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Gopal Prasad

Publications and source records attributed to Gopal Prasad.

At least 19 recordsLinked to original sources
arXiv2019

Fake projective planes

A fake projective plane is a smooth complex surface which is not the complex projective plane but has the same Betti numbers as the complex projective plane. The first example of such a surface was constructed by David Mumford in 1979 using p-adic uniformization. Two more examples were found by Ishida and Kato by related method. Keum has recently given an example which is possibly different from the three known earlier. It is an interesting problem in complex algebraic geometry to determine all fake projective planes. Using the arithmeticity of the fundamental group of fake projective planes, the formula for the covolume of principal arithmetic subgroups given by the first-named auhor, and some number theoretic estimates, we give a classification of fake projective planes in this paper. Twenty eight distinct classes are found. The construction given in the paper appears to be more direct and more natural. It does not use p-adic uniformization.

math.AG↗
arXiv2017

A new approach to unramified descent in Bruhat-Tits theory

The purpose of the paper is to present an alternative approach to unramified descent in Bruhat-Tits theory of reductive groups over nonarchimedean local fields. This approach appears to be conceptually simpler than the approach in the papers of Bruhat and Tits.

math.RT↗
arXiv2017

Generic elements of a Zariski-dense subgroup form an open subset

Let G be a semi-simple algebraic group over a finitely generated field K of characteristic zero, and let Γ< G(K) be a finitely generated Zariski-dense subgroup. In this note we prove that the set of K-generic elements of Γ(whose existence was established earlier in [9]) is open in the profinite topology of Γ. We then extend this result to the fields of positive characteristic, and also prove the existence of generic elements in this case.

math.GR↗
arXiv2015

On the congruence kernel for simple algebraic groups

This paper contains several results about the structure of the congruence kernel C^(S)(G) of an absolutely almost simple simply connected algebraic group G over a global field K with respect to a set of places S of K. In particular, we show that C^(S)(G) is always trivial if S contains a generalized arithmetic progression. We also give a criterion for the centrality of C^(S)(G) in the general situation in terms of the existence of commuting lifts of the groups G(K_v) for v \notin S in the S-arithmetic completion \widehat{G}^(S). This result enables one to give simple proofs of the centrality in a number of cases. Finally, we show that if K is a number field and $G$ is K-isotropic then C^(S)(G) as a normal subgroup of \widehat{G}^(S) is almost generated by a single element.

math.GR↗
arXiv2013

Weakly commensurable groups, with applications to differential geometry

The article contains a survey of our results on weakly commensurable arithmetic and general Zariski-dense subgroups, length-commensurable and isospectral locally symmetric spaces and of related problems in the theory of semi-simple agebraic groups. We have included a discussion of very recent results and conjectures on absolutely almost simple algebraic groups having the same maximal tori and finite-dimensional division algebras having the same maximal subfields.

math.GR↗
arXiv2013

Arithmetic Groups (Banff, Alberta, April 14-19, 2013)

We present detailed summaries of the talks that were given during a week-long workshop on Arithmetic Groups at the Banff International Research Station in April 2013. The vast majority of these reports are based on abstracts that were kindly provided by the speakers. Video recordings of many of the lectures are available online.

math.GR↗
arXiv2011

On the fields generated by the lengths of closed geodesics in locally symmetric spaces

This paper is the next installment of our analysis of length-commensurable locally symmetric spaces begun in Publ. math. IHES 109(2009), 113-184. For a Riemannian manifold $M$, we let $L(M)$ be the weak length spectrum of $M$, i.e. the set of lengths of all closed geodesics in $M$, and let $\mathcal{F}(M)$ denote the subfield of $\mathbb{R}$ generated by $L(M)$. Let now $M_i$ be an arithmetically defined locally symmetric space associated with a simple algebraic $\mathbb{R}$-group $G_i$ for $i = 1, 2$. Assuming Schanuel's conjecture from transcendental number theory, we prove (under some minor technical restrictions) the following dichotomy: either $M_1$ and $M_2$ are length-commensurable, i.e. $\mathbb{Q} \cdot L(M_1) = \mathbb{Q} \cdot L(M_2)$, or the compositum $\mathcal{F}(M_1)\mathcal{F}(M_2)$ has infinite transcendence degree over $\mathcal{F}(M_i)$ for at least one $i = 1$ or $2$ (which means that the sets $L(M_1)$ and $L(M_2)$ are very different).

math.DG↗
arXiv2009

Addendum to `Fake Projective Planes'

The addendum updates the results presented in the paper `Fake Projective Plane, Invent Math 168, 321-370 (2007)' and makes some additions and corrections. The fake projective planes are classified into twenty six classes. Together with a recent work of Donald Cartwright and Tim Steger, there is now a complete list of fake projective planes. There are precisely one hundred fake projective planes as complex surfaces classified up to biholomorphism.

math.AG↗
arXiv2009

Local-global principles for embedding of fields with involution into simple algebras with involution

In this paper we prove local-global principles for embedding of fields with involution into central simple algebras with involution over a global field. These should be of interest in study of classical groups over global fields. We deduce from our results that in a group of type D_n, n>4 even, two weakly commensurable Zariski-dense S-arithmetic subgroups are actually commensurable. A consequence of this result is that given an absolutely simple algebraic K-group G of type D_n, n>4 even, K a number field, any K-form G' of G having the same set of isomorphism classes of maximal K-tori as G, is necessarily K-isomorphic to G. These results lead to results about isolength and isospectral compact hyperbolic spaces of dimension 2n-1 with n even.

math.NT↗
arXiv2008

Number theoretic techniques in the theory of Lie groups and differential geometry

The purpose of this article is to present a survey of our recent results on length commensurable and isospectral locally symmetric spaces. The geometric questions led us to the notion of "weak commensurability" of two Zariski-dense subgroups in a semi-simple Lie group. We have shown that for arithmetic subgroups, weak commensurability has surprisingly strong consequences. Our proofs make use of p-adic techniques and results from algebraic and transcendental number theory.

math.DG↗
arXiv2008

Arithmetic fake projective spaces and arithmetic fake grassmannians

We show that if n>5, PU(n-1,1) does not contain a cocompact arithmetic subgroup with the same Euler-Poincare characteristic (in the sense of C.T.C. Wall) as the complex projective space of dimension n-1, and show that if n=5, there are at least four such subgroups, which are in fact torsion-free. This, in particular, leads to examples of a fake projective space of dimension 4. Analogous results for arithmetic fake grassmannians Gr(m,n) with n>3 odd are also obtained.

math.AG↗