Twists and residual modular Galois representations

Size: px
Start display at page:

Download "Twists and residual modular Galois representations"

Transcription

1 Twists and residual modular Galois representations Samuele Anni University of Warwick Building Bridges, Bristol 10 th July 2014

2 Modular curves and Modular Forms 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Local representation 4 Twist 5 Examples

3 Modular curves and Modular Forms Let us fix a positive integer n Z >0. Definition The congruence subgroup Γ 1 (n) of SL 2 (Z ) is the subgroup given by {( ) } a b Γ 1 (n) = SL c d 2 (Z ) : n a 1, n c. The integer n is called level of the congruence subgroup.

4 Modular curves and Modular Forms Over the upper half plane: H = {z C Im(z) > 0} we can define an action of Γ 1 (n) via fractional transformations: Γ 1 (n) H H (γ, z) γ(z) = az + b cz + d ( ) a b where γ =. c d Moreover, if n 4 then Γ 1 (n) acts freely on H. Escher, Reducing Lizards Tessellation

5 Modular curves and Modular Forms Definition We define the modular curve Y 1 (n) C to be the non-compact Riemann surface obtained giving on Γ 1 (n)\h the complex structure induced by the quotient map. Let X 1 (n) C be the compactification of Y 1 (n) C. Fact: Y 1 (n) C can be defined algebraically over Q (in fact over Z [1/n]).

6 Twists and residual modular Galois representations Modular curves and Modular Forms The group GL + 2 (Q ) acts on H via fractional transformation, and its action has a particular behaviour with respect to Γ 1 (n). Γ 1(n) Γ 1(n) Proposition For every g GL + 2 (Q ), the discrete groups gγ 1 (n)g 1 and Γ 1 (n) are commensurable g 1 Γ 1(n)g H Y 1 (n) C g H Y 1 (n) C

7 Modular curves and Modular Forms We define operators on Y 1 (n) through the correspondences given before: the Hecke ( ) operators T p for every prime p, using 1 0 g = GL + 0 p 2 (Q ) ; the diamond ( ) operators d for every d (Z /nz ), using a b g = Γ c d 0 (n), where Γ 0 (n) is the set of matrices in SL 2 (Z ) which are upper triangular modulo n.

8 Modular curves and Modular Forms For n 5 and k positive integers, let l be a prime not dividing n. Following Katz, we define the space of mod l cusp forms as mod l cusp forms S(n, k) Fl = H 0 (X 1 (n) Fl, ω k ( Cusps)). S(n, k) Fl is a finite dimensional F l -vector space, equipped with Hecke operators T n (n 1) and diamond operators d for every d (Z /nz ). Analogous definition in characteristic zero and over any ring where n is invertible.

9 Modular curves and Modular Forms One may think that mod l modular forms come from reduction of characteristic zero modular forms mod l: S(n, k) Z [1/n] S(n, k) Fl. Unfortunately, this map is not surjective for k = 1. Even worse: given a character ɛ: (Z /nz ) C the map S(n, k, ɛ) OK S(n, k, ɛ) F is not always surjective even if k > 1. O K is the ring of integers of the number field where ɛ is defined, F is the residue field at l and ɛ is the reduction of ɛ S(n, k, ɛ) OK = {f S(n, k) OK d (Z /nz ), d f = ɛ(d)f }.

10 Modular curves and Modular Forms Definition The Hecke algebra T(n, k) of S(n, k) C is the Z -subalgebra of End C (S(Γ 1 (n), k) C ) generated by Hecke operators T p for every prime p and by diamond operators d for every d (Z /nz ). Fact: T(n, k) is finitely generated as Z -module. Given a character ɛ: (Z /nz ) C, we associate a Hecke algebra T ɛ (n, k) to each S(n, k, ɛ) C : S(n, k, ɛ) C = {f S(n, k) C d (Z /nz ), d f = ɛ(d)f }.

11 Residual modular Galois representations 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Local representation 4 Twist 5 Examples

12 Residual modular Galois representations Theorem (Deligne, Shimura) Let n and k be positive integers. Let F be a finite field of characteristic l, with l not dividing n, and f : T(n, k) F a surjective morphism of rings. Then there is a continuous semi-simple representation: ρ f : Gal(Q /Q ) GL 2 (F), unramified outside nl, such that for all p not dividing nl we have: Trace(ρ f (Frob p )) = f (T p ) and det(ρ f (Frob p )) = f ( p )p k 1 in F. Such a ρ f is unique up to isomorphism. Computing ρ f is difficult, but theoretically it can be done in polynomial time in n, k, #F: Edixhoven, Couveignes, de Jong, Merkl, Bruin, Bosman (#F 32); Mascot, Zeng, Tian (#F 41).

