Modularity of Galois representations. imaginary quadratic fields
|
|
- Gervais George
- 5 years ago
- Views:
Transcription
1 over imaginary quadratic fields Krzysztof Klosin (joint with T. Berger) City University of New York December 3, 2011
2 Notation F =im. quadr. field, p # Cl F d K, fix p p
3 Notation F =im. quadr. field, p # Cl F d K, fix p p Σ = finite set of finite places of F, p, p Σ, G Σ = Gal(F Σ /F );
4 Notation F =im. quadr. field, p # Cl F d K, fix p p Σ = finite set of finite places of F, p, p Σ, G Σ = Gal(F Σ /F ); O= ring of integers in a finite extension of Q p, ϖ=uniformizer, F = O/ϖ;
5 Notation F =im. quadr. field, p # Cl F d K, fix p p Σ = finite set of finite places of F, p, p Σ, G Σ = Gal(F Σ /F ); O= ring of integers in a finite extension of Q p, ϖ=uniformizer, F = O/ϖ; Ψ= (unramified) Hecke character of -type z z, Ψ p : G Σ O the associated Galois character, χ 0 = Ψ p (mod ϖ)
6 Main Theorem Theorem (Berger-K., 2011) Suppose that dim F Ext 1 F[G Σ ] (χ 0, 1) = 1.
7 Main Theorem Theorem (Berger-K., 2011) Suppose that dim F Ext 1 F[G Σ ] (χ 0, 1) = 1. Let ρ : G Σ GL 2 (Q p ) be continuous and irreducible. Suppose that:
8 Main Theorem Theorem (Berger-K., 2011) Suppose that dim F Ext 1 F[G Σ ] (χ 0, 1) = 1. Let ρ : G Σ GL 2 (Q p ) be continuous and irreducible. Suppose that: det ρ = Ψ p
9 Main Theorem Theorem (Berger-K., 2011) Suppose that dim F Ext 1 F[G Σ ] (χ 0, 1) = 1. Let ρ : G Σ GL 2 (Q p ) be continuous and irreducible. Suppose that: det ρ = Ψ p ρ ss = 1 χ 0
10 Main Theorem Theorem (Berger-K., 2011) Suppose that dim F Ext 1 F[G Σ ] (χ 0, 1) = 1. Let ρ : G Σ GL 2 (Q p ) be continuous and irreducible. Suppose that: det ρ = Ψ p ρ ss = 1 χ 0 ρ is crystalline (or ordinary if p splits) ρ is minimally ramified.
11 Main Theorem Theorem (Berger-K., 2011) Suppose that dim F Ext 1 F[G Σ ] (χ 0, 1) = 1. Let ρ : G Σ GL 2 (Q p ) be continuous and irreducible. Suppose that: det ρ = Ψ p ρ ss = 1 χ 0 ρ is crystalline (or ordinary if p splits) ρ is minimally ramified. Then ρ is modular, i.e., L(ρ γ, s) = L(s, π) for some automorphic representation π of GL 2 (A F ).
12 Remarks to the Main Theorem 1 At the moment this is the only known modularity result for an imaginary quadratic field.
13 Remarks to the Main Theorem 1 At the moment this is the only known modularity result for an imaginary quadratic field. 2 This is similar to a result of Skinner and Wiles which applies to Q or a totally real field, but their method fails for F =imaginary quadratic. An important step in their method is the existence of an ordinary, minimal deformation [ ] ρ = : G 0 Q GL 2 (O) of the residual representation [ ] 1 ρ 0 = = 1 χ 0 χ 0 0 (uses Kummer theory).
14 Remarks to the Main Theorem 1 At the moment this is the only known modularity result for an imaginary quadratic field. 2 This is similar to a result of Skinner and Wiles which applies to Q or a totally real field, but their method fails for F =imaginary quadratic. An important step in their method is the existence of an ordinary, minimal deformation [ ] ρ = : G 0 Q GL 2 (O) of the residual representation [ ] 1 ρ 0 = = 1 χ 0 χ 0 0 (uses Kummer theory). But Theorem (Berger-K.) No such deformation ρ exists for F.
15 Remarks to the Main Theorem We do not follow [SW]-strategy. Instead we develop a commutative algebra criterion that allows one to reduce the problem of modularity of all deformations of ρ 0 to that of modularity of the reducible deformations of ρ 0.
16 Remarks to the Main Theorem We do not follow [SW]-strategy. Instead we develop a commutative algebra criterion that allows one to reduce the problem of modularity of all deformations of ρ 0 to that of modularity of the reducible deformations of ρ 0. The condition dim F Ext 1 F[G Σ ] (χ 0, 1) = 1 is (probably) essential (work in progress).
