fds353876,
Author = {Charollois, P and Dasgupta, S},
Title = {Integral Eisenstein cocycles on $\mathbf{GL}_n$, I:
Sczech’s cocycle and $p$-adic $L$-functions of totally
real fields},
Journal = {Cambridge Journal of Mathematics},
Volume = {2},
Number = {1},
Pages = {49-90},
Publisher = {International Press of Boston},
Year = {2014},
url = {http://dx.doi.org/10.4310/cjm.2014.v2.n1.a2},
Doi = {10.4310/cjm.2014.v2.n1.a2},
Key = {fds353876}
}
@article{fds338514,
Author = {Dasgupta, S},
Title = {A conjectural product formula for Brumer-Stark units over
real quadratic fields},
Journal = {Journal of Number Theory},
Volume = {133},
Number = {3},
Pages = {915-925},
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 Brumer-Stark 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 = {ℒ-invariants and Shimura curves},
Journal = {Algebra and Number Theory},
Volume = {6},
Number = {3},
Pages = {455-485},
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 Darmon-style ℒ-invariants from modular forms on
Shimura curves that are special at p. In this paper, we show
that these ℒ-invariants are preserved by the
Jacquet-Langlands 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
Stark-Heegner points in terms of a certain Abel-Jacobi 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 Gross-Stark
conjecture},
Journal = {Annals of Mathematics},
Volume = {174},
Number = {1},
Pages = {439-484},
Publisher = {Annals of Mathematics, Princeton U},
Year = {2011},
Month = {January},
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 p-adic analogue of
Stark's conjecture that relates the value of the derivative
of the p-adic L-function associated to χ and the p-adic
logarithm of a p-unit 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 L-invariants of χ and χ-1
holds. This condition on L-invariants 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 Shintani-type formula for Gross-Stark units over function
fields},
Journal = {Journal of Mathematical Sciences},
Volume = {16},
Number = {3},
Pages = {415-440},
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 p-unit u in H
whose p-adic absolute values are related in a precise way to
the partial zeta-functions of the extension H/F. Gross later
refined this conjecture by proposing a formula for the
p-adic 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 p-adic completion of
H. In this article we state and prove a function field
analogue of this Shintani-type 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 gross-stark units for totally
real fields},
Journal = {Duke Mathematical Journal},
Volume = {143},
Number = {2},
Pages = {225-279},
Publisher = {Duke University Press},
Year = {2008},
Month = {June},
url = {http://dx.doi.org/10.1215/00127094-2008-019},
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 p-unit 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 Brumer-Stark conjecture.
Gross [12, Conjecture 7.6] proposed a refinement of the
Brumer-Stark 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 = {10.1215/00127094-2008-019},
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 = {553-574},
Publisher = {Canadian Mathematical Society},
Year = {2007},
Month = {January},
url = {http://dx.doi.org/10.4153/CJM-2007-023-0},
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 p-adic 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
Z-valued rather than only Zp ∩ Q-valued; 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/CJM-2007-023-0},
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 = {301-346},
Publisher = {Annals of Mathematics, Princeton U},
Year = {2006},
Month = {January},
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 = {Stark-Heegner points on modular Jacobians},
Journal = {Annales Scientifiques de l'Ecole Normale
Superieure},
Volume = {38},
Number = {3},
Pages = {427-469},
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 Stark-Heegner
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)p-new of the Jacobian of X0(Np). This theorem
generalizes a conjecture of Mazur, Tate, and Teitelbaum on
the p-adic periods of elliptic curves, which was proven by
Greenberg and Stevens. As a by-product of our theorem, we
obtain an efficient method of calculating the p-adic periods
of J0(Np)p-new. © 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 = {691-707},
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 upper-triangular matrices over R all of whose diagonal
entries are 1. For i = 1,2,...,n - 1, let Si denote the
matrix whose only nonzero off-diagonal 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),
2971-2975] 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 = {17-28},
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 = {524-534},
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/m-2]
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}
}
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Mathematics Department