13 Residual modular Galois representations Question Can we compute the image of a residual modular Galois representation without computing the representation? Theorem (A.) There is a polynomial time algorithm which takes as input: n and k be positive integers; l be a prime number not dividing n, such that 2 k l + 1; a character ɛ : (Z /nz ) C ; a morphism of ring f : T ɛ (n, k) F l, and in particular the images of all diamond operators and of the T p operators up to a bound B, and gives as output the image of the associated Galois representation ρ f, up to conjugacy as subgroup of GL 2 (F l ) without computing ρ f.

14 Residual modular Galois representations In almost all cases, the bound B is the Sturm Bound for Γ 0 (n) and weight k: k 12 n ( ) k n log log n p 12 p n prime In the cases when this bound is not enough, then the Sturm Bound for Γ 0 (nq 2 ) and weight k, where q is the smallest odd prime not dividing n, is the required bound.

15 Residual modular Galois representations Some of the problems studied: ρ f can arise from lower level or weight, i.e. there exists g S(m, j) Fl with m n or j k such that ρ g = ρf ρ f can arise as twist of a representation of lower conductor, i.e. there exist g S(m, j) Fl with m n or j k and a Dirichlet character χ such that ρ g χ = ρ f

16 Residual modular Galois representations One of the principal ingredients: Theorem (Khare, Wintenberger, Dieulefait, Kisin), Serre s Conjecture Let l be a prime number and let ρ: Gal(Q /Q ) GL 2 (F l ) be an odd, absolutely irreducible, continuous representation. Then ρ is modular of level N(ρ), weight k(ρ) and character ɛ(ρ). N(ρ) (the level) is the Artin conductor away from l. k(ρ) (the weight) is given by a recipe in terms of ρ Il. ɛ(ρ): (Z /N(ρ)Z ) F l is given by: det ρ = ɛ(ρ)χ k(ρ) 1.

17 Residual modular Galois representations Setting ( ) n and k be positive integers; l be a prime number not dividing n, such that 2 k l + 1; ɛ : (Z /nz ) C be a character; f : T ɛ (n, k) F l be a morphism of rings; ρ f : G Q GL 2 (F l ) be the unique, up to isomorphism, continuous semi-simple representation attached to f ; ɛ : (Z /nz ) F l be the character defined by ɛ(a) = f ( a ) for all a (Z /nz ). Let p be a prime dividing nl. Let us denote by G p = Gal(Q p /Q p ) G Q the decomposition subgroup at p; I p the inertia subgroup, I t the tame inertia subgroup; G i,p, with i Z >0, the higher ramification subgroups (I p = G 0,p ). Notation: given a residual representation ρ, we will denote as N p (ρ) the valuation at p of the Artin conductor of ρ.

18 Local representation 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Local representation Local representation at l Local representation at primes dividing the level 4 Twist 5 Examples

19 Local representation Local representation at l Local representation at l Theorem (Deligne) Assume setting ( ). Suppose that f (T l ) 0. Then ρ f Gl is reducible, and up to conjugation in GL 2 (F l ), we have ( ) ρ f Gl χ k 1 = l λ(ɛ(l)/f (T l )) 0 λ(f (T l )) where λ(a) is the unramified character of G l taking Frob l G l /I l to a, for any a F l.

20 Local representation Local representation at l Theorem (Fontaine) Assume setting ( ). Suppose that f (T l ) = 0. Then ρ f Gl and up to conjugation in GL 2 (F l ), we have ( ) ρ f Il ϕ k 1 0 = 0 ϕ k 1 is irreducible, where ϕ, ϕ: I t F l are the two fundamental characters of level 2.

21 Local representation Local representation at primes dividing the level Local representation at primes dividing the level Theorem (Gross-Vignéras, Serre: Conjecture 3.2.6? ) Let ρ : G Q GL(V ) be a continuous, odd, irreducible representation of the absolute Galois group over Q to a 2-dimensional F l -vector space V. Let n = N(ρ) and k = k(ρ), let f S(n, k) Fl be an eigenform such that ρ f = ρ. Let p be a prime divisor of ln. (1) If f (T p ) 0, then there exists a stable line D V for the action of G p, the decomposition subgroup at p, such that the inertia group at p acts trivially on V /D. Moreover, f (T p ) is equal to the eigenvalue of Frob p which acts on V /D. (2) If f (T p ) = 0, then there exists no stable line D V as in (1).

22 Twist 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Local representation 4 Twist Local representation and conductor Twisting by Dirichlet characters 5 Examples

23 Twist Local representation and conductor Proposition (A.) Assume setting ( ) and that ρ f is irreducible and it does not arise from lower level. Let p be a prime dividing n such that f (T p ) 0. Then ρ f Gp is decomposable if and only if ρ f Ip is decomposable. This proposition is proved using representation theory.