17 Remarks to the Main Theorem We do not follow [SW]-strategy. Instead we develop a commutative algebra criterion that allows one to reduce the problem of modularity of all deformations of ρ 0 to that of modularity of the reducible deformations of ρ 0. The condition dim F Ext 1 F[G Σ ] (χ 0, 1) = 1 is (probably) essential (work in progress). The unramifiedness of Ψ-condition can be replaced by demanding that H 2 c (S Kf, Z p ) tors = 0.
18 Method [ ] 1 Let ρ 0 = : G 0 χ Σ GL 2 (F) be a non-semisimple 0 residual representation.
19 Method [ ] 1 Let ρ 0 = : G 0 χ Σ GL 2 (F) be a non-semisimple 0 residual representation. We study the crystalline (or ordinary if p splits) deformations of ρ 0.
20 Method [ ] 1 Let ρ 0 = : G 0 χ Σ GL 2 (F) be a non-semisimple 0 residual representation. We study the crystalline (or ordinary if p splits) deformations of ρ 0. There exists a universal couple (R Σ, ρ Σ : G Σ GL 2 (R Σ ))
21 Method [ ] 1 Let ρ 0 = : G 0 χ Σ GL 2 (F) be a non-semisimple 0 residual representation. We study the crystalline (or ordinary if p splits) deformations of ρ 0. There exists a universal couple (R Σ, ρ Σ : G Σ GL 2 (R Σ )) One gets a surjection φ : R Σ T Σ
22 Method [ ] 1 Let ρ 0 = : G 0 χ Σ GL 2 (F) be a non-semisimple 0 residual representation. We study the crystalline (or ordinary if p splits) deformations of ρ 0. There exists a universal couple (R Σ, ρ Σ : G Σ GL 2 (R Σ )) One gets a surjection φ : R Σ T Σ Goal: Show that φ is an isomorphism.
23 Ideal of reducibility I re := the smallest ideal I of R Σ such that for χ 1, χ 2 characters. tr ρ Σ = χ 1 + χ 2 (mod I )
24 Ideal of reducibility I re := the smallest ideal I of R Σ such that for χ 1, χ 2 characters. tr ρ Σ = χ 1 + χ 2 (mod I ) R Σ /I re controls the reducible deformations Key idea: Reduce the problem to that of modularity of reducible lifts.
25 Commutative algebra criterion Theorem (Berger-K.) Let R, S be commutative rings. Choose r R such that n r n R = 0. Let A be a domain and suppose that S is a finitely generated free module over A. Suppose we have a commutative diagrams of ring maps: R φ S R/rR φ S/φ(r)S.
26 Commutative algebra criterion Theorem (Berger-K.) Let R, S be commutative rings. Choose r R such that n r n R = 0. Let A be a domain and suppose that S is a finitely generated free module over A. Suppose we have a commutative diagrams of ring maps: R φ S R/rR φ S/φ(r)S. If rk A S/φ(r)S = 0, then φ is an isomorphism.
27 Commutative algebra criterion Theorem (Berger-K.) Let R, S be commutative rings. Choose r R such that n r n R = 0. Let A be a domain and suppose that S is a finitely generated free module over A. Suppose we have a commutative diagrams of ring maps: R φ S R/rR φ S/φ(r)S. If rk A S/φ(r)S = 0, then φ is an isomorphism. The rank condition can be replaced by a condition rr r 2 R = φ(r)s φ(r) 2 S and then the theorem gives an alternative to the criterion of Wiles and Lenstra.
