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020 a| 9781470460198q| (paperback);

020 z| 9781470473242q| (pdf);

035 a| (WaSeSS)ssj0002824546;

040 a| LBSORb| engc| LBSORd| WaSeSS;

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100 1 a| Fonseca, Tiago J.?| UNAUTHORIZED;

245 10 a| Higher Ramanujan equations and periods of Abelian varietiesh| [electronic resource] /c| Tiago J. Fonseca.;

260 a| Providence, RI :b| American Mathematical Society,c| 2023.;

263 a| 2307;

300 a| pages cm;

490 0 a| Memoirs of the American Mathematical Society,x| 0065-9266 ;v| Volume 281, Number 1391;

504 a| Includes bibliographical references and index.;

505 0 a| Symplectic vector bundles over schemes -- Symplectic-Hodge bases of principally polarized abelian schemes -- Abelian schemes with real multiplication -- The moduli stacks Bg and BF -- The tangent bundles of Bg and BF : higher Ramanujan vector fields -- Integral solution of the higher Ramanujan equation -- Representability of Bg and BF by a scheme -- The case of elliptic curves: explicit equations -- Analytic families of complex tori, abelian varieties, and their uniformization -- Analytic moduli spaces of complex abelian varieties with a symplectic-Hodge basis -- The analytic higher Ramanujan equations -- Values of Pg and PF : periods of abelian varieties -- An algebraic independence conjecture on the values of phF -- Group-theoretic description of the higher Ramanujan vector fields -- Zariski-density of leaves of the higher Ramanujan foliation.;

506 a| Available only to authorized users.;

520 a| "We describe higher dimensional generalizations of Ramanujan's classical differential relations satisfied by the Eisenstein series E2, E4, E6. Such "higher Ramanujan equations" are given geometrically in terms of vector fields living on certain moduli stacks classifying abelian schemes equipped with suitable frames of their first de Rham cohomology. These vector fields are canonically constructed by means of the Gauss-Manin connection and the Kodaira-Spencer isomorphism. Using Mumford's theory of degenerating families of abelian varieties, we construct remarkable solutions of these differential equations generalizing (E2,E4,E6), which are also shown to be defined over Z. This geometric framework taking account of integrality issues is mainly motivated by questions in Transcendental Number Theory regarding an extension of Nesterenko's celebrated theorem on the algebraic independence of values of Eisenstein series. In this direction, we discuss the precise relation between periods of abelian varieties and the values of the above referred solutions of the higher Ramanujan equations, thereby linking the study of such differential equations to Grothendieck's Period Conjecture. Working in the complex analytic category, we prove "functional" transcendence results, such as the Zariski-density of every leaf of the holomorphic foliation induced by the higher Ramanujan equations"--c| Provided by publisher.;

538 a| Mode of access: World Wide Web;

650 0 a| Abelian varieties.=| ^A68270;

650 0 a| Foliations (Mathematics)=| ^A69917;

650 0 a| Holomorphic functions.=| ^A58602;

650 0 a| Moduli theory.=| ^A121775;

650 0 a| Transcendental numbers.=| ^A62412;

650 7 a| Dynamical systems and ergodic theory -- Complex dynamical systems -- Holomorphic foliations and vector fields.2| msc;

650 7 a| Algebraic geometry -- Families, fibrations -- Stacks and moduli problems.2| msc;

650 7 a| Algebraic geometry -- Abelian varieties and schemes -- None of the above, but in this section.2| msc;

650 7 a| Number theory -- Diophantine approximation, transcendental number theory -- Transcendence (general theory).2| msc;

650 7 a| Number theory -- Arithmetic algebraic geometry (Diophantine geometry) -- Arithmetic aspects of modular and Shimura varieties.2| msc;

655 0 a| Electronic books.=| ^A491897;

856 40 z| Full text available from Ebook Central - Academic Completeu| https://ebookcentral.proquest.com/lib/eastcarolina/detail.action?docID=30330535;

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Higher Ramanujan equations and periods of Abelian varieties / Tiago J. Fonseca.

Author/creator	Fonseca, Tiago J.
Format	Electronic
Publication Info	Providence, RI : American Mathematical Society, 2023.
Description	pages cm
Supplemental Content	Full text available from Ebook Central - Academic Complete
Subjects	Abelian varieties. Foliations (Mathematics) Holomorphic functions. Moduli theory. Transcendental numbers.

Series	Memoirs of the American Mathematical Society, 0065-9266 ; Volume 281, Number 1391
Contents	Symplectic vector bundles over schemes -- Symplectic-Hodge bases of principally polarized abelian schemes -- Abelian schemes with real multiplication -- The moduli stacks Bg and BF -- The tangent bundles of Bg and BF : higher Ramanujan vector fields -- Integral solution of the higher Ramanujan equation -- Representability of Bg and BF by a scheme -- The case of elliptic curves: explicit equations -- Analytic families of complex tori, abelian varieties, and their uniformization -- Analytic moduli spaces of complex abelian varieties with a symplectic-Hodge basis -- The analytic higher Ramanujan equations -- Values of Pg and PF : periods of abelian varieties -- An algebraic independence conjecture on the values of phF -- Group-theoretic description of the higher Ramanujan vector fields -- Zariski-density of leaves of the higher Ramanujan foliation.
Abstract	"We describe higher dimensional generalizations of Ramanujan's classical differential relations satisfied by the Eisenstein series E2, E4, E6. Such "higher Ramanujan equations" are given geometrically in terms of vector fields living on certain moduli stacks classifying abelian schemes equipped with suitable frames of their first de Rham cohomology. These vector fields are canonically constructed by means of the Gauss-Manin connection and the Kodaira-Spencer isomorphism. Using Mumford's theory of degenerating families of abelian varieties, we construct remarkable solutions of these differential equations generalizing (E2,E4,E6), which are also shown to be defined over Z. This geometric framework taking account of integrality issues is mainly motivated by questions in Transcendental Number Theory regarding an extension of Nesterenko's celebrated theorem on the algebraic independence of values of Eisenstein series. In this direction, we discuss the precise relation between periods of abelian varieties and the values of the above referred solutions of the higher Ramanujan equations, thereby linking the study of such differential equations to Grothendieck's Period Conjecture. Working in the complex analytic category, we prove "functional" transcendence results, such as the Zariski-density of every leaf of the holomorphic foliation induced by the higher Ramanujan equations"-- Provided by publisher.
Bibliography note	Includes bibliographical references and index.
Access restriction	Available only to authorized users.
Technical details	Mode of access: World Wide Web
Genre/form	Electronic books.
LCCN	2023012627
ISBN	9781470460198 (paperback)
ISBN	(pdf)

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