24 Twist Local representation and conductor Proposition (A.) Assume setting ( ) and that ρ f is irreducible and it does not arise from lower level. Let p be a prime dividing n, such that f (T p ) 0. Then: (a) ρ f Ip is decomposable if and only if N p (ρ f ) = N p (ɛ); (b) ρ f Ip is indecomposable if and only if N p (ρ f ) = 1 + N p (ɛ).

25 Twist Local representation and conductor Sketch of the proof The valuation of N(ρ f ) at p is given by: N p (ρ f ) = i 0 1 [G 0,p : G i,p ] dim(v /V G i,p ) = dim(v /V Ip ) + b(v ), where V is the two-dimensional F l -vector space underlying the representation, V G i,p is its subspace of invariants under G i,p, and b(v ) is the wild part of the conductor. Since f (T p ) 0, the representation restricted to the decomposition group at p is reducible. Hence, after conjugation, ( ) ( ) ( ) ρ f Gp ɛ1 χ k 1 = l, ρ 0 ɛ f Ip ɛ1 Ip ɛ Ip = = where ɛ 1 and ɛ 2 are characters of G p with ɛ 2 unramified, χ l is the mod l cyclotomic character and belongs to F l.

26 Twist Local representation and conductor Remark If ρ f Ip is indecomposable then the image of inertia at p is of order divisible by l and so the image cannot be exceptional, hence it is big.

27 Twist Twisting by Dirichlet characters Let n be a positive integer. Any Dirichlet character of conductor n can be decomposed into local characters, one for each prime divisor of n. With no loss of generality, we reduce ourselves to study twists of modular Galois representations with Dirichlet characters with prime power conductor.

28 Twist Twisting by Dirichlet characters Question What is the conductor of the twist? Shimura gave an upper bound: lcm(cond(χ) 2, n) where n is the level of the form and χ is the character used for twisting.

29 Twist Twisting by Dirichlet characters Proposition (A.) Assume setting ( ). Let p be a prime not dividing nl. Let χ : (Z /p i Z ) F l, for i > 0, be a non-trivial character. Then N p (ρ f χ) = 2N p (χ).

30 Twist Twisting by Dirichlet characters Proposition (A.) Assume setting ( ) and that ρ f is irreducible and it does not arise from lower level. Let p be a prime dividing n and suppose that f (T p ) 0. Let χ : (Z /p i Z ) F l, for i > 0, be a non-trivial character. Then N p (ρ f χ) = N p (χɛ) + N p (χ).

31 Twist Twisting by Dirichlet characters It is also possible to know what is the system of eigenvalues associated to the twist: Proposition (A.) Assume setting ( ). Suppose that ρ f is irreducible and that N(ρ f ) = n. Let p be a prime dividing n and suppose that f (T p ) 0. Let χ from (Z /p i Z ) to F l, with i > 0, be a non-trivial character. Then (a) if ρ f Ip is decomposable then the representation ρ f χ restricted to G p, the decomposition group at p, admits a stable line with unramified quotient if and only if N p (ρ f χ) = N p (ρ f ); (b) if ρ f Ip is indecomposable then the representation ρ f χ restricted to G p does not admit any stable line with unramified quotient.

32 Twist Twisting by Dirichlet characters Proposition (A.) Assume setting ( ). Suppose that ρ f is irreducible and that N(ρ f ) = n. Let p be a prime dividing n and suppose that f (T p ) = 0. Then: (a) if ρ f Gp is reducible then there exists a mod l modular form g of weight k and level at most np and a non-trivial character χ : (Z /p i Z ) F l with i > 0 such that g(t p ) 0 and ρ g = ρf χ; (b) if ρ f Gp is irreducible then for any non-trivial character χ : (Z /p i Z ) F l with i > 0 the representation ρ f χ restricted to G p does not admit any stable line with unramified quotient.

33 Examples 1 Modular curves and Modular Forms 2 Residual modular Galois representations 3 Local representation 4 Twist 5 Examples Example 1 Example 2 Example 3

34 Examples Example 1 Example 1 Let n = 135 = Let ɛ be the Dirichlet character modulo 135 of conductor 5 mapping 56 1, 82 ζ S(135, 3, ɛ) new C two Galois orbits: the two Hecke eigenvalue fields are: Q (x x x x ) and Q (x x x x ). In the first one, applying the reduction map for all prime ideals above 7, we obtain the following eigenvalue systems defined over F 7 2 = F 7 [x]/[x 2 + 6x + 4] = F 7 [α]:

35 Examples Example 1 p f 1 (T p ) f 2 (T p ) f 3 (T p ) 2 α 6α α α + 4 2α + 3 5α α + 3 6α + 4 α α 6α α α α 6α α + 4 6α + 4 α α + 4 4α + 3 3α f 1,f 2 and f 3 are mod 7 modular forms of level 135 and weight 3. It is easy to verify that the corresponding representations are of minimal level and weight.