28 Applying the commutative algebra criterion Corollary Set S = T Σ, R = R Σ.
29 Applying the commutative algebra criterion Corollary Set S = T Σ, R = R Σ. Suppose I re = rr and #T Σ /φ(i re )T Σ <.
30 Applying the commutative algebra criterion Corollary Set S = T Σ, R = R Σ. Suppose I re = rr and #T Σ /φ(i re )T Σ <. If the map φ : R Σ T Σ induces an isomorphism then φ is an isomorphism. R Σ /I re = TΣ /φ(i re ),
31 Applying the commutative algebra criterion Corollary Set S = T Σ, R = R Σ. Suppose I re = rr and #T Σ /φ(i re )T Σ <. If the map φ : R Σ T Σ induces an isomorphism then φ is an isomorphism. R Σ /I re = TΣ /φ(i re ), Upshot: To show R Σ = T Σ it suffices to prove:
32 Applying the commutative algebra criterion Corollary Set S = T Σ, R = R Σ. Suppose I re = rr and #T Σ /φ(i re )T Σ <. If the map φ : R Σ T Σ induces an isomorphism then φ is an isomorphism. R Σ /I re = TΣ /φ(i re ), Upshot: To show R Σ = T Σ it suffices to prove: I re is principal,
33 Applying the commutative algebra criterion Corollary Set S = T Σ, R = R Σ. Suppose I re = rr and #T Σ /φ(i re )T Σ <. If the map φ : R Σ T Σ induces an isomorphism then φ is an isomorphism. R Σ /I re = TΣ /φ(i re ), Upshot: To show R Σ = T Σ it suffices to prove: I re is principal, R Σ /I re = TΣ /φ(i re ), i.e., that every reducible deformation of ρ 0 is modular.
34 Applying the commutative algebra criterion Corollary Set S = T Σ, R = R Σ. Suppose I re = rr and #T Σ /φ(i re )T Σ <. If the map φ : R Σ T Σ induces an isomorphism then φ is an isomorphism. R Σ /I re = TΣ /φ(i re ), Upshot: To show R Σ = T Σ it suffices to prove: I re is principal, R Σ /I re = TΣ /φ(i re ), i.e., that every reducible deformation of ρ 0 is modular. Remark: The criterion removes the condition p B 2,ω k 2 from a modularity result for residually reducible Galois representations over Q due to Calegari.
35 Principality of the ideal of reducibility Theorem (Bellaïche-Chenevier, Calegari) If dim F Ext 1 F[G Σ ] (χ 0, 1) = dim F Ext 1 F[G Σ ] (1, χ 0) = 1, then I re is principal.
36 Principality of the ideal of reducibility Theorem (Bellaïche-Chenevier, Calegari) If dim F Ext 1 F[G Σ ] (χ 0, 1) = dim F Ext 1 F[G Σ ] (1, χ 0) = 1, then I re is principal. Theorem (Berger-K.) Let A be a Noetherian local ring with 2 A. Set S = A[G Σ ]. Let ρ : S M 2 (A) be an A-algebra map with ρ = ρ 0 mod m A.
37 Principality of the ideal of reducibility Theorem (Bellaïche-Chenevier, Calegari) If dim F Ext 1 F[G Σ ] (χ 0, 1) = dim F Ext 1 F[G Σ ] (1, χ 0) = 1, then I re is principal. Theorem (Berger-K.) Let A be a Noetherian local ring with 2 A. Set S = A[G Σ ]. Let ρ : S M 2 (A) be an A-algebra map with ρ = ρ 0 mod m A. If A is reduced, infinite, but #A/I re,a <, then I re,a is principal.
38 Modularity of reducible deformations Goal is to show that φ : R Σ /I re T Σ /φ(i re ) is an isomorphism.
39 Modularity of reducible deformations Goal is to show that φ : R Σ /I re T Σ /φ(i re ) is an isomorphism. Two steps: Show #R Σ /I re #O/L value (Iwasawa Main Conjecture - Rubin).
40 Modularity of reducible deformations Goal is to show that φ : R Σ /I re T Σ /φ(i re ) is an isomorphism. Two steps: Show #R Σ /I re #O/L value (Iwasawa Main Conjecture - Rubin). Show #T Σ /φ(i re ) #O/L value (congruences - Berger).
41 Higher-dimensional context Let F be a number field, G Σ = Gal(F Σ /F );
42 Higher-dimensional context Let F be a number field, G Σ = Gal(F Σ /F ); τ j : G Σ GL nj (F) be absolutely irreducible;
43 Higher-dimensional context Let F be a number field, G Σ = Gal(F Σ /F ); τ j : G Σ GL nj (F) be absolutely irreducible; [ ] τ1 ρ 0 = : G 0 τ Σ GL n1 +n 2 (F) be non-semisimple. 2
44 Higher-dimensional context Let F be a number field, G Σ = Gal(F Σ /F ); τ j : G Σ GL nj (F) be absolutely irreducible; [ ] τ1 ρ 0 = : G 0 τ Σ GL n1 +n 2 (F) be non-semisimple. 2 Study crystalline deformations of ρ 0, get (R Σ, ρ Σ ).
