Local behaviour of Galois representations
|
|
- Andrew Short
- 5 years ago
- Views:
Transcription
1 Local behaviour of Galois representations Devika Sharma Weizmann Institute of Science, Israel 23rd June, 2017 Devika Sharma (Weizmann) 23rd June, / 14
2 The question Let p be a prime. Let f ř 8 ně1 a npf qq n be a normalized eigenform in Skě2 new pn, ɛq. Devika Sharma (Weizmann) 23rd June, / 14
3 The question Let p be a prime. Let f ř 8 ně1 a npf qq n be a normalized eigenform in Skě2 new pn, ɛq. Let ρ f : G Q ÝÑ GL 2 pk f,pq. Devika Sharma (Weizmann) 23rd June, / 14
4 The question Let p be a prime. Let f ř 8 ně1 a npf qq n be a normalized eigenform in Skě2 new pn, ɛq. Let ρ f : G Q ÝÑ GL 2 pk f,pq. Let G p Ă G Q be the decomposition group at p. Devika Sharma (Weizmann) 23rd June, / 14
5 The question Let p be a prime. Let f ř 8 ně1 a npf qq n be a normalized eigenform in Skě2 new pn, ɛq. Let ρ f : G Q ÝÑ GL 2 pk f,pq. Let G p Ă G Q be the decomposition group at p. Assume f is p-ordinary, i.e., p a p pf q. Devika Sharma (Weizmann) 23rd June, / 14
6 The question Let p be a prime. Let f ř 8 ně1 a npf qq n be a normalized eigenform in Skě2 new pn, ɛq. Let ρ f : G Q ÝÑ GL 2 pk f,pq. Let G p Ă G Q be the decomposition group at p. Assume f is p-ordinary, i.e., p a p pf q. Then Wiles showed, η ν ρ f Gp 0 η 1, with η 1 unramified. Devika Sharma (Weizmann) 23rd June, / 14
7 The question Let p be a prime. Let f ř 8 ně1 a npf qq n be a normalized eigenform in Skě2 new pn, ɛq. Let ρ f : G Q ÝÑ GL 2 pk f,pq. Let G p Ă G Q be the decomposition group at p. Assume f is p-ordinary, i.e., p a p pf q. Then Wiles showed, η ν ρ f Gp 0 η 1, with η 1 unramified. Natural to ask, when does ρ f Gp split? Devika Sharma (Weizmann) 23rd June, / 14
8 Current status Guess: ρ f Gp splits ðñ f has complex multiplication (CM). Devika Sharma (Weizmann) 23rd June, / 14
9 Current status Guess: ρ f Gp splits ðñ f has complex multiplication (CM). f has CM ùñ ρ f Gp splits. Devika Sharma (Weizmann) 23rd June, / 14
10 Current status Guess: ρ f Gp splits ðñ f has complex multiplication (CM). f has CM ùñ ρ f Gp splits. f is non-cm?? ùñ ρ f Gp is non-split. Devika Sharma (Weizmann) 23rd June, / 14
11 We ask for more Devika Sharma (Weizmann) 23rd June, / 14
12 We ask for more Every p-ordinary form f is part of a family of modular forms where f f k. H f : tf k0 : k 0 ě 1u Devika Sharma (Weizmann) 23rd June, / 14
13 We ask for more Every p-ordinary form f is part of a family of modular forms where f f k. H f : tf k0 : k 0 ě 1u Theorem For p 3, every member of a non-cm family H f is non-split Devika Sharma (Weizmann) 23rd June, / 14
14 We ask for more Every p-ordinary form f is part of a family of modular forms where f f k. H f : tf k0 : k 0 ě 1u Theorem For p 3, every member of a non-cm family H f is non-split if condition pc 1q condition pc 2q are satisfied. Devika Sharma (Weizmann) 23rd June, / 14
15 Set up Let p 3. Devika Sharma (Weizmann) 23rd June, / 14
16 Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Devika Sharma (Weizmann) 23rd June, / 14
17 Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Let ρ : ρ f : G S Ñ GL 2 pf p q be the corresponding residual representation. Devika Sharma (Weizmann) 23rd June, / 14
18 Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Let ρ : ρ f : G S Ñ GL 2 pf p q be the corresponding residual representation. The Galois action on the p-torsion points on E also gives ρ. Devika Sharma (Weizmann) 23rd June, / 14
19 Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Let ρ : ρ f : G S Ñ GL 2 pf p q be the corresponding residual representation. The Galois action on the p-torsion points on E also gives ρ. Assume that Devika Sharma (Weizmann) 23rd June, / 14
20 Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Let ρ : ρ f : G S Ñ GL 2 pf p q be the corresponding residual representation. The Galois action on the p-torsion points on E also gives ρ. Assume that p N, Devika Sharma (Weizmann) 23rd June, / 14
21 Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Let ρ : ρ f : G S Ñ GL 2 pf p q be the corresponding residual representation. The Galois action on the p-torsion points on E also gives ρ. Assume that p N, ρ is surjective. Devika Sharma (Weizmann) 23rd June, / 14
22 Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Let ρ : ρ f : G S Ñ GL 2 pf p q be the corresponding residual representation. The Galois action on the p-torsion points on E also gives ρ. Assume that p N, ρ is surjective. This implies that H f is a non-cm family. Devika Sharma (Weizmann) 23rd June, / 14
23 Notation Devika Sharma (Weizmann) 23rd June, / 14
24 Notation H : field cut out by the projective image of ρ Devika Sharma (Weizmann) 23rd June, / 14
25 Notation H : field cut out by the projective image of ρ Cl H : class group of H, and Ă Cl H : Cl H {Cl p H Devika Sharma (Weizmann) 23rd June, / 14