36 Examples Example 1 Let us focus on f 1. Since 5 divides the level and f 1 (T 5 ) 0: ( ) ( ρ f1 G5 ɛ1 χ 2 = 7 ɛ 1 2 = ɛ ) f 1 χ ɛ 2 0 ɛ 2 The character of f 1 has conductor 5 hence the representation is reducible and decomposable, so = 0. Moreover ɛ 2 (5) = f 1 (T 5 ). If we twist by ɛ 1 f 1 then we have: ( ) (ρ f1 ɛ 1 f 1 ) G5 ɛ 1 = 2 χ ɛ 2 ɛ 1 f 1 hence we know the eigenvalue at 5. The computation on the conductor gives that the conductor is 135 and (ɛ 1 2 χ2 7 )(5) = 4ɛ 1 2 (5) = 5α + 5.

37 Examples Example 1 p f 1 (T p ) f 2 (T p ) f 3 (T p ) ρ f1 ɛ 1 f 1 2 α 6α α + 6 α α + 4 2α + 3 5α + 5 5α α + 3 6α + 4 α + 3 α α 6α α + 6 α α α 6α + 1 6α α + 4 6α + 4 α + 3 α α + 4 4α + 3 3α 3α α + 6 α + 6 6α 6α 41 5α + 1 2α + 6 5α + 1 5α α α 6α + 1 6α α 5α 2α + 5 2α + 5

38 Examples Example 1 The level is also divisible by 3. But f 1 (T 3 ) = 0. The local representation at 3 is irreducible: in order to prove this we have to check all lover levels and possible twist. In this case it is easy since the newforms space is empty in most cases. The argument is similar for all levels, so we will just show what happens for level 15. p f 1 (T p ) g 1 (T p ) g 2 (T p ) g 3 (T p ) g 4 (T p ) 2 α 6α + 6 α + 5 4α + 1 3α α α + 6 α 6α α + 4 3α + 5 4α + 1 2α + 1 5α α + 5 3α α + 3 6α + 1 α α 6α α 4α + 5 3α

39 Examples Example 1 It is possible to show that the image of the Galois representation up to conjugation is ρ f1 (G Q ) =< α > SL 2 (F 7 ) GL 2 (F 7 2) (f (T p )) 2 (ɛ f (p)p 2 ) 1 6, 0, 6, 4, 5, 4, 5, 3, 1, 1, 6, 4, 3

40 Examples Example 2 Example 2 Let us consider S(40, 2, τ) new C where τ is the quadratic character conductor 40 mapping 31 1,21 1, As before, by reduction get mod 7 eigenvalue systems, for example f i for i = 1, 2, 3, 4: p f 1 (T p ) f 2 (T p ) f 3 (T p ) f 4 (T p )

41 Examples Example 2 Let χ be the character over F 7 of conductor 8 such that χɛ f1 conductor 4. Since f 1 (T 2 ) 0, then we have that: has N 2 (ρ f χ) = N 2 (χɛ f1 ) + N 2 (χ) = = 5 6. Hence, let us check the twist and the eigenvalue systems at level 160: N 2 (ρ f χ) = = 160: p g 1 (T p ) g 2 (T p ) g 3 (T p ) g 4 (T p ) ρ f1 χ ρ f2 χ ρ f3 χ ρ f4 χ

42 Examples Example 2 All the systems given by g i are reduction of the same form. Also in this case it is possible to prove that the image of the Galois representation is big: in this case, it is isomorphic to GL 2 (F 7 ), up to conjugation.

43 Examples Example 3 Example 3 S(7, 3) new C, S(49, 3) new C The Hecke eigenvalue fields in newformspace at level 49 are given by: Q (x 2 3x + 9) Q (x 4 4x x 2 44x + 167) Q (x 8 + 4x 7 6x 6 96x x x 3 514x x ) Number Field with defining polynomial x x x x x x Number Field with defining polynomial x x x x x x while in level 7... the Hecke eigenvalue field is Q.

44 Examples Example 3 Let F 5 2 = F 5 [x]/[x 2 + 2x + 4] = F 5 [β]. Let χ be Dirichlet character over F β. modulo 49 of conductor 7 mapping p f (T p ) g(t p ) ρ f χ 2 2 β β β+1 3β β 4β In this case we have that Pρ f = A4 and ρ f (G Q ) = F 5 π 1 (A 4 ), where A 4 is the alternating group of 4 elements.

45 Examples Example 3 Twists and residual modular Galois representations Samuele Anni University of Warwick Building Bridges, Bristol 10 th July 2014 Thanks!

Residual modular Galois representations: images and applications

Residual 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 information

15 Elliptic curves and Fermat s last theorem

15 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 information

Cover Page. The handle holds various files of this Leiden University dissertation.