45 Higher-dimensional context Let F be a number field, G Σ = Gal(F Σ /F ); τ j : G Σ GL nj (F) be absolutely irreducible; [ ] τ1 ρ 0 = : G 0 τ Σ GL n1 +n 2 (F) be non-semisimple. 2 Study crystalline deformations of ρ 0, get (R Σ, ρ Σ ). Then I re is principal for essentially self-dual deformations (Berger-K., 2011);
46 Higher-dimensional context Let F be a number field, G Σ = Gal(F Σ /F ); τ j : G Σ GL nj (F) be absolutely irreducible; [ ] τ1 ρ 0 = : G 0 τ Σ GL n1 +n 2 (F) be non-semisimple. 2 Study crystalline deformations of ρ 0, get (R Σ, ρ Σ ). Then I re is principal for essentially self-dual deformations (Berger-K., 2011); Commutative algebra criterion still works;
47 Higher-dimensional context One needs to prove R Σ /I re = T Σ /φ(i re ).
48 Higher-dimensional context One needs to prove R Σ /I re = T Σ /φ(i re ). This uses the Bloch-Kato conjecture for the module Hom( τ 2, τ 1 ), where τ j are the unique lifts of τ j to O,
49 Higher-dimensional context One needs to prove R Σ /I re = T Σ /φ(i re ). This uses the Bloch-Kato conjecture for the module Hom( τ 2, τ 1 ), where τ j are the unique lifts of τ j to O, Congruences among automorphic forms on higher-rank groups (results of Agarwal, Böcherer, Dummigan, Schulze-Pillot and K. on congruences to the Yoshida lifts on Sp 4 allow us to prove that certain 4-dimensional Galois represenentations arise from Siegel modular forms.)
50 Another modularity result N=square-free, k=even, p > k 4, F = Q. Assume that every prime l N satisfies l 1 mod p. Let f S 2 (N), g S k (N), Σ = {l N, p}. Assume that ρ f and ρ g are absolutely irreducible. Theorem (Berger-K.) Suppose: dim F H 1 Σ (Q, Hom(ρ g, ρ f (k/2 1))) = 1; R ρf (k/2 1) = R ρg = O; the B-K conjecture holds for H 1 Σ (Q, Hom(ρ g, ρ f (k/2 1))). Let ρ : G Q,Σ GL 4 (Q p ) be continuous, irreducible and such that: ρ ss = ρ f (k/2 1) ρ g ; ρ is crystalline at p and essentially self-dual. Then ρ is modular. More precisely, there exists a Siegel modular form of weight k/2 + 1, level Γ 0 (N) and trivial character such that ρ = ρ F.
51 Thank you. For papers and preprints visit klosin
ON LIFTING AND MODULARITY OF REDUCIBLE RESIDUAL GALOIS REPRESENTATIONS OVER IMAGINARY QUADRATIC FIELDS
ON LIFTING AND MODULARITY OF REDUCIBLE RESIDUAL GALOIS REPRESENTATIONS OVER IMAGINARY QUADRATIC FIELDS TOBIAS BERGER 1 AND KRZYSZTOF KLOSIN 2 Abstract. In this paper we study deformations of mod p Galois
More informationA DEFORMATION PROBLEM FOR GALOIS REPRESENTATIONS OVER IMAGINARY QUADRATIC FIELDS
Journal of the Inst. of Math. Jussieu (2009) 8(4), 669 692 c Cambridge University Press 669 doi:10.1017/s1474748009000036 Printed in the United Kingdom A DEFORMATION PROBLEM FOR GALOIS REPRESENTATIONS
More informationRing theoretic properties of Hecke algebras and Cyclicity in Iwasawa theory
Ring theoretic properties of Hecke algebras and Cyclicity in Iwasawa theory Haruzo Hida Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A. A talk in June, 2016 at Banff conference center.
