Introduction Endomorphisms and Cohomology Connections to Monodromy Representation Theory formulation Tate Cycles Splitting of Abelian varieties V. Kumar Murty University of Toronto GEOCRYPT October 2013 V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction Endomorphisms and Cohomology A classical problem Connections to Monodromy A geometric analogue Representation Theory formulation Tate Cycles A question about polynomials ◮ Question: Given an irreducible polynomial f ( T ) ∈ Z [ T ], and a prime p , does it necessarily remain irreducible modulo a given prime p ? ◮ Answer: Obviously not. ◮ Question: Given an irreducible polynomial f ( T ) ∈ Z [ T ], is there a prime p such that f ( T ) (mod p ) is irreducible? ◮ Answer: Not necessarily! ◮ A simple example is f ( T ) = T 4 + 1 . V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction Endomorphisms and Cohomology A classical problem Connections to Monodromy A geometric analogue Representation Theory formulation Tate Cycles Factorization of T 4 + 1 (mod p ) ◮ If p ≡ 1 (mod 4), there is an a such that a 2 ≡ − 1 (mod p ). ◮ With this a , we have T 4 + 1 = ( T 2 + a )( T 2 − a ) (mod p ) . V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction Endomorphisms and Cohomology A classical problem Connections to Monodromy A geometric analogue Representation Theory formulation Tate Cycles Factorization of T 4 + 1 (mod p ) ◮ If p ≡ 7 (mod 8), there is a b such that b 2 ≡ 2 (mod p ). ◮ With this b , we have T 4 + 1 = ( T 2 + 1) 2 − 2 T 2 = ( T 2 − bT + 1)( T 2 + bT + 1) (mod p ) . ◮ If p ≡ 3 (mod 8), there is a c such that c 2 ≡ − 2 (mod p ) and T 4 + 1 = ( T 2 − cT − 1)( T 2 + cT − 1) (mod p ) . V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction Endomorphisms and Cohomology A classical problem Connections to Monodromy A geometric analogue Representation Theory formulation Tate Cycles Splitting of polynomials ◮ Expressed another way, this means that an irreducible polynomial f ( T ) ∈ Z [ T ] may become reducible (mod p ) for every prime p . ◮ On the other hand, if there is a prime p such that f ( T ) (mod p ) is irreducible, then there infinitely many such primes ◮ In fact a positive density of primes. We shall see why later. V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction Endomorphisms and Cohomology A classical problem Connections to Monodromy A geometric analogue Representation Theory formulation Tate Cycles What is behind this? ◮ The answer comes from algebraic number theory. ◮ Let f be a normal polynomial and let E be the splitting field of f . Let O be the ring of integers. ◮ Dedekind’s theorem: For all but finitely many p , the factorization of f (mod p ) is identical to the splitting of the ideal p O in the Dedekind domain O . ◮ In other words, f ( T ) = f 1 ( T ) e 1 · · · f r ( T ) e r (mod p ) p O = p e 1 1 · · · p e r r . V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction Endomorphisms and Cohomology A classical problem Connections to Monodromy A geometric analogue Representation Theory formulation Tate Cycles The Frobenius automorphism ◮ We have f ( T ) = f 1 ( T ) e 1 · · · f r ( T ) e r (mod p ) p O = p e 1 1 · · · p e r r ◮ To each p = p i , there is an automorphism Frob p in the Galois group of E / Q . ◮ For most primes p , this is the unique automorphism σ which satisfies σ ( x ) ≡ x p (mod p ) . V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction Endomorphisms and Cohomology A classical problem Connections to Monodromy A geometric analogue Representation Theory formulation Tate Cycles The Frobenius automorphism ◮ This automorphism Frob p is an element which is of order equal to deg f i . ◮ In particular, if f is irreducible (mod p ), then p O stays prime in E and the order of Frob p is n = deg f . ◮ Thus, Frob p generates Gal ( E / Q ) and so, this group must be cyclic. ◮ In the case of T 4 + 1, the splitting field is Q ( ζ 8 ) which has Galois group Z / 2 Z × Z / 2 Z . V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction Endomorphisms and Cohomology A classical problem Connections to Monodromy A geometric analogue Representation Theory formulation Tate Cycles A converse ◮ Conversely, suppose that f is normal and generates a cyclic extension. ◮ Then there are a positive density of primes p such that f (mod p ) is irreducible. ◮ This follows from the Chebotarev density theorem. V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction Endomorphisms and Cohomology A classical problem Connections to Monodromy A geometric analogue Representation Theory formulation Tate Cycles The general case ◮ What about non-normal polynomials? Consider the general case: the Galois group of f is the symmetric group S n where n = deg f . ◮ Then f (mod p ) factors according to the cycle structure of the conjugacy class of Frobenius automorphisms over p . ◮ In particular, f (mod p ) will be irreducible whenever the Frob p is an n -cycle. ◮ To find such a prime, we need only check p ≪ (log d f ) 2 (if we believe the Generalized Riemann Hypothesis). V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction Endomorphisms and Cohomology A classical problem Connections to Monodromy A geometric analogue Representation Theory formulation Tate Cycles A geometric analogue ◮ We now ask for a geometric analogue of this question. A natural place to start is in the setting of Abelian varieties. ◮ Examples: elliptic curves (dimension 1), Jacobian of a curve of genus g (dimension g ), and many more. ◮ Complete reducibility: any Abelian variety is isogenous to a product of simple (absolutely simple) Abelian varieties and this factorization is essentially unique. V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction Endomorphisms and Cohomology A classical problem Connections to Monodromy A geometric analogue Representation Theory formulation Tate Cycles The Geometric Question ◮ Given a simple or absolutely simple Abelian variety over a number field, is there a prime (infinitely many primes, a positive density of primes ...) for which the reduction A p modulo p is simple or absolutely simple? ◮ Answer: No. V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction Endomorphisms and Cohomology A classical problem Connections to Monodromy A geometric analogue Representation Theory formulation Tate Cycles Cryptographic motivation ◮ To use Abelian varieties over finite fields as the basis of a discrete-log based crypto system, the usual problems of ◮ point counting ◮ efficient arithmetic have to be solved. ◮ One way to study good candidates for cryptographically useful Abelian varieties is to begin with one over Q or over a number field and reduce mod p (or v ). ◮ In doing this, we encounter the problem that a simple Abelian variety over a number field may split mod v for every prime v . This doesn’t happen in the elliptic curve case. ◮ Efficient arithmetic also poses new challenges: develop a more abstract approach less dependent on equations. (Work in progress with Pramath Sastry) V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction Endomorphisms and Cohomology A classical problem Connections to Monodromy A geometric analogue Representation Theory formulation Tate Cycles The case of elliptic curves ◮ We expect that if there is a prime p such that E ( F p ) is cyclic, then there are infinitely many such primes. ◮ Work of Serre, Ram Murty, ... ◮ The existence of such a prime can be formulated in terms of a global condition. ◮ We also expect that the order E ( F p ) should be (nearly) a prime for infinitely many primes, but this is not yet known. ◮ The best that we know is (assuming GRH) the existence of infinitely many p for which | E ( F p ) | has a bounded number of factors (Miri-KM, Weng, Cojocaru,...) V. Kumar Murty University of Toronto Splitting of Abelian varieties
Introduction The endomorphism algebra Endomorphisms and Cohomology Abelian surfaces with quaternionic multiplication Connections to Monodromy Two digressions Representation Theory formulation Endomorphisms and cohomology Tate Cycles Complex multiplication The endomorphism algebra ◮ Returning to the above question: does there exist a prime at which the reduction stays simple, the answer depends on the