26 Notation H : field cut out by the projective image of ρ Cl H : class group of H, and Cl Ă H : Cl H {Cl p H E : global units of H, and E r : E{E p Devika Sharma (Weizmann) 23rd June, / 14
27 Notation H : field cut out by the projective image of ρ Cl H : class group of H, and Cl Ă H : Cl H {Cl p H E : global units of H, and E r : E{E p U p : ź P p U P, where U P are local units at P p in H Devika Sharma (Weizmann) 23rd June, / 14
28 Notation H : field cut out by the projective image of ρ Cl H : class group of H, and Cl Ă H : Cl H {Cl p H E : global units of H, and E r : E{E p U p : ź P p U P, where U P are local units at P p in H ru p : U p {U p p Devika Sharma (Weizmann) 23rd June, / 14
29 Notation H : field cut out by the projective image of ρ Cl H : class group of H, and Cl Ă H : Cl H {Cl p H E : global units of H, and E r : E{E p U p : ź P p U P, where U P are local units at P p in H ru p : U p {U p p ru p,0 Ă r U p Devika Sharma (Weizmann) 23rd June, / 14
30 Notation H : field cut out by the projective image of ρ Cl H : class group of H, and Cl Ă H : Cl H {Cl p H E : global units of H, and E r : E{E p U p : ź P p U P, where U P are local units at P p in H ru p : U p {U p p ru p,0 Ă r U p Remark The spaces Ă Cl H, r E, r U p and r U p,0 are all F p rg Q s-modules. Devika Sharma (Weizmann) 23rd June, / 14
31 Notation Definition Let W be the space M 2 pf p q with the conjugation action of G Q via ρ. Devika Sharma (Weizmann) 23rd June, / 14
32 Notation Definition Let W be the space M 2 pf p q with the conjugation action of G Q via ρ. Let W 0 be the submodule of trace 0 matrices in W. Devika Sharma (Weizmann) 23rd June, / 14
33 Notation Definition Let W be the space M 2 pf p q with the conjugation action of G Q via ρ. Let W 0 be the submodule of trace 0 matrices in W. For a finite dimensional F p rg Q s-module V, Devika Sharma (Weizmann) 23rd June, / 14
34 Notation Definition Let W be the space M 2 pf p q with the conjugation action of G Q via ρ. Let W 0 be the submodule of trace 0 matrices in W. For a finite dimensional F p rg Q s-module V, let V Ad : sum of all J-H factors isomorphic to W 0. Devika Sharma (Weizmann) 23rd June, / 14
35 Conditions The conditions are pc1q Cl Ă Ad H is trivial, and Devika Sharma (Weizmann) 23rd June, / 14
36 Conditions The conditions are pc1q Cl Ă Ad H is trivial, and pc2q the composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero. Devika Sharma (Weizmann) 23rd June, / 14
37 Conditions The conditions are pc1q Cl Ă Ad H is trivial, and pc2q the composition r E Ad Ñ r U Ad p Corollary r U Ad p,0 is non-zero. Let f P S 2 pnq, for N ď 1, 000, as above. Then every member of H f is non-split. Devika Sharma (Weizmann) 23rd June, / 14
38 Conditions The conditions are pc1q Cl Ă Ad H is trivial, and pc2q the composition r E Ad Ñ r U Ad p Corollary r U Ad p,0 is non-zero. Let f P S 2 pnq, for N ď 1, 000, as above. Then every member of H f is non-split. Remark When N 118, hypothesis pc2q fails. Devika Sharma (Weizmann) 23rd June, / 14
39 Conditions The conditions are pc1q Cl Ă Ad H is trivial, and pc2q the composition r E Ad Ñ r U Ad p Corollary r U Ad p,0 is non-zero. Let f P S 2 pnq, for N ď 1, 000, as above. Then every member of H f is non-split. Remark When N 118, hypothesis pc2q fails. We give an alternative argument to deal with such cases. Devika Sharma (Weizmann) 23rd June, / 14
40 Conditions pc1q and pc2q Devika Sharma (Weizmann) 23rd June, / 14
41 Conditions pc1q and pc2q Note that W 0 is an irreducible GL 2 pf p q-module, while the conditions (C1) and (C2) are over PGL 2 pf p q. This is because scalars act trivially on W 0. Devika Sharma (Weizmann) 23rd June, / 14
42 Conditions pc1q and pc2q Note that W 0 is an irreducible GL 2 pf p q-module, while the conditions (C1) and (C2) are over PGL 2 pf p q. This is because scalars act trivially on W 0. This works well in computations as the order of GL 2 pf p q is 48, where as PGL 2 pf p q is 24! Devika Sharma (Weizmann) 23rd June, / 14
43 Condition (C1) Devika Sharma (Weizmann) 23rd June, / 14
44 Condition (C1) ĂCl Ad H is trivial Devika Sharma (Weizmann) 23rd June, / 14
45 Condition (C1) ĂCl Ad H is trivial The field H is the splitting field of the 3-division polynomial Φ 3. Devika Sharma (Weizmann) 23rd June, / 14
46 Condition (C1) ĂCl Ad H is trivial The field H is the splitting field of the 3-division polynomial Φ 3. It can be explicitly generated by composing Φ 3, with the polynomial x 3 disc E and x 2 ` 3. Devika Sharma (Weizmann) 23rd June, / 14
47 Condition (C1) ĂCl Ad H is trivial The field H is the splitting field of the 3-division polynomial Φ 3. It can be explicitly generated by composing Φ 3, with the polynomial x 3 disc E and x 2 ` 3. In most examples, p 3 Cl H ùñ pc1q holds. Devika Sharma (Weizmann) 23rd June, / 14