Cover 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 information

Universität Regensburg Mathematik

Universität 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 information

Endomorphism algebras of semistable abelian varieties over Q of GL(2)-type

Endomorphism 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 information

Galois Representations

Galois 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 information

Calculation and arithmetic significance of modular forms

Calculation and arithmetic significance of modular forms Calculation and arithmetic significance of modular forms Gabor Wiese 07/11/2014 An elliptic curve Let us consider the elliptic curve given by the (affine) equation y 2 + y = x 3 x 2 10x 20 We show its

More information

Congruences, graphs and modular forms

Congruences, 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 information

Higher genus curves and the inverse Galois problem

Higher genus curves and the inverse Galois problem Higher genus curves and the inverse Galois problem Samuele Anni joint work with Pedro Lemos and Samir Siksek University of Warwick Congreso de Jóvenes Investigadores de la Real Sociedad Matemática Española

More information

Modularity of Abelian Varieties

Modularity 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 information

Non CM p-adic analytic families of modular forms

Non 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 information

Mod p Galois representations attached to modular forms

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 information

Workshop on Serre s Modularity Conjecture: the level one case

Workshop 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 information

Honours 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 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 information

Computing coefficients of modular forms

Computing 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 information

EKNATH GHATE AND VINAYAK VATSAL. 1. Introduction

EKNATH 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 information

GALOIS REPRESENTATIONS WITH CONJECTURAL CONNECTIONS TO ARITHMETIC COHOMOLOGY

GALOIS REPRESENTATIONS WITH CONJECTURAL CONNECTIONS TO ARITHMETIC COHOMOLOGY GALOIS REPRESENTATIONS WITH CONJECTURAL CONNECTIONS TO ARITHMETIC COHOMOLOGY AVNER ASH, DARRIN DOUD, AND DAVID POLLACK Abstract. In this paper we extend a conjecture of Ash and Sinnott relating niveau

More information

Serre s conjecture. Alex J. Best. June 2015

Serre s conjecture. Alex J. Best. June 2015 Serre s conjecture Alex J. Best Contents June 2015 1 Introduction 2 2 Background 2 2.1 Modular forms........................... 2 2.2 Galois representations...................... 6 3 Obtaining Galois representations

More information

MA 162B LECTURE NOTES: THURSDAY, FEBRUARY 26

MA 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 information

Up 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. 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 information

The Arithmetic of Noncongruence Modular Forms. Winnie Li. Pennsylvania State University, U.S.A. and National Center for Theoretical Sciences, Taiwan

The Arithmetic of Noncongruence Modular Forms. Winnie Li. Pennsylvania State University, U.S.A. and National Center for Theoretical Sciences, Taiwan The Arithmetic of Noncongruence Modular Forms Winnie Li Pennsylvania State University, U.S.A. and National Center for Theoretical Sciences, Taiwan 1 Modular forms A modular form is a holomorphic function

More information

On 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 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 information

On the arithmetic of modular forms

On 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 information

Cusp forms and the Eichler-Shimura relation

Cusp forms and the Eichler-Shimura relation Cusp forms and the Eichler-Shimura relation September 9, 2013 In the last lecture we observed that the family of modular curves X 0 (N) has a model over the rationals. In this lecture we use this fact

More information

Galois Representations

Galois 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 information

Arakelov theory and height bounds

Arakelov theory and height bounds Abstract Arakelov theory and height bounds Peter Bruin Berlin, 7 November 2009 In the work of Edixhoven, Couveignes et al. (see [5] and [4]) on computing two-dimensional Galois representations associated

More information

Recent Work on Serre s Conjectures

Recent Work on Serre s Conjectures Recent Work on s UC Berkeley UC Irvine May 23, 2007 Prelude In 1993 1994, I was among the number theorists who lectured to a variety of audiences about the proof of Fermat s Last Theorem that Andrew Wiles

More information

Overview of the proof

Overview 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 information

MATH G9906 RESEARCH SEMINAR IN NUMBER THEORY (SPRING 2014) LECTURE 1 (FEBRUARY 7, 2014) ERIC URBAN

MATH 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 information

Ring theoretic properties of Hecke algebras and Cyclicity in Iwasawa theory

Ring 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 information

From K3 Surfaces to Noncongruence Modular Forms. Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015

From K3 Surfaces to Noncongruence Modular Forms. Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015 From K3 Surfaces to Noncongruence Modular Forms Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015 Winnie Li Pennsylvania State University 1 A K3 surface

More information

9 Artin representations

9 Artin representations 9 Artin representations Let K be a global field. We have enough for G ab K. Now we fix a separable closure Ksep and G K := Gal(K sep /K), which can have many nonabelian simple quotients. An Artin representation