More informationIRREDUCIBILITY OF AUTOMORPHIC GALOIS REPRESENTATIONS OF GL(n), n AT MOST 5. FRANK CALEGARI AND TOBY GEE
IRREDUCIBILITY OF AUTOMORPHIC GALOIS REPRESENTATIONS OF GL(n), n AT MOST 5. FRANK CALEGARI AND TOBY GEE Abstract. Let π be a regular, algebraic, essentially self-dual cuspidal automorphic representation
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 informationCritical p-adic L-functions and applications to CM forms Goa, India. August 16, 2010
Critical p-adic L-functions and applications to CM forms Goa, India Joël Bellaïche August 16, 2010 Objectives Objectives: 1. To give an analytic construction of the p-adic L-function of a modular form
More informationAnti-cyclotomic Cyclicity and Gorenstein-ness of Hecke algebras
Anti-cyclotomic Cyclicity and Gorenstein-ness of Hecke algebras Haruzo Hida Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A. A talk in June, 2016 at Rubinfest at Harvard. The author
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 informationp-adic gauge theory in number theory K. Fujiwara Nagoya Tokyo, September 2007
p-adic gauge theory in number theory K. Fujiwara Nagoya Tokyo, September 2007 Dirichlet s theorem F : totally real field, O F : the integer ring, [F : Q] = d. p: a prime number. Dirichlet s unit theorem:
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 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 informationThe Birch & Swinnerton-Dyer conjecture. Karl Rubin MSRI, January
The Birch & Swinnerton-Dyer conjecture Karl Rubin MSRI, January 18 2006 Outline Statement of the conjectures Definitions Results Methods Birch & Swinnerton-Dyer conjecture Suppose that A is an abelian
More informationTop exterior powers in Iwasawa theory
joint with Ted Chinburg, Ralph Greenberg, Mahesh Kakde, Romyar Sharifi and Martin Taylor Maurice Auslander International Conference April 28, 2018 Iwasawa theory (started by Kenkichi Iwasawa 1950 s). Study
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 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 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 informationIsogeny invariance of the BSD conjecture
Isogeny invariance of the BSD conjecture Akshay Venkatesh October 30, 2015 1 Examples The BSD conjecture predicts that for an elliptic curve E over Q with E(Q) of rank r 0, where L (r) (1, E) r! = ( p
More informationMinimal modularity lifting for GL 2 over an arbitrary number field
Minimal modularity lifting for GL 2 over an arbitrary number field David Hansen July 4, 2013 Abstract We prove a modularity lifting theorem for minimally ramified deformations of two-dimensional odd Galois
More informationIntroductory comments on the eigencurve
Introductory comments on the eigencurve Handout # 5: March 8, 2006 (These are brief indications, hardly more than an annotated list, of topics mentioned in my lectures. ) 1 The basic Hecke diagram As before
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 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 informationIwasawa algebras and duality
Iwasawa algebras and duality Romyar Sharifi University of Arizona March 6, 2013 Idea of the main result Goal of Talk (joint with Meng Fai Lim) Provide an analogue of Poitou-Tate duality which 1 takes place
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 informationIWASAWA INVARIANTS OF GALOIS DEFORMATIONS
IWASAWA INVARIANTS OF GALOIS DEFORMATIONS TOM WESTON Abstract. Fix a residual ordinary representation ρ : G F GL n(k) of the absolute Galois group of a number field F. Generalizing work of Greenberg Vatsal
More informationTriple product p-adic L-functions for balanced weights and arithmetic properties
Triple product p-adic L-functions for balanced weights and arithmetic properties Marco A. Seveso, joint with Massimo Bertolini, Matthew Greenberg and Rodolfo Venerucci 2013 Workshop on Iwasawa theory and
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 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 informationThe Local Langlands Conjectures for n = 1, 2
The Local Langlands Conjectures for n = 1, 2 Chris Nicholls December 12, 2014 1 Introduction These notes are based heavily on Kevin Buzzard s excellent notes on the Langlands Correspondence. The aim is
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 informationl-adic MODULAR DEFORMATIONS AND WILES S MAIN CONJECTURE
l-adic MODULAR DEFORMATIONS AND WILES S MAIN CONJECTURE FRED DIAMOND AND KENNETH A. RIBET 1. Introduction Let E be an elliptic curve over Q. The Shimura-Taniyama conjecture asserts that E is modular, i.e.,
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 informationUp to twist, there are only finitely many potentially p-ordinary abelian varieties over. conductor
Up to twist, there are only finitely many potentially p-ordinary abelian varieties over Q of GL(2)-type with fixed prime-to-p conductor Haruzo Hida Department of Mathematics, UCLA, Los Angeles, CA 90095-1555,
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 informationK-groups of rings of algebraic integers and Iwasawa theory
K-groups of rings of algebraic integers and Iwasawa theory Miho AOKI Okayama University of Science 1 F/Q : finite abelian extension O F : the ring of algebraic integers of F K n (O F ) (n 0) H i (O F,
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 informationImaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups Universiteit Leiden, Université Bordeaux 1 July 12, 2012 - UCSD - X - a Question Let K be a number field and G K = Gal(K/K) the absolute
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 informationThe Asymptotic Fermat Conjecture
The Asymptotic Fermat Conjecture Samir Siksek (Warwick) joint work with Nuno Freitas (Warwick), and Haluk Şengün (Sheffield) 23 August 2018 Motivation Theorem (Wiles, Taylor Wiles 1994) Semistable elliptic
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 informationKIDA S FORMULA AND CONGRUENCES
KIDA S FORMULA AND CONGRUENCES ROBERT POLLACK AND TOM WESTON 1. Introduction Let f be a modular eigenform of weight at least two and let F be a finite abelian extension of Q. Fix an odd prime p at which
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 informationON THE EIGENVARIETY OF HILBERT MODULAR FORMS AT CLASSICAL PARALLEL WEIGHT ONE POINTS WITH DIHEDRAL PROJECTIVE IMAGE. 1.