48 Condition (C1) ĂCl Ad H is trivial The field H is the splitting field of the 3-division polynomial Φ 3. It can be explicitly generated by composing Φ 3, with the polynomial x 3 disc E and x 2 ` 3. In most examples, p 3 Cl H ùñ pc1q holds. When p 3 Cl H, we compute the J-H factors to deduce that pc1q holds. Devika Sharma (Weizmann) 23rd June, / 14
49 Condition (C1) ĂCl Ad H is trivial The field H is the splitting field of the 3-division polynomial Φ 3. It can be explicitly generated by composing Φ 3, with the polynomial x 3 disc E and x 2 ` 3. In most examples, p 3 Cl H ùñ pc1q holds. When p 3 Cl H, we compute the J-H factors to deduce that pc1q holds. This uses PARI/GP! Devika Sharma (Weizmann) 23rd June, / 14
50 Condition pc 2q Devika Sharma (Weizmann) 23rd June, / 14
51 Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Devika Sharma (Weizmann) 23rd June, / 14
52 Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 Devika Sharma (Weizmann) 23rd June, / 14
53 Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 r U Ad p» 5W 0 Devika Sharma (Weizmann) 23rd June, / 14
54 Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 r U Ad p» 5W 0 r U Ad p,0» W 0 Devika Sharma (Weizmann) 23rd June, / 14
55 Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 r U Ad p» 5W 0 r U Ad p,0» W 0 To check that pc2q holds, Devika Sharma (Weizmann) 23rd June, / 14
56 Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 r U Ad p» 5W 0 r U Ad p,0» W 0 To check that pc2q holds, we find e P E satisfying Devika Sharma (Weizmann) 23rd June, / 14
57 Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 r U Ad p» 5W 0 r U Ad p,0» W 0 To check that pc2q holds, we find e P E satisfying I pp pq fixes e, for some P p, and Devika Sharma (Weizmann) 23rd June, / 14
58 Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 r U Ad p» 5W 0 r U Ad p,0» W 0 To check that pc2q holds, we find e P E satisfying I pp pq fixes e, for some P p, and e P 1 ` P n, for n ď 2 Devika Sharma (Weizmann) 23rd June, / 14
59 Elliptic curves satisfying conditions pc 1q and pc 2q A f Af pa, bq for A f Cl H e e lies in 89.a1 89 p 1323, 28134q 2 e 2 4 e2 5 e 2 6 e 7e8 2e ` P a p12528, q 2 3 e2 4e4 3 e4 4 e6 5 e 4 7 e4 8 e2 9 1 ` P b p 1323, 65178q 158.b p 4563, q 2 3 e1 2e 2e 3 e e ` P c p , q 158.e p 11691, q b p , q e 5 e e e2 8 e 2 9 e ` P e p 27243, q Devika Sharma (Weizmann) 23rd June, / 14
60 Exception: Example 118 Devika Sharma (Weizmann) 23rd June, / 14
61 Exception: Example 118 In this example, Cl H 2 implies that pc1q holds, Devika Sharma (Weizmann) 23rd June, / 14
62 Exception: Example 118 In this example, Cl H 2 implies that pc1q holds, while Condition pc2q fails! Devika Sharma (Weizmann) 23rd June, / 14
63 Exception: Example 118 In this example, Cl H 2 implies that pc1q holds, while Condition pc2q fails! Alternative condition pc2 1 q: This involves showing that a particular totally ramified Z{3-extension K 3 over Q 3 is distinct from the cyclotomic Z{3-extension over Q 3. Devika Sharma (Weizmann) 23rd June, / 14
64 Exception: Example 118 In this example, Cl H 2 implies that pc1q holds, while Condition pc2q fails! Alternative condition pc2 1 q: This involves showing that a particular totally ramified Z{3-extension K 3 over Q 3 is distinct from the cyclotomic Z{3-extension over Q 3. Checking the alternative condition includes explicitly computing the norm subgroup corresponding to K 3 {Q 3. This uses PARI-GP extensively. Devika Sharma (Weizmann) 23rd June, / 14
65 Thank You. Devika Sharma (Weizmann) 23rd June, / 14
Mod p Galois representations attached to modular forms
Mod p Galois representations attached to modular forms Ken Ribet UC Berkeley April 7, 2006 After Serre s article on elliptic curves was written in the early 1970s, his techniques were generalized and extended
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics MOD p REPRESENTATIONS ON ELLIPTIC CURVES FRANK CALEGARI Volume 225 No. 1 May 2006 PACIFIC JOURNAL OF MATHEMATICS Vol. 225, No. 1, 2006 MOD p REPRESENTATIONS ON ELLIPTIC
More informationMA 162B LECTURE NOTES: THURSDAY, FEBRUARY 26
MA 162B LECTURE NOTES: THURSDAY, FEBRUARY 26 1. Abelian Varieties of GL 2 -Type 1.1. Modularity Criteria. Here s what we ve shown so far: Fix a continuous residual representation : G Q GLV, where V is
More informationSERRE S CONJECTURE AND BASE CHANGE FOR GL(2)
SERRE S CONJECTURE AND BASE CHANGE OR GL(2) HARUZO HIDA 1. Quaternion class sets A quaternion algebra B over a field is a simple algebra of dimension 4 central over a field. A prototypical example is the
More informationWorkshop on Serre s Modularity Conjecture: the level one case
Workshop on Serre s Modularity Conjecture: the level one case UC Berkeley Monte Verità 13 May 2009 Notation We consider Serre-type representations of G = Gal(Q/Q). They will always be 2-dimensional, continuous
More informationNUNO FREITAS AND ALAIN KRAUS
ON THE DEGREE OF THE p-torsion FIELD OF ELLIPTIC CURVES OVER Q l FOR l p NUNO FREITAS AND ALAIN KRAUS Abstract. Let l and p be distinct prime numbers with p 3. Let E/Q l be an elliptic curve with p-torsion