More information

NUNO FREITAS AND ALAIN KRAUS

NUNO 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 information

The Galois Representation Associated to Modular Forms (Part I)

The Galois Representation Associated to Modular Forms (Part I) The Galois Representation Associated to Modular Forms (Part I) Modular Curves, Modular Forms and Hecke Operators Chloe Martindale May 20, 2015 Contents 1 Motivation and Background 1 2 Modular Curves 2

More information

Raising the Levels of Modular Representations Kenneth A. Ribet

Raising 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 information

SERRE S CONJECTURE AND BASE CHANGE FOR GL(2)

SERRE 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 information

2-ADIC PROPERTIES OF CERTAIN MODULAR FORMS AND THEIR APPLICATIONS TO ARITHMETIC FUNCTIONS

2-ADIC PROPERTIES OF CERTAIN MODULAR FORMS AND THEIR APPLICATIONS TO ARITHMETIC FUNCTIONS 2-ADIC PROPERTIES OF CERTAIN MODULAR FORMS AND THEIR APPLICATIONS TO ARITHMETIC FUNCTIONS KEN ONO AND YUICHIRO TAGUCHI Abstract. It is a classical observation of Serre that the Hecke algebra acts locally

More information

Hecke Operators for Arithmetic Groups via Cell Complexes. Mark McConnell. Center for Communications Research, Princeton

Hecke 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 information

Séminaire BOURBAKI Novembre ème année, , n o p-adic FAMILIES OF MODULAR FORMS [after Hida, Coleman, and Mazur]

Séminaire BOURBAKI Novembre ème année, , n o p-adic FAMILIES OF MODULAR FORMS [after Hida, Coleman, and Mazur] Séminaire BOURBAKI Novembre 2009 62ème année, 2009-2010, n o 1013 p-adic FAMILIES OF MODULAR FORMS [after Hida, Coleman, and Mazur] by Matthew EMERTON INTRODUCTION The theory of p-adic families of modular

More information

On the equality case of the Ramanujan Conjecture for Hilbert modular forms

On 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 information

SERRE S CONJECTURES BRYDEN CAIS

SERRE S CONJECTURES BRYDEN CAIS SERRE S CONJECTURES BRYDEN CAIS 1. Introduction The goal of these notes is to provide an introduction to Serre s conjecture concerning odd irreducible 2- dimensional mod p Galois representations. The primary

More information

Galois groups with restricted ramification

Galois 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 information

Pacific Journal of Mathematics

Pacific 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 information

VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

VARIATION 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 information

The Local Langlands Conjectures for n = 1, 2

The 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 information

Wildly ramified Galois representations and a generalization of a conjecture of Serre

Wildly 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 information

ORAL QUALIFYING EXAM QUESTIONS. 1. Algebra

ORAL 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 information

Hecke fields and its growth

Hecke fields and its growth Hecke fields and its growth Haruzo Hida Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A. Kyushu university talk on August 1, 2014 and PANT talk on August 5, 2014. The author is partially

More information

Diophantine equations and semistable elliptic curves over totally real fields

Diophantine 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 information

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS

FORMAL 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 information

COUNTING MOD l SOLUTIONS VIA MODULAR FORMS

COUNTING 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 information

A Lift of Cohomology Eigenclasses of Hecke Operators

A Lift of Cohomology Eigenclasses of Hecke Operators Brigham Young University BYU ScholarsArchive All Theses and Dissertations 2010-05-24 A Lift of Cohomology Eigenclasses of Hecke Operators Brian Francis Hansen Brigham Young University - Provo Follow this

More information

Overview. exp(2πiq(x)z) x Z m

Overview. exp(2πiq(x)z) x Z m Overview We give an introduction to the theory of automorphic forms on the multiplicative group of a quaternion algebra over Q and over totally real fields F (including Hilbert modular forms). We know

More information

POTENTIAL MODULARITY AND APPLICATIONS

POTENTIAL 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 information

Computer methods for Hilbert modular forms

Computer methods for Hilbert modular forms Computer methods for Hilbert modular forms John Voight University of Vermont Workshop on Computer Methods for L-functions and Automorphic Forms Centre de Récherche Mathématiques (CRM) 22 March 2010 Computer

More information

Computational Arithmetic of Modular Forms. (Modulformen II)

Computational Arithmetic of Modular Forms. (Modulformen II) Computational Arithmetic of Modular Forms (Modulformen II) Wintersemester 2007/2008 Universität Duisburg-Essen Gabor Wiese gabor.wiese@uni-due.de Version of 4th February 2008 2 Preface This lecture is

More information

Problems on Growth of Hecke fields

Problems 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 information

The Galois representation associated to modular forms pt. 2 Erik Visse

The Galois representation associated to modular forms pt. 2 Erik Visse The Galois representation associated to modular forms pt. 2 Erik Visse May 26, 2015 These are the notes from the seminar on local Galois representations held in Leiden in the spring of 2015. The website

