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Moduli Spaces of Abelian Varieties Associated to Mod-p Galois Representations.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Moduli Spaces of Abelian Varieties Associated to Mod-p Galois Representations./
作者:	Chidambaram, Shiva.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2021,
面頁冊數:	89 p.
附註:	Source: Dissertations Abstracts International, Volume: 83-02, Section: B.
Contained By:	Dissertations Abstracts International83-02B.
標題:	Mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28414824
ISBN:	9798516956119

Moduli Spaces of Abelian Varieties Associated to Mod-p Galois Representations.
Chidambaram, Shiva.

Moduli Spaces of Abelian Varieties Associated to Mod-p Galois Representations. - Ann Arbor : ProQuest Dissertations & Theses, 2021 - 89 p.

Source: Dissertations Abstracts International, Volume: 83-02, Section: B.

Thesis (Ph.D.)--The University of Chicago, 2021.

This item is not available from ProQuest Dissertations & Theses.

This thesis consists of four research papers stapled together. In this work, we study moduli spaces of principally polarised abelian varieties of dimension g > 1 with p-torsion structure for prime p. In particular, given a Galois representation p̅: GQ → GSp(2g,Fp ) with cyclotomic similitude character, we study various rationality aspects of the twist \uD835\uDCD0g(p̅) of the Siegel modular variety \uD835\uDCD0g(p) of genus g and level p.Using a description of the cohomology of the compactification \uD835\uDCD02*(3) given by Hoffman and Weintraub, we show that the variety \uD835\uDCD02(p̅) is not rational in general. When p̅ is surjective, the minimal degree of a rational cover is 6. Boxer, Calegari, Gee, and Pilloni have shown the existence of a rational cover \uD835\uDCD0ω2(p̅) of degree 6. We find explicit formulae parametrizing the pullback \uD835\uDCDC2ω(p̅) of \uD835\uDCD0ω2(p̅ ) under the Torelli map \uD835\uDCDC2 → \uD835\uDCD02 . This describes the universal family of genus 2 curves with a rational Weierstrass point, having fixed 3-torsion of Jacobian. This exploits Shioda's work on Mordell-Weil lattices and the invariant theory of the complex reflection group C3 x Sp4(F3). We also outline how similar results can be obtained for (g,p)=(2,2),(3,2),(4,2).By making use of the modularity lifting theorem for abelian surfaces proved by Boxer, Calegari, Gee. and Pilloni, we produce some explicit examples of modular abelian surfaces A with EndC(A) = Z. Using the explicit formulae describing families of abelian surfaces with fixed 3-torsion, and transferring modularity in the family yields infinitely many such examples.When g = 1 and p > 5, the existence of mod-p Galois representations not arising from elliptic curves over Q is known. For g > 1 and (g,p) ≠ (2,2), (2,3), (3,2), we investigate a local obstruction to the existence of rational points on \uD835\uDCD0g(p̅), and thus construct Galois representations p̅: GQ → GSp(2g,Fp ) with cyclotomic similitude character, that do not arise from the p-torsion of any g-dimensional abelian variety over Q. This is accomplished by solving embedding problems with local conditions at suitably chosen auxiliary primes ɭ ≠ p, with the help of Galois cohomological machinery.

ISBN: 9798516956119Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Abelian surfaces

Moduli Spaces of Abelian Varieties Associated to Mod-p Galois Representations.
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245 1 0 $a Moduli Spaces of Abelian Varieties Associated to Mod-p Galois Representations.
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300 $a 89 p.
500 $a Source: Dissertations Abstracts International, Volume: 83-02, Section: B.
500 $a Includes supplementary digital materials.
500 $a Advisor: Calegari, Frank.
502 $a Thesis (Ph.D.)--The University of Chicago, 2021.
506 $a This item is not available from ProQuest Dissertations & Theses.
506 $a This item must not be sold to any third party vendors.
520 $a This thesis consists of four research papers stapled together. In this work, we study moduli spaces of principally polarised abelian varieties of dimension g > 1 with p-torsion structure for prime p. In particular, given a Galois representation p̅: GQ → GSp(2g,Fp ) with cyclotomic similitude character, we study various rationality aspects of the twist \uD835\uDCD0g(p̅) of the Siegel modular variety \uD835\uDCD0g(p) of genus g and level p.Using a description of the cohomology of the compactification \uD835\uDCD02*(3) given by Hoffman and Weintraub, we show that the variety \uD835\uDCD02(p̅) is not rational in general. When p̅ is surjective, the minimal degree of a rational cover is 6. Boxer, Calegari, Gee, and Pilloni have shown the existence of a rational cover \uD835\uDCD0ω2(p̅) of degree 6. We find explicit formulae parametrizing the pullback \uD835\uDCDC2ω(p̅) of \uD835\uDCD0ω2(p̅ ) under the Torelli map \uD835\uDCDC2 → \uD835\uDCD02 . This describes the universal family of genus 2 curves with a rational Weierstrass point, having fixed 3-torsion of Jacobian. This exploits Shioda's work on Mordell-Weil lattices and the invariant theory of the complex reflection group C3 x Sp4(F3). We also outline how similar results can be obtained for (g,p)=(2,2),(3,2),(4,2).By making use of the modularity lifting theorem for abelian surfaces proved by Boxer, Calegari, Gee. and Pilloni, we produce some explicit examples of modular abelian surfaces A with EndC(A) = Z. Using the explicit formulae describing families of abelian surfaces with fixed 3-torsion, and transferring modularity in the family yields infinitely many such examples.When g = 1 and p > 5, the existence of mod-p Galois representations not arising from elliptic curves over Q is known. For g > 1 and (g,p) ≠ (2,2), (2,3), (3,2), we investigate a local obstruction to the existence of rational points on \uD835\uDCD0g(p̅), and thus construct Galois representations p̅: GQ → GSp(2g,Fp ) with cyclotomic similitude character, that do not arise from the p-torsion of any g-dimensional abelian variety over Q. This is accomplished by solving embedding problems with local conditions at suitably chosen auxiliary primes ɭ ≠ p, with the help of Galois cohomological machinery.
590 $a School code: 0330.
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
653 $a Abelian surfaces
653 $a Burkhardt quartic
653 $a Mod-p Galois representation
653 $a Siegel modular variety
653 $a Torsion representation
690 $a 0405
690 $a 0642
710 2 $a The University of Chicago. $b Mathematics. $3 2049825
773 0 $t Dissertations Abstracts International $g 83-02B.
790 $a 0330
791 $a Ph.D.
792 $a 2021
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28414824