More informationResidual modular Galois representations: images and applications
Residual modular Galois representations: images and applications Samuele Anni University of Warwick London Number Theory Seminar King s College London, 20 th May 2015 Mod l modular forms 1 Mod l modular
More information15 Elliptic curves and Fermat s last theorem
15 Elliptic curves and Fermat s last theorem Let q > 3 be a prime (and later p will be a prime which has no relation which q). Suppose that there exists a non-trivial integral solution to the Diophantine
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik On projective linear groups over finite fields as Galois groups over the rational numbers Gabor Wiese Preprint Nr. 14/2006 On projective linear groups over finite fields
More informationTwists and residual modular Galois representations
Twists and residual modular Galois representations Samuele Anni University of Warwick Building Bridges, Bristol 10 th July 2014 Modular curves and Modular Forms 1 Modular curves and Modular Forms 2 Residual
More informationFORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS
Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ
More information10 l-adic representations
0 l-adic representations We fix a prime l. Artin representations are not enough; l-adic representations with infinite images naturally appear in geometry. Definition 0.. Let K be any field. An l-adic Galois
More informationFinding Galois representations corresponding to certain Hecke eigenclasses
International Journal of Number Theory c World Scientific Publishing Company Finding Galois representations corresponding to certain Hecke eigenclasses Meghan De Witt Department of Mathematics University
More informationSome algebraic number theory and the reciprocity map
Some algebraic number theory and the reciprocity map Ervin Thiagalingam September 28, 2015 Motivation In Weinstein s paper, the main problem is to find a rule (reciprocity law) for when an irreducible
More informationRaising the Levels of Modular Representations Kenneth A. Ribet
1 Raising the Levels of Modular Representations Kenneth A. Ribet 1 Introduction Let l be a prime number, and let F be an algebraic closure of the prime field F l. Suppose that ρ : Gal(Q/Q) GL(2, F) is
More informationREDUCTION OF ELLIPTIC CURVES OVER CERTAIN REAL QUADRATIC NUMBER FIELDS
MATHEMATICS OF COMPUTATION Volume 68, Number 228, Pages 1679 1685 S 0025-5718(99)01129-1 Article electronically published on May 21, 1999 REDUCTION OF ELLIPTIC CURVES OVER CERTAIN REAL QUADRATIC NUMBER
More informationModularity of Galois representations. imaginary quadratic fields
over imaginary quadratic fields Krzysztof Klosin (joint with T. Berger) City University of New York December 3, 2011 Notation F =im. quadr. field, p # Cl F d K, fix p p Notation F =im. quadr. field, p
More informationVARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES
VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON 1. Introduction Let ρ : G Q GL 2 (k) be an absolutely irreducible modular Galois representation over a finite
More informationEndomorphism algebras of semistable abelian varieties over Q of GL(2)-type
of semistable abelian varieties over Q of GL(2)-type UC Berkeley Tatefest May 2, 2008 The abelian varieties in the title are now synonymous with certain types of modular forms. (This is true because we
More informationProblems on Growth of Hecke fields
Problems on Growth of Hecke fields Haruzo Hida Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A. A list of conjectures/problems related to my talk in Simons Conference in January 2014
More informationReciprocity maps with restricted ramification
Reciprocity maps with restricted ramification Romyar Sharifi UCLA January 6, 2016 1 / 19 Iwasawa theory for modular forms Let be an p odd prime and f a newform of level N. Suppose that f is ordinary at
More informationEKNATH GHATE AND VINAYAK VATSAL. 1. Introduction
ON THE LOCAL BEHAVIOUR OF ORDINARY Λ-ADIC REPRESENTATIONS EKNATH GHATE AND VINAYAK VATSAL 1. Introduction In this paper we study the local behaviour of the Galois representations attached to ordinary Λ-adic
More informationLecture 5: Schlessinger s criterion and deformation conditions
Lecture 5: Schlessinger s criterion and deformation conditions Brandon Levin October 30, 2009 1. What does it take to be representable? We have been discussing for several weeks deformation problems, and
More informationGalois groups with restricted ramification
Galois groups with restricted ramification Romyar Sharifi Harvard University 1 Unique factorization: Let K be a number field, a finite extension of the rational numbers Q. The ring of integers O K of K
More informationGalois Representations
Galois Representations Samir Siksek 12 July 2016 Representations of Elliptic Curves Crash Course E/Q elliptic curve; G Q = Gal(Q/Q); p prime. Fact: There is a τ H such that E(C) = C Z + τz = R Z R Z. Easy
