國立虎尾科技大學 |

The adelic image of Galois attached to J(,o)(37).

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	The adelic image of Galois attached to J(,o)(37)./
Author:	Kaskel, Bruce Ephraim.
Description:	1 online resource (48 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 58-06, Section: B.
Contained By:	Dissertations Abstracts International58-06B.
Subject:	Mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9780591096125

The adelic image of Galois attached to J(,o)(37).
Kaskel, Bruce Ephraim.

The adelic image of Galois attached to J(,o)(37). - 1 online resource (48 pages)

Source: Dissertations Abstracts International, Volume: 58-06, Section: B.

Thesis (Ph.D.)--University of California, Berkeley, 1996.

Includes bibliographical references

Fix a prime N and let J denote J0(N), the Jacobian of the modular curve X0(N). If N=11 or N≥17 then J is non-trivial. The Hecke algebra T acts on J and we let T=T\\otimes{\\\\bf Z}.$ The adelic Tate module Ta J=\\lim\\limits\\sb\\longleftarrow J\\lbrack n\\rbrack$ is then a T-module and the Galois action on Ta J gives rise to a representation Gal(Q/Q)→AutT(Ta J). One knows in some cases that Ta J is a free T-module (this might well be true generally but work of Mazur shows only that this is true locally at primes $\\ell\ot=2).$ In these cases, by choosing a T-basis of Ta J, one may consider $\\rho$ as a map into GL2(T). Let G be the image of $\\rho.$ This image can be thought of as Gal(Q/Q). If J is one dimensional (an elliptic curve), then T = Z and a theorem of Serre says that G is of finite index in GL2(Z). This is the case when N = 11, 17 or 19. A calculation of G for $J\\sb0(11)$ was done by Lang and Trotter in Frobenius Distributions in GL2-Extensions. The abelian variety J=J0(37) is 2-dimensional and Mazur's work shows that Ta J is a free T-module of rank 2. We calculate G in this case using the fact that J0(37) is isogenous to a product of two elliptic curves. The result is an explicit description of G which enables one to check if G contains certain elements. For example, we show that G contains an element $\au$ which (for $\\ell\ot=5)$ acts on the $\\ell$-adic Tate module ${\\cal T}a\\sb{\\ell}J$ by multiplication by $-$5. Alternatively, one can think of $\au$ as belonging to Gal(Q/Q) and that for any $P\\in J\\sb{\\rm tors}$ with order prime to 5, $\au(P)=-5P.$ The general strategy is to first find the image of each $\\ell$-adic representation attached to J. The most difficult calculations are those for $\\ell$ = 2, 3. After finding all the $\\ell$-adic images, we determine G as a subgroup of the product of these local images.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9780591096125Subjects--Topical Terms:

527692
Mathematics.
Subjects--Index Terms:

number theoryIndex Terms--Genre/Form:

554714
Electronic books.

The adelic image of Galois attached to J(,o)(37).
LDR:03108ntm a22003377 4500 001 1151132
005 20241104055719.5
006 m o d
007 cr bn ---uuuuu
008 250605s1996 xx obm 000 0 eng d
020 $a 9780591096125
035 $a (MiAaPQ)AAI9703169
035 $a AAI9703169
040 $a MiAaPQ $b eng $c MiAaPQ $d NTU
100 1 $a Kaskel, Bruce Ephraim. $3 1477831
245 1 4 $a The adelic image of Galois attached to J(,o)(37).
264 0 $c 1996
300 $a 1 online resource (48 pages)
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
500 $a Source: Dissertations Abstracts International, Volume: 58-06, Section: B.
500 $a Publisher info.: Dissertation/Thesis.
500 $a Advisor: Coleman, Robert F.
502 $a Thesis (Ph.D.)--University of California, Berkeley, 1996.
504 $a Includes bibliographical references
520 $a Fix a prime N and let J denote J0(N), the Jacobian of the modular curve X0(N). If N=11 or N≥17 then J is non-trivial. The Hecke algebra T acts on J and we let T=T\\otimes{\\\\bf Z}.$ The adelic Tate module Ta J=\\lim\\limits\\sb\\longleftarrow J\\lbrack n\\rbrack$ is then a T-module and the Galois action on Ta J gives rise to a representation Gal(Q/Q)→AutT(Ta J). One knows in some cases that Ta J is a free T-module (this might well be true generally but work of Mazur shows only that this is true locally at primes $\\ell\ot=2).$ In these cases, by choosing a T-basis of Ta J, one may consider $\\rho$ as a map into GL2(T). Let G be the image of $\\rho.$ This image can be thought of as Gal(Q/Q). If J is one dimensional (an elliptic curve), then T = Z and a theorem of Serre says that G is of finite index in GL2(Z). This is the case when N = 11, 17 or 19. A calculation of G for $J\\sb0(11)$ was done by Lang and Trotter in Frobenius Distributions in GL2-Extensions. The abelian variety J=J0(37) is 2-dimensional and Mazur's work shows that Ta J is a free T-module of rank 2. We calculate G in this case using the fact that J0(37) is isogenous to a product of two elliptic curves. The result is an explicit description of G which enables one to check if G contains certain elements. For example, we show that G contains an element $\au$ which (for $\\ell\ot=5)$ acts on the $\\ell$-adic Tate module ${\\cal T}a\\sb{\\ell}J$ by multiplication by $-$5. Alternatively, one can think of $\au$ as belonging to Gal(Q/Q) and that for any $P\\in J\\sb{\\rm tors}$ with order prime to 5, $\au(P)=-5P.$ The general strategy is to first find the image of each $\\ell$-adic representation attached to J. The most difficult calculations are those for $\\ell$ = 2, 3. After finding all the $\\ell$-adic images, we determine G as a subgroup of the product of these local images.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2024
538 $a Mode of access: World Wide Web
650 4 $a Mathematics. $3 527692
653 $a number theory
655 7 $a Electronic books. $2 local $3 554714
690 $a 0405
710 2 $a ProQuest Information and Learning Co. $3 1178819
710 2 $a University of California, Berkeley. $3 1183587
773 0 $t Dissertations Abstracts International $g 58-06B.
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=9703169 $z click for full text (PQDT)