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Samit Dasgupta

Professor of Mathematics
Mathematics

Selected Publications


On the Brumer-Stark conjecture

Journal Article Annals of Mathematics · January 1, 2023 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 an ... Full text Open Access Cite

On constant terms of Eisenstein series

Journal Article Acta Arithmetica · January 1, 2021 Full text Cite

On the characteristic polynomial of the gross regulator matrix

Journal Article Transactions of the American Mathematical Society · January 1, 2019 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 define ... Full text Cite

On the Gross-Stark Conjecture

Journal Article Annals of Mathematics · November 1, 2018 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 ... Full text Cite

Sylvester’s problem and mock heegner points

Journal Article Proceedings of the American Mathematical Society · January 1, 2018 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. ... Full text Cite

Partial zeta values, Gross's tower of fields conjecture, and Gross-Stark units

Journal Article Journal of the European Mathematical Society · January 1, 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 Gross-Stark units. This is achieved by introducing an integral versi ... Full text Cite

The Eisenstein cocycle and Gross’s tower of fields conjecture

Journal Article Annales Mathematiques du Quebec · August 1, 2016 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 ... Full text Cite

Factorization of p-adic Rankin L-series

Journal Article Inventiones Mathematicae · July 1, 2016 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 ... Full text Cite

The p-adic L-functions of evil Eisenstein series

Journal Article Compositio Mathematica · June 18, 2015 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. ... Full text Cite

Integral Eisenstein cocycles on GLn, II: Shintani's method

Journal Article Commentarii Mathematici Helvetici · January 1, 2015 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 corol ... Full text Cite

P-adic L-functions and Euler systems: A tale in two trilogies

Chapter · January 1, 2014 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. ... Full text Cite

A conjectural product formula for Brumer-Stark units over real quadratic fields

Journal Article Journal of Number Theory · March 1, 2013 Following methods of Hayes, we state a conjectural product formula for ratios of Brumer-Stark units over real quadratic fields. © 2012 Elsevier Inc. ... Full text Cite

ℒ-invariants and Shimura curves

Journal Article Algebra and Number Theory · January 1, 2012 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 ... Full text Cite

Hilbert modular forms and the Gross-Stark conjecture

Journal Article Annals of Mathematics · January 1, 2011 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 ... Full text Cite

A Shintani-type formula for Gross-Stark units over function fields

Journal Article Journal of Mathematical Sciences · December 1, 2009 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 ... Cite

Shintani zeta functions and gross-stark units for totally real fields

Journal Article Duke Mathematical Journal · June 1, 2008 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 a ... Full text Cite

Computations of elliptic units for real quadratic fields

Journal Article Canadian Journal of Mathematics · January 1, 2007 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 conjectur ... Full text Cite

Elliptic units for real quadratic fields

Journal Article Annals of Mathematics · January 1, 2006 Full text Cite

Stark-Heegner points on modular Jacobians

Journal Article Annales Scientifiques de l'Ecole Normale Superieure · May 1, 2005 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 ellip ... Full text Cite

On the Brumer-Stark conjecture

Journal Article Annals of Mathematics · January 1, 2023 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 an ... Full text Open Access Cite

On constant terms of Eisenstein series

Journal Article Acta Arithmetica · January 1, 2021 Full text Cite

On the characteristic polynomial of the gross regulator matrix

Journal Article Transactions of the American Mathematical Society · January 1, 2019 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 define ... Full text Cite

On the Gross-Stark Conjecture

Journal Article Annals of Mathematics · November 1, 2018 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 ... Full text Cite

Sylvester’s problem and mock heegner points

Journal Article Proceedings of the American Mathematical Society · January 1, 2018 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. ... Full text Cite

Partial zeta values, Gross's tower of fields conjecture, and Gross-Stark units

Journal Article Journal of the European Mathematical Society · January 1, 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 Gross-Stark units. This is achieved by introducing an integral versi ... Full text Cite

The Eisenstein cocycle and Gross’s tower of fields conjecture

Journal Article Annales Mathematiques du Quebec · August 1, 2016 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 ... Full text Cite

Factorization of p-adic Rankin L-series

Journal Article Inventiones Mathematicae · July 1, 2016 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 ... Full text Cite

The p-adic L-functions of evil Eisenstein series

Journal Article Compositio Mathematica · June 18, 2015 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. ... Full text Cite

Integral Eisenstein cocycles on GLn, II: Shintani's method

Journal Article Commentarii Mathematici Helvetici · January 1, 2015 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 corol ... Full text Cite

P-adic L-functions and Euler systems: A tale in two trilogies

Chapter · January 1, 2014 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. ... Full text Cite

A conjectural product formula for Brumer-Stark units over real quadratic fields

Journal Article Journal of Number Theory · March 1, 2013 Following methods of Hayes, we state a conjectural product formula for ratios of Brumer-Stark units over real quadratic fields. © 2012 Elsevier Inc. ... Full text Cite

ℒ-invariants and Shimura curves

Journal Article Algebra and Number Theory · January 1, 2012 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 ... Full text Cite

Hilbert modular forms and the Gross-Stark conjecture

Journal Article Annals of Mathematics · January 1, 2011 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 ... Full text Cite

A Shintani-type formula for Gross-Stark units over function fields

Journal Article Journal of Mathematical Sciences · December 1, 2009 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 ... Cite

Shintani zeta functions and gross-stark units for totally real fields

Journal Article Duke Mathematical Journal · June 1, 2008 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 a ... Full text Cite

Computations of elliptic units for real quadratic fields

Journal Article Canadian Journal of Mathematics · January 1, 2007 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 conjectur ... Full text Cite

Elliptic units for real quadratic fields

Journal Article Annals of Mathematics · January 1, 2006 Full text Cite

Stark-Heegner points on modular Jacobians

Journal Article Annales Scientifiques de l'Ecole Normale Superieure · May 1, 2005 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 ellip ... Full text Cite

A presentation for the unipotent group over rings with identity

Journal Article Journal of Algebra · March 15, 2001 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 c ... Full text Cite

Transversals of additive Latin squares

Journal Article Israel Journal of Mathematics · January 1, 2001 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 ... Full text Cite

On the size of minimum super arrovian domains

Journal Article SIAM Journal on Discrete Mathematics · January 1, 1999 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 Kel ... Full text Cite