More informationGalois Representations
9 Galois Representations This book has explained the idea that all elliptic curves over Q arise from modular forms. Chapters 1 and introduced elliptic curves and modular curves as Riemann surfaces, and
More informationEXTENSIONS AND THE EXCEPTIONAL ZERO OF THE ADJOINT SQUARE L-FUNCTIONS
EXTENSIONS AND THE EXCEPTIONAL ZERO OF THE ADJOINT SQUARE L-FUNCTIONS HARUZO HIDA Take a totally real field F with integer ring O as a base field. We fix an identification ι : Q p = C Q. Fix a prime p>2,
More informationORAL QUALIFYING EXAM QUESTIONS. 1. Algebra
ORAL QUALIFYING EXAM QUESTIONS JOHN VOIGHT Below are some questions that I have asked on oral qualifying exams (starting in fall 2015). 1.1. Core questions. 1. Algebra (1) Let R be a noetherian (commutative)
More informationThe Kronecker-Weber Theorem
The Kronecker-Weber Theorem November 30, 2007 Let us begin with the local statement. Theorem 1 Let K/Q p be an abelian extension. Then K is contained in a cyclotomic extension of Q p. Proof: We give the
More information5 Dedekind extensions
18.785 Number theory I Fall 2016 Lecture #5 09/22/2016 5 Dedekind extensions In this lecture we prove that the integral closure of a Dedekind domain in a finite extension of its fraction field is also
More informationCOUNTING MOD l SOLUTIONS VIA MODULAR FORMS
COUNTING MOD l SOLUTIONS VIA MODULAR FORMS EDRAY GOINS AND L. J. P. KILFORD Abstract. [Something here] Contents 1. Introduction 1. Galois Representations as Generating Functions 1.1. Permutation Representation
More informationABSTRACT ALGEBRA 1 COURSE NOTES, LECTURE 11: SYLOW THEORY.
ABSTRACT ALGEBRA 1 COURSE NOTES, LECTURE 11: SYLOW THEORY. ANDREW SALCH Here s a quick definition we could have introduced a long time ago: Definition 0.1. If n is a positive integer, we often write C
More informationIntroduction to Shimura Curves
Introduction to Shimura Curves Alyson Deines Today 1 Motivation In class this quarter we spent a considerable amount of time studying the curves X(N = Γ(N \ H, X 0 (N = Γ 0 (N \ H, X 1 (NΓ 1 (N \ H, where
More informationA BRIEF INTRODUCTION TO LOCAL FIELDS
A BRIEF INTRODUCTION TO LOCAL FIELDS TOM WESTON The purpose of these notes is to give a survey of the basic Galois theory of local fields and number fields. We cover much of the same material as [2, Chapters
More informationIndependence of Heegner Points Joseph H. Silverman (Joint work with Michael Rosen)
Independence of Heegner Points Joseph H. Silverman (Joint work with Michael Rosen) Brown University Cambridge University Number Theory Seminar Thursday, February 22, 2007 0 Modular Curves and Heegner Points
More informationNON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES
NON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES MATTHEW BOYLAN Abstract Let pn be the ordinary partition function We show, for all integers r and s with s 1 and 0 r < s, that #{n : n r mod
More informationLecture 11: Clifford algebras
Lecture 11: Clifford algebras In this lecture we introduce Clifford algebras, which will play an important role in the rest of the class. The link with K-theory is the Atiyah-Bott-Shapiro construction
More informationALGEBRA PH.D. QUALIFYING EXAM SOLUTIONS October 20, 2011
ALGEBRA PH.D. QUALIFYING EXAM SOLUTIONS October 20, 2011 A passing paper consists of four problems solved completely plus significant progress on two other problems; moreover, the set of problems solved
More informationPseudo-Modularity and Iwasawa Theory
Pseudo-Modularity and Iwasawa Theory Preston Wake and Carl Wang Erickson April 11, 2015 Preston Wake and Carl Wang Erickson Pseudo-Modularity and Iwasawa Theory April 11, 2015 1 / 15 The Ordinary Eigencurve
More information14 From modular forms to automorphic representations
14 From modular forms to automorphic representations We fix an even integer k and N > 0 as before. Let f M k (N) be a modular form. We would like to product a function on GL 2 (A Q ) out of it. Recall
More informationComputing coefficients of modular forms
Computing coefficients of modular forms (Work in progress; extension of results of Couveignes, Edixhoven et al.) Peter Bruin Mathematisch Instituut, Universiteit Leiden Théorie des nombres et applications
More informationSOLVING FERMAT-TYPE EQUATIONS x 5 + y 5 = dz p
MATHEMATICS OF COMPUTATION Volume 79, Number 269, January 2010, Pages 535 544 S 0025-5718(09)02294-7 Article electronically published on July 22, 2009 SOLVING FERMAT-TYPE EQUATIONS x 5 + y 5 = dz p NICOLAS
More informationOn the generation of the coefficient field of a newform by a single Hecke eigenvalue
On the generation of the coefficient field of a newform by a single Hecke eigenvalue Koopa Tak-Lun Koo and William Stein and Gabor Wiese November 2, 27 Abstract Let f be a non-cm newform of weight k 2