ON THE EIGENVARIETY OF HILBERT MODULAR FORMS AT CLASSICAL PARALLEL WEIGHT ONE POINTS WITH DIHEDRAL PROJECTIVE IMAGE SHAUNAK V. DEO Abstract. We show that the p-adic eigenvariety constructed by Andreatta-
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 informationA MOD-p ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET. March 7, 2017
A MOD-p ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET DIPENDRA PRASAD March 7, 2017 Abstract. Following the natural instinct that when a group operates on a number field then every term in the
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 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 informationOn the equality case of the Ramanujan Conjecture for Hilbert modular forms
On the equality case of the Ramanujan Conjecture for Hilbert modular forms Liubomir Chiriac Abstract The generalized Ramanujan Conjecture for unitary cuspidal automorphic representations π on GL 2 posits
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 informationModularity of Abelian Varieties
1 Modularity of Abelian Varieties This is page 1 Printer: Opaque this 1.1 Modularity Over Q Definition 1.1.1 (Modular Abelian Variety). Let A be an abelian variety over Q. Then A is modular if there exists
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 informationMODULARITY LIFTING BEYOND THE TAYLOR WILES METHOD
MODULARITY LIFTING BEYOND THE TAYLOR WILES METHOD FRANK CALEGARI AND DAVID GERAGHTY Abstract. We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods
More informationSEMISTABLE MODULARITY LIFTING OVER IMAGINARY QUADRATIC FIELDS
SEMISTABLE MODULARITY LIFTING OVER IMAGINARY QUADRATIC FIELDS FRANK CALEGARI 1. Introduction In this paper, we prove non-minimal modularity lifting theorem for ordinary Galois representations over imaginary
More informationHIDA THEORY NOTES TAKEN BY PAK-HIN LEE
HIDA THEORY NOTES TAKEN BY PAK-HIN LEE Abstract. These are notes from the (ongoing) Student Number Theory Seminar on Hida theory at Columbia University in Fall 2017, which is organized by David Hansen
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 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 informationPart II Galois Theory
Part II Galois Theory Theorems Based on lectures by C. Birkar Notes taken by Dexter Chua Michaelmas 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after
More informationExercises on chapter 4
Exercises on chapter 4 Always R-algebra means associative, unital R-algebra. (There are other sorts of R-algebra but we won t meet them in this course.) 1. Let A and B be algebras over a field F. (i) Explain
More informationMaximal Class Numbers of CM Number Fields
Maximal Class Numbers of CM Number Fields R. C. Daileda R. Krishnamoorthy A. Malyshev Abstract Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis
More informationCongruences Between Selmer Groups 1
Journal of Number Theory 80, 135153 (2000) Article ID jnth.1999.2445, available online at http:www.idealibrary.com on Congruences Between Selmer Groups 1 Li Guo Department of Mathematics and Computer Science,
More informationKida s Formula and Congruences
Documenta Math. 615 Kida s Formula and Congruences To John Coates, for his 60 th birthday Robert Pollack and Tom Weston Received: August 30, 2005 Revised: June 21, 2006 Abstract. We consider a generalization
More informationNEW EXAMPLES OF p-adically RIGID AUTOMORPHIC FORMS
NEW EXAMPLES OF p-adically RIGID AUTOMORPHIC FORMS JOËL BELLAÏCHE Abstract. In this paper, we prove two p-adic rigidity results for automorphic forms for the quasi-split unitary group in three variables
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 informationL-functions and Arithmetic. Conference Program. Title: On the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication
Monday, June 13 9:15-9:30am Barry Mazur, Harvard University Welcoming remarks 9:30-10:30am John Coates, University of Cambridge Title: On the conjecture of Birch and Swinnerton-Dyer for elliptic curves
