Samit Dasgupta

Transversals of additive Latin squares

AbstractLetA={a1, …,ak} andB={b1, …,bk} be two subsets of an Abelian groupG, k≤|G|. Snevily conjectured that, whenG is of odd order, there is a permutationπ ∈Sksuch that the sums αi+bi, 1≤i≤k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even whenA is a sequence ofk<|G| elements, i.e., by allowing repeated elements inA. 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)a and

Computations of Elliptic Units for Real Quadratic Fields

\mathbb{Z}_{p^a }

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

in the casek<p, and verify Snevily’s conjecture for every cyclic group of odd order.

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

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 . Stark has proposed a conjecture stating the existence of a p-unit 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 . Gross proposed a refinement of Stark’s conjecture which gives a conjectural formula for the image of Stark’s unit in F× p /E, where Fp denotes the completion of F at p and E denotes the topological closure of the group of totally positive units E of F . We propose a further refinement of Gross’ conjecture by proposing a conjectural formula for the exact value of Stark’s unit in F× p .

The p-adic L-functions of evil Eisenstein series

Let K be a real quadratic field, and p a rational prime which is inert in K. Let α be a modular unit onΓ0(N). In an earlier joint article with Henri Darmon, we presented the definition of an element u(α, τ ) ∈ K× p attached to α and each τ ∈ K. We conjectured that the p-adic number u(α, τ ) lies in a specific ring class extension ofK 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) = ∆(z)2∆(4z)/∆(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 u(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. Received by the editors September 27, 2004; revised August 15, 2005. AMS subject classification: 11R37, 11R11, 11Y40. c ©Canadian Mathematical Society 2007. 553

Integral Eisenstein cocycles on GL n , II: Shintani's method

Let E be an elliptic curve over Q attached to a newform f of weight two on Γ0(N). Let K be a real quadratic field, and let p||N be a prime of multiplicative reduction for E which is inert in K, so that the p-adic completion Kp of K is the quadratic unramified extension of Qp. Subject to the condition that all the primes dividing M := N/p are split in K, the article [Dar] proposes an analytic construction of “Stark–Heegner points” in E(Kp), and conjectures that these points are defined over specific class fields of K. More precisely, let

Integral Eisenstein cocycles on

We define a cocycle on Gln using Shintanis method. It is closely related to cocycles defined earlier by 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 Shintanis proof of the Klingen-Siegel rationality theorem on partial zeta functions of totally real fields. Next we prove that the cohomology class represented by our Shintani cocycle is essentially equal to that represented by the Eisenstein cocycle introduced by Sczech. This generalizes a result of Sczech and Solomon in the case n=2. Finally we introduce an integral version of our Shintani cocycle by smoothing at an auxiliary prime ell. Applying the formalism of the first paper in this series, we prove that certain specializations of the smoothed class yield the p-adic L-functions of totally real fields. Combining our cohomological construction with a theorem of Spiess, we show that the order of vanishing of these p-adic L-functions is at least as large as the one predicted by a conjecture of Gross.

\mathbf {GL}_n

© Cambridge University Press 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.