More informationClass groups and Galois representations
and Galois representations UC Berkeley ENS February 15, 2008 For the J. Herbrand centennaire, I will revisit a subject that I studied when I first came to Paris as a mathematician, in 1975 1976. At the
More informationTHE PARAMODULAR CONJECTURE ARMAND BRUMER
THE PARAMODULAR CONJECTURE ARMAND BRUMER (Joint work with Ken Kramer and Magma) Modular Forms and Curves of Low Genus: Computational Aspects @ ICERM Sept. 30, 2015 B&Kramer: Certain abelian varieties bad
More informationFlag Varieties. Matthew Goroff November 2, 2016
Flag Varieties Matthew Goroff November 2, 2016 1. Grassmannian Variety Definition 1.1: Let V be a k-vector space of dimension n. The Grassmannian Grpr, V q is the set of r-dimensional subspaces of V. It
More informationFinite Subgroups of Gl 2 (C) and Universal Deformation Rings
Finite Subgroups of Gl 2 (C) and Universal Deformation Rings University of Missouri Conference on Geometric Methods in Representation Theory November 21, 2016 Goal Goal : Find connections between fusion
More informationComputing complete lists of Artin representations given a group, character, and conductor bound
Computing complete lists of Artin representations given a group, character, and conductor bound John Jones Joint work with David Roberts 7/27/15 Computing Artin Representations John Jones SoMSS 1 / 21
More informationImage of big Galois representations and modular forms mod p
Image of big Galois representations and modular forms mod p Notes by Tony Feng for a talk by Joel Bellaiche June 14, 016 1 Introduction This will be a talk about modular forms mod p, specifically: the
More informationCover Page. The handle holds various files of this Leiden University dissertation.
Cover Page The handle http://hdl.handle.net/1887/22043 holds various files of this Leiden University dissertation. Author: Anni, Samuele Title: Images of Galois representations Issue Date: 2013-10-24 Chapter
More informationarxiv:math/ v2 [math.nt] 29 Apr 2003
Modularity of rigid Calabi-Yau threefolds over Q Luis Dieulefait and Jayanta Manoharmayum arxiv:math/0304434v2 [math.nt] 29 Apr 2003 Abstract We prove modularity for a huge class of rigid Calabi-Yau threefolds
More informationAWS 2018, Problem Session (Algebraic aspects of Iwasawa theory)
AWS 2018, Problem Session (Algebraic aspects of Iwasawa theory) Kâzım Büyükboduk March 3-7, 2018 Contents 1 Commutative Algebra 1 2 Classical Iwasawa Theory (of Tate motives) 2 3 Galois cohomology and
More informationNorm Groups with Tame Ramification
ECTURE 3 orm Groups with Tame Ramification et K be a field with char(k) 6=. Then K /(K ) ' {continuous homomorphisms Gal(K) Z/Z} ' {degree étale algebras over K} which is dual to our original statement
More informationOn the generalized Fermat equation x 2l + y 2m = z p
On the generalized Fermat equation x 2l + y 2m = z p Samuele Anni joint work with Samir Siksek University of Warwick University of Debrecen, 29 th Journées Arithmétiques; 6 th July 2015 Generalized Fermat
More informationLecture 4: Examples of automorphic forms on the unitary group U(3)
Lecture 4: Examples of automorphic forms on the unitary group U(3) Lassina Dembélé Department of Mathematics University of Calgary August 9, 2006 Motivation The main goal of this talk is to show how one
More informationWildly ramified Galois representations and a generalization of a conjecture of Serre
Wildly ramified Galois representations and a generalization of a conjecture of Serre Darrin Doud Brigham Young University Department of Mathematics 292 TMCB Provo, UT 84602 November 22, 2004 Abstract Serre
More informationHONDA-TATE THEOREM FOR ELLIPTIC CURVES
HONDA-TATE THEOREM FOR ELLIPTIC CURVES MIHRAN PAPIKIAN 1. Introduction These are the notes from a reading seminar for graduate students that I organised at Penn State during the 2011-12 academic year.
More informationOn The Structure of Certain Galois Cohomology Groups
Documenta Math. 335 On The Structure of Certain Galois Cohomology Groups To John Coates on the occasion of his 60th birthday Ralph Greenberg Received: October 31, 2005 Revised: February 14, 2006 Abstract.
More informationALGEBRA PH.D. QUALIFYING EXAM September 27, 2008
ALGEBRA PH.D. QUALIFYING EXAM September 27, 2008 A passing paper consists of four problems solved completely plus significant progress on two other problems; moreover, the set of problems solved completely
More informationKIDA S FORMULA AND CONGRUENCES
University of Massachusetts Amherst ScholarWorks@UMass Amherst Mathematics and Statistics Department Faculty Publication Series Mathematics and Statistics 2005 KIDA S FORMULA AND CONGRUENCES R Pollack
More informationGraduate Preliminary Examination
Graduate Preliminary Examination Algebra II 18.2.2005: 3 hours Problem 1. Prove or give a counter-example to the following statement: If M/L and L/K are algebraic extensions of fields, then M/K is algebraic.