More information

Reciprocity maps with restricted ramification

Reciprocity 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 information

Lecture 4: Examples of automorphic forms on the unitary group U(3)

Lecture 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 information

Modulformen und das inverse Galois-Problem

Modulformen und das inverse Galois-Problem Modulformen und das inverse Galois-Problem Gabor Wiese Université du Luxembourg Vortrag auf der DMV-Jahrestagung 2012 in Saarbrücken 19. September 2012 Modulformen und das inverse Galois-Problem p.1/19

More information

9 Artin representations

9 Artin representations 9 Artin representations Let K be a global field. We have enough for G ab K. Now we fix a separable closure Ksep and G K := Gal(K sep /K), which can have many nonabelian simple quotients. An Artin representation

More information

On the generalized Fermat equation x 2l + y 2m = z p

On 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 information

p-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 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 information

Computing weight one modular forms over C and F p.

Computing weight one modular forms over C and F p. Computing weight one modular forms over C and F p. Kevin Buzzard May 17, 2016 Abstract We report on a systematic computation of weight one cuspidal eigenforms for the group Γ 1(N) in characteristic zero

More information

Gabor Wiese. 1st August Abstract

Gabor Wiese. 1st August Abstract Mod p Modular Forms Gabor Wiese 1st August 2006 Abstract These are notes and background information for my lectures at the MSRI Summer Graduate Workshop in Computational Number Theory, 31st July to 11th

More information

l-adic MODULAR DEFORMATIONS AND WILES S MAIN CONJECTURE

l-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 information

Ordinary forms and their local Galois representations

Ordinary forms and their local Galois representations Ordinary forms and their local Galois representations Eknath Ghate Abstract. We describe what is known about the local splitting behaviour of Galois representations attached to ordinary cuspidal eigenforms.

More information

(www.math.uni-bonn.de/people/harder/manuscripts/buch/), files chap2 to chap

(www.math.uni-bonn.de/people/harder/manuscripts/buch/), files chap2 to chap The basic objects in the cohomology theory of arithmetic groups Günter Harder This is an exposition of the basic notions and concepts which are needed to build up the cohomology theory of arithmetic groups

More information

Some algebraic number theory and the reciprocity map

Some 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 information

10 l-adic representations

10 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 information

Computing Hilbert modular forms

Computing Hilbert modular forms Computing Hilbert modular forms John Voight Dartmouth College Curves and Automorphic Forms Arizona State University 10 March 2014 Hilbert modular forms Let F be a totally real field with [F : Q] = n and

More information

Proven Cases of a Generalization of Serre's Conjecture

Proven 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 information

Cover Page. The handle holds various files of this Leiden University dissertation.

Cover 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 information

ON THE MODULARITY OF ELLIPTIC CURVES OVER Q: WILD 3-ADIC EXERCISES.

ON THE MODULARITY OF ELLIPTIC CURVES OVER Q: WILD 3-ADIC EXERCISES. JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0894-047(XX)0000-0 ON THE MODULARITY OF ELLIPTIC CURVES OVER Q: WILD -ADIC EXERCISES. CHRISTOPHE BREUIL, BRIAN CONRAD,

More information

THE PARAMODULAR CONJECTURE ARMAND BRUMER

THE 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 information

Three-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 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 information

NON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES

NON-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 information

HECKE OPERATORS ON CERTAIN SUBSPACES OF INTEGRAL WEIGHT MODULAR FORMS.

HECKE OPERATORS ON CERTAIN SUBSPACES OF INTEGRAL WEIGHT MODULAR FORMS. HECKE OPERATORS ON CERTAIN SUBSPACES OF INTEGRAL WEIGHT MODULAR FORMS. MATTHEW BOYLAN AND KENNY BROWN Abstract. Recent works of Garvan [2] and Y. Yang [7], [8] concern a certain family of half-integral

More information

The Galois Representation Attached to a Hilbert Modular Form

The Galois Representation Attached to a Hilbert Modular Form The Galois Representation Attached to a Hilbert Modular Form Gabor Wiese Essen, 17 July 2008 Abstract This talk is the last one in the Essen seminar on quaternion algebras. It is based on the paper by

More information

Kleine AG: Travaux de Shimura

Kleine AG: Travaux de Shimura Kleine AG: Travaux de Shimura Sommer 2018 Programmvorschlag: Felix Gora, Andreas Mihatsch Synopsis This Kleine AG grew from the wish to understand some aspects of Deligne s axiomatic definition of Shimura

More information

Lemma 1.1. The field K embeds as a subfield of Q(ζ D ).

Lemma 1.1. The field K embeds as a subfield of Q(ζ D ). Math 248A. Quadratic characters associated to quadratic fields The aim of this handout is to describe the quadratic Dirichlet character naturally associated to a quadratic field, and to express it in terms

More information

Image of big Galois representations and modular forms mod p

Image 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 information

The Kronecker-Weber Theorem

The 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 information

IRREDUCIBILITY 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 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 information

Lifting Galois Representations, and a Conjecture of Fontaine and Mazur

Lifting Galois Representations, and a Conjecture of Fontaine and Mazur Documenta Math. 419 Lifting Galois Representations, and a Conjecture of Fontaine and Mazur Rutger Noot 1 Received: May 11, 2001 Revised: November 16, 2001 Communicated by Don Blasius Abstract. Mumford

More information

Class groups and Galois representations

Class 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 information

MAZUR TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS

MAZUR TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS MAZUR TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS ROBERT POLLACK AND TOM WESTON Abstract. We establish formulae for the Iwasawa invariants of Mazur Tate elements of cuspidal eigenforms, generalizing known

More information

On the Langlands Program

On 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 information

Growth of Hecke fields over a slope 0 family

Growth of Hecke fields over a slope 0 family Growth of Hecke fields over a slope 0 family Haruzo Hida Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A. A conference talk on January 27, 2014 at Simons Conference (Puerto Rico). The

More information

Modular congruences, Q-curves, and the diophantine equation x 4 +y 4 = z p

Modular congruences, Q-curves, and the diophantine equation x 4 +y 4 = z p arxiv:math/0304425v1 [math.nt] 27 Apr 2003 Modular congruences, Q-curves, and the diophantine equation x 4 +y 4 = z p Luis V. Dieulefait Centre de Recerca Matemática Apartat 50, E-08193 Bellaterra, Spain

More information

Proof. We omit the proof of the case when f is the reduction of a characteristic zero eigenform (cf. Theorem 4.1 in [DS74]), so assume P (X) has disti

Proof. We omit the proof of the case when f is the reduction of a characteristic zero eigenform (cf. Theorem 4.1 in [DS74]), so assume P (X) has disti Some local (at p) properties of residual Galois representations Johnson Jia, Krzysztof Klosin March 5, 26 1 Preliminary results In this talk we are going to discuss some local properties of (mod p) Galois

More information

HAS LARGE INDEX. Frank Calegari and Matthew Emerton

HAS LARGE INDEX. Frank Calegari and Matthew Emerton Mathematical Research Letters 11, 125 137 (2004) THE HECKE ALGEBRA T k HAS LARGE INDEX Frank Calegari and Matthew Emerton Abstract. Let T k (N) new denote the Hecke algebra acting on newformsof weight

More information

l-adic Representations

l-adic Representations l-adic Representations S. M.-C. 26 October 2016 Our goal today is to understand l-adic Galois representations a bit better, mostly by relating them to representations appearing in geometry. First we ll

More information

ON MODULAR GALOIS REPRESENTATIONS MODULO PRIME POWERS.

ON MODULAR GALOIS REPRESENTATIONS MODULO PRIME POWERS. ON MODULAR GALOIS REPRESENTATIONS MODULO PRIME POWERS. IMIN CHEN, IAN KIMING, GABOR WIESE Abstract. We study modular Galois representations mod p m. We show that there are three progressively weaker notions

More information

HIGHLY REDUCIBLE GALOIS REPRESENTATIONS ATTACHED TO THE HOMOLOGY OF GL(n, Z)

HIGHLY REDUCIBLE GALOIS REPRESENTATIONS ATTACHED TO THE HOMOLOGY OF GL(n, Z) HIGHLY REDUCIBLE GALOIS REPRESENTATIONS ATTACHED TO THE HOMOLOGY OF GL(n, Z) AVNER ASH AND DARRIN DOUD Abstract Let n 1 and F an algebraic closure of a finite field of characteristic p > n + 1 Let ρ :

More information

A MOD-p ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET. March 7, 2017

A 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 information

MODULAR FORMS AND SOME CASES OF THE INVERSE GALOIS PROBLEM

MODULAR FORMS AND SOME CASES OF THE INVERSE GALOIS PROBLEM MODULAR FORMS AND SOME CASES OF THE INVERSE GALOIS PROBLEM DAVID ZYWINA Abstract. We prove new cases of the inverse Galois problem by considering the residual Galois representations arising from a fixed

More information

EXERCISES 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) (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 information

Refinement of Tate s Discriminant Bound and Non-Existence Theorems for Mod p Galois Representations

Refinement of Tate s Discriminant Bound and Non-Existence Theorems for Mod p Galois Representations Documenta Math. 641 Refinement of Tate s Discriminant Bound and Non-Existence Theorems for Mod p Galois Representations Dedicated to Professor Kazuya Kato on the occasion of his fiftieth birthday Hyunsuk

More information