More informationMod 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 informationLocal behaviour of Galois representations
Local behaviour of Galois representations Devika Sharma Weizmann Institute of Science, Israel 23rd June, 2017 Devika Sharma (Weizmann) 23rd June, 2017 1 / 14 The question Let p be a prime. Let f ř 8 ně1
More informationDifferent Constructions of Universal Deformation Rings
Different Constructions of Universal Deformation Rings Math 847: A Proof of Fermat s Last Theorem, Spring 2013 Rachel Davis Contents 1 Introduction 1 1.1 Representability.......................... 2 2
More information(Not only on the Paramodular Conjecture)
Experiments on Siegel modular forms of genus 2 (Not only on the Paramodular Conjecture) Modular Forms and Curves of Low Genus: Computational Aspects ICERM October 1st, 2015 Experiments with L-functions
More informationREVIEW OF GALOIS DEFORMATIONS
REVIEW OF GALOIS DEFORMATIONS SIMON RUBINSTEIN-SALZEDO 1. Deformations and framed deformations We'll review the study of Galois deformations. Here's the setup. Let G be a pronite group, and let ρ : G GL
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 informationTRILINEAR FORMS AND TRIPLE PRODUCT EPSILON FACTORS WEE TECK GAN
TRILINAR FORMS AND TRIPL PRODUCT PSILON FACTORS W TCK GAN Abstract. We give a short and simple proof of a theorem of Dipendra Prasad on the existence and non-existence of invariant trilinear forms on a
More informationCongruences, graphs and modular forms
Congruences, graphs and modular forms Samuele Anni (IWR - Universität Heidelberg) joint with Vandita Patel (University of Warwick) (work in progress) FoCM 2017, UB, 12 th July 2017 The theory of congruences
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 informationCOURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA
COURSE SUMMARY FOR MATH 504, FALL QUARTER 2017-8: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties
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 informationAre ζ-functions able to solve Diophantine equations?
Are ζ-functions able to solve Diophantine equations? An introduction to (non-commutative) Iwasawa theory Mathematical Institute University of Heidelberg CMS Winter 2007 Meeting Leibniz (1673) L-functions
More informationHecke Operators for Arithmetic Groups via Cell Complexes. Mark McConnell. Center for Communications Research, Princeton
Hecke Operators for Arithmetic Groups via Cell Complexes 1 Hecke Operators for Arithmetic Groups via Cell Complexes Mark McConnell Center for Communications Research, Princeton Hecke Operators for Arithmetic
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 informationAutomorphic Galois representations and Langlands correspondences
Automorphic Galois representations and Langlands correspondences II. Attaching Galois representations to automorphic forms, and vice versa: recent progress Bowen Lectures, Berkeley, February 2017 Outline
More informationThese warmup exercises do not need to be written up or turned in.
18.785 Number Theory Fall 2017 Problem Set #3 Description These problems are related to the material in Lectures 5 7. Your solutions should be written up in latex and submitted as a pdf-file via e-mail
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 informationThe GL 2 main conjecture for elliptic curves without complex multiplication. by Otmar Venjakob
The GL 2 main conjecture for elliptic curves without complex multiplication by Otmar Venjakob Arithmetic of elliptic curves E elliptic curve over Q : E : y 2 + A 1 xy + A 3 y = x 3 + A 2 x 2 + A 4 x +
More informationON 3-CLASS GROUPS OF CERTAIN PURE CUBIC FIELDS. Frank Gerth III
Bull. Austral. ath. Soc. Vol. 72 (2005) [471 476] 11r16, 11r29 ON 3-CLASS GROUPS OF CERTAIN PURE CUBIC FIELDS Frank Gerth III Recently Calegari and Emerton made a conjecture about the 3-class groups of
More informationON THE EIGENVARIETY OF HILBERT MODULAR FORMS AT CLASSICAL PARALLEL WEIGHT ONE POINTS WITH DIHEDRAL PROJECTIVE IMAGE. 1.