More informationLECTURE NOTES AMRITANSHU PRASAD
LECTURE NOTES AMRITANSHU PRASAD Let K be a field. 1. Basic definitions Definition 1.1. A K-algebra is a K-vector space together with an associative product A A A which is K-linear, with respect to which
More informationMorava K-theory of BG: the good, the bad and the MacKey
Morava K-theory of BG: the good, the bad and the MacKey Ruhr-Universität Bochum 15th May 2012 Recollections on Galois extensions of commutative rings Let R, S be commutative rings with a ring monomorphism
More informationThe study of 2-dimensional p-adic Galois deformations in the l p case
The study of 2-dimensional p-adic Galois deformations in the l p case Vincent Pilloni April 22, 2008 Abstract We introduce the language and give the classical results from the theory of deformations :
More informationGalois theory (Part II)( ) Example Sheet 1
Galois theory (Part II)(2015 2016) Example Sheet 1 c.birkar@dpmms.cam.ac.uk (1) Find the minimal polynomial of 2 + 3 over Q. (2) Let K L be a finite field extension such that [L : K] is prime. Show that
More informationON THE MODIFIED MOD p LOCAL LANGLANDS CORRESPONDENCE FOR GL 2 (Q l )
ON THE MODIFIED MOD p LOCAL LANGLANDS CORRESPONDENCE FOR GL 2 (Q l ) DAVID HELM We give an explicit description of the modified mod p local Langlands correspondence for GL 2 (Q l ) of [EH], Theorem 5.1.5,
More informationOn the arithmetic of modular forms
On the arithmetic of modular forms Gabor Wiese 15 June 2017 Modular forms There are five fundamental operations: addition, subtraction, multiplication, division, and modular forms. Martin Eichler (1912-1992)
More informationALGEBRA 11: Galois theory
Galois extensions Exercise 11.1 (!). Consider a polynomial P (t) K[t] of degree n with coefficients in a field K that has n distinct roots in K. Prove that the ring K[t]/P of residues modulo P is isomorphic
More informationMA 162B LECTURE NOTES: FRIDAY, JANUARY 30
MA 162B LECTURE NOTES: FRIDAY, JANUARY 30 1. Examples of Cohomology Groups (cont d) 1.1. H 2 and Projective Galois Representations. Say we have a projective Galois representation ρ : G P GL(V ) where either
More informationModules over Principal Ideal Domains
Modules over Principal Ideal Domains Let henceforth R denote a commutative ring with 1. It is called a domain iff it has no zero-divisors, i.e. if ab = 0 then either a or b is zero. Or equivalently, two
More informationON TAMELY RAMIFIED IWASAWA MODULES FOR THE CYCLOTOMIC p -EXTENSION OF ABELIAN FIELDS
Itoh, T. Osaka J. Math. 51 (2014), 513 536 ON TAMELY RAMIFIED IWASAWA MODULES FOR THE CYCLOTOMIC p -EXTENSION OF ABELIAN FIELDS TSUYOSHI ITOH (Received May 18, 2012, revised September 19, 2012) Abstract
More informationA PROOF OF BURNSIDE S p a q b THEOREM
A PROOF OF BURNSIDE S p a q b THEOREM OBOB Abstract. We prove that if p and q are prime, then any group of order p a q b is solvable. Throughout this note, denote by A the set of algebraic numbers. We
More informationNumber Theory Fall 2016 Problem Set #6
18.785 Number Theory Fall 2016 Problem Set #6 Description These problems are related to the material covered in Lectures 10-12. Your solutions are to be written up in latex (you can use the latex source
More information5 Dedekind extensions
18.785 Number theory I Fall 2017 Lecture #5 09/20/2017 5 Dedekind extensions In this lecture we prove that the integral closure of a Dedekind domain in a finite extension of its fraction field is also
More informationClass numbers of cubic cyclic. By Koji UCHIDA. (Received April 22, 1973)
J. Math. Vol. 26, Soc. Japan No. 3, 1974 Class numbers of cubic cyclic fields By Koji UCHIDA (Received April 22, 1973) Let n be any given positive integer. It is known that there exist real. (imaginary)
More informationALGEBRA EXERCISES, PhD EXAMINATION LEVEL
ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)
More informationCover Page. The handle holds various files of this Leiden University dissertation.
Cover Page The handle http://hdl.handle.net/1887/20310 holds various files of this Leiden University dissertation. Author: Jansen, Bas Title: Mersenne primes and class field theory Date: 2012-12-18 Chapter
More informationp-adic fields Chapter 7
Chapter 7 p-adic fields In this chapter, we study completions of number fields, and their ramification (in particular in the Galois case). We then look at extensions of the p-adic numbers Q p and classify
More informationPrimes of the Form x 2 + ny 2
Primes of the Form x 2 + ny 2 Steven Charlton 28 November 2012 Outline 1 Motivating Examples 2 Quadratic Forms 3 Class Field Theory 4 Hilbert Class Field 5 Narrow Class Field 6 Cubic Forms 7 Modular Forms
More informationCourse 2316 Sample Paper 1
Course 2316 Sample Paper 1 Timothy Murphy April 19, 2015 Attempt 5 questions. All carry the same mark. 1. State and prove the Fundamental Theorem of Arithmetic (for N). Prove that there are an infinity
More informationBjorn Poonen. April 30, 2006
in in University of California at Berkeley April 30, 2006 of gonality in Let X be a smooth, projective, geometrically integral curve of genus g over a field k. Define the gonality of X as γ k (X ) := min{
More informationThe Patching Argument
The Patching Argument Brandon Levin August 2, 2010 1 Motivating the Patching Argument My main references for this talk were Andrew s overview notes and Kisin s paper Moduli of Finite Flat Group Schemes.