ON THE EIGENVARIETY OF HILBERT MODULAR FORMS AT CLASSICAL PARALLEL WEIGHT ONE POINTS WITH DIHEDRAL PROJECTIVE IMAGE SHAUNAK V. DEO Abstract. We show that the p-adic eigenvariety constructed by Andreatta-
More informationConductors in p-adic families
Ramanujan J (2017) 44:359 366 DOI 10.1007/s11139-016-9836-7 Conductors in p-adic families Jyoti Prakash Saha 1 Received: 28 November 2015 / Accepted: 10 August 2016 / Published online: 31 October 2016
More informationKAGAWA Takaaki. March, 1998
Elliptic curves with everywhere good reduction over real quadratic fields KAGAWA Takaaki March, 1998 A dissertation submitted for the degree of Doctor of Science at Waseda University Acknowledgments I
More informationMATH G9906 RESEARCH SEMINAR IN NUMBER THEORY (SPRING 2014) LECTURE 1 (FEBRUARY 7, 2014) ERIC URBAN
MATH G9906 RESEARCH SEMINAR IN NUMBER THEORY (SPRING 014) LECTURE 1 (FEBRUARY 7, 014) ERIC URBAN NOTES TAKEN BY PAK-HIN LEE 1. Introduction The goal of this research seminar is to learn the theory of p-adic
More informationTAYLOR-WILES PATCHING LEMMAS À LA KISIN. Contents 1. Patching lemmas 1 References 9
TAYLOR-WILES PATCHING LEMMAS À LA KISIN HARUZO HIDA Contents 1. Patching lemmas 1 References 9 We present an exposition of Kisin s patching argument which simplified and generalized earlier argument by
More informationGenus 2 Curves of p-rank 1 via CM method
School of Mathematical Sciences University College Dublin Ireland and Claude Shannon Institute April 2009, GeoCrypt Joint work with Laura Hitt, Michael Naehrig, Marco Streng Introduction This talk is about
More informationA NOTE ON POTENTIAL DIAGONALIZABILITY OF CRYSTALLINE REPRESENTATIONS
A NOTE ON POTENTIAL DIAGONALIZABILITY OF CRYSTALLINE REPRESENTATIONS HUI GAO, TONG LIU 1 Abstract. Let K 0 /Q p be a finite unramified extension and G K0 denote the Galois group Gal(Q p /K 0 ). We show
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 informationEVEN GALOIS REPRESENTATIONS
EVEN GALOIS REPRESENTATIONS FRANK CALEGARI 0.1. Introduction. These are edited notes for some talks I gave during the Fontaine Trimester at the Institut Henri Poincaré in March 2010. The exposition of
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik Higher congruence companion forms Rajender Adibhatla and Jayanta Manoharmayum Preprint Nr. 26/2011 HIGHER CONGRUENCE COMPANION FORMS RAJENDER ADIBHATLA AND JAYANTA MANOHARMAYUM
More informationOverview of the proof
of the proof UC Berkeley CIRM 16 juillet 2007 Saturday: Berkeley CDG Sunday: CDG MRS Gare Saint Charles CIRM Monday: Jet lag Jet lag = Slides Basic setup and notation G = Gal(Q/Q) We deal with 2-dimensional
More informationOn the Langlands Program
On the Langlands Program John Rognes Colloquium talk, May 4th 2018 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2018 to Robert P. Langlands of the Institute for
More informationKUMMER S CRITERION ON CLASS NUMBERS OF CYCLOTOMIC FIELDS
KUMMER S CRITERION ON CLASS NUMBERS OF CYCLOTOMIC FIELDS SEAN KELLY Abstract. Kummer s criterion is that p divides the class number of Q(µ p) if and only if it divides the numerator of some Bernoulli number
More informationNon CM p-adic analytic families of modular forms
Non CM p-adic analytic families of modular forms Haruzo Hida Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A. The author is partially supported by the NSF grant: DMS 1464106. Abstract:
More informationON p-adic REPRESENTATIONS OF Gal(Q p /Q p ) WITH OPEN IMAGE
ON p-adic REPRESENTATIONS OF Gal(Q p /Q p ) WITH OPEN IMAGE KEENAN KIDWELL 1. Introduction Let p be a prime. Recently Greenberg has given a novel representation-theoretic criterion for an absolutely irreducible
More informationPOTENTIAL MODULARITY AND APPLICATIONS
POTENTIAL MODULARITY AND APPLICATIONS ANDREW SNOWDEN Contents 1. Introduction 1 2. Review of compatible systems 2 3. Potential modularity 3 4. Putting representations into compatible systems 5 5. Lifting
More informationThree-dimensional imprimitive representations of PSL 2 (Z) and their associated vector-valued modular forms
Three-dimensional imprimitive representations of PSL 2 (Z) and their associated vector-valued modular forms U-M Automorphic forms workshop, March 2015 1 Definition 2 3 Let Γ = PSL 2 (Z) Write ( 0 1 S =
More informationThe Main Conjecture. 1 Introduction. J. Coates, R. Sujatha. November 13, 2009
The Main Conjecture J. Coates, R. Sujatha November 13, 2009 1 Introduction In his important papers [16], [17], Wiles proved in most cases the so-called main conjecture for the cyclotomic Z p -extension
More information