More informationHonours Research Project: Modular forms and Galois representations mod p, and the nilpotent action of Hecke operators mod 2
Honours Research Project: Modular forms and Galois representations mod p, and the nilpotent action of Hecke operators mod 2 Mathilde Gerbelli-Gauthier May 20, 2014 Abstract We study Hecke operators acting
More informationPractice Algebra Qualifying Exam Solutions
Practice Algebra Qualifying Exam Solutions 1. Let A be an n n matrix with complex coefficients. Define tr A to be the sum of the diagonal elements. Show that tr A is invariant under conjugation, i.e.,
More informationThus, the integral closure A i of A in F i is a finitely generated (and torsion-free) A-module. It is not a priori clear if the A i s are locally
Math 248A. Discriminants and étale algebras Let A be a noetherian domain with fraction field F. Let B be an A-algebra that is finitely generated and torsion-free as an A-module with B also locally free
More informationOn the modular curve X 0 (23)
On the modular curve X 0 (23) René Schoof Abstract. The Jacobian J 0(23) of the modular curve X 0(23) is a semi-stable abelian variety over Q with good reduction outside 23. It is simple. We prove that
More informationNumber Theory Fall 2016 Problem Set #3
18.785 Number Theory Fall 2016 Problem Set #3 Description These problems are related to the material covered in Lectures 5-7. Your solutions are to be written up in latex and submitted as a pdf-file via
More informationExistence of Taylor-Wiles Primes
Existence of Taylor-Wiles Primes Michael Lipnowski Introduction Let F be a totally real number field, ρ = ρ f : G F GL 2 (k) be ( an odd residually ) modular representation (odd meaning that complex conjugation
More informationProven Cases of a Generalization of Serre's Conjecture
Brigham Young University BYU ScholarsArchive All Theses and Dissertations 2006-07-07 Proven Cases of a Generalization of Serre's Conjecture Jonathan H. Blackhurst Brigham Young University - Provo Follow
More information7 Orders in Dedekind domains, primes in Galois extensions
18.785 Number theory I Lecture #7 Fall 2015 10/01/2015 7 Orders in Dedekind domains, primes in Galois extensions 7.1 Orders in Dedekind domains Let S/R be an extension of rings. The conductor c of R (in
More informationTunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society
Tunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society Grothendieck Messing deformation theory for varieties of K3 type Andreas Langer and Thomas Zink
More informationON GALOIS GROUPS OF ABELIAN EXTENSIONS OVER MAXIMAL CYCLOTOMIC FIELDS. Mamoru Asada. Introduction
ON GALOIS GROUPS OF ABELIAN ETENSIONS OVER MAIMAL CYCLOTOMIC FIELDS Mamoru Asada Introduction Let k 0 be a finite algebraic number field in a fixed algebraic closure Ω and ζ n denote a primitive n-th root
More informationDiophantine equations and semistable elliptic curves over totally real fields
Diophantine equations and semistable elliptic curves over totally real fields Samuele Anni (IWR - Universität Heidelberg) joint with Samir Siksek (University of Warwick) Journées Algophantiennes Bordelaises
More informationThe Kpπ,1q-property for marked curves over finite fields
The Kpπ,1q-property for marked curves over finite fields Philippe Lebacque and Alexander Schmidt arxiv:1401.1979v2 [math.nt] 24 Mar 2015 Abstract We investigate the Kpπ,1q-property for p of smooth, marked
More informationTwisted L-Functions and Complex Multiplication
Journal of umber Theory 88, 104113 (2001) doi:10.1006jnth.2000.2613, available online at http:www.idealibrary.com on Twisted L-Functions and Complex Multiplication Abdellah Sebbar Department of Mathematics
More informationON 2-CLASS FIELD TOWERS OF SOME IMAGINARY QUADRATIC NUMBER FIELDS
ON 2-CLASS FIELD TOWERS OF SOME IMAGINARY QUADRATIC NUMBER FIELDS FRANZ LEMMERMEYER Abstract. We construct an infinite family of imaginary quadratic number fields with 2-class groups of type (2, 2, 2 whose
More informationAlgebraic number theory Revision exercises
Algebraic number theory Revision exercises Nicolas Mascot (n.a.v.mascot@warwick.ac.uk) Aurel Page (a.r.page@warwick.ac.uk) TA: Pedro Lemos (lemos.pj@gmail.com) Version: March 2, 20 Exercise. What is the
More informationDONG QUAN NGOC NGUYEN
REPRESENTATION OF UNITS IN CYCLOTOMIC FUNCTION FIELDS DONG QUAN NGOC NGUYEN Contents 1 Introduction 1 2 Some basic notions 3 21 The Galois group Gal(K /k) 3 22 Representation of integers in O, and the
More informationEXERCISES IN MODULAR FORMS I (MATH 726) (2) Prove that a lattice L is integral if and only if its Gram matrix has integer coefficients.
EXERCISES IN MODULAR FORMS I (MATH 726) EYAL GOREN, MCGILL UNIVERSITY, FALL 2007 (1) We define a (full) lattice L in R n to be a discrete subgroup of R n that contains a basis for R n. Prove that L is
More informationDieudonné Modules and p-divisible Groups
Dieudonné Modules and p-divisible Groups Brian Lawrence September 26, 2014 The notion of l-adic Tate modules, for primes l away from the characteristic of the ground field, is incredibly useful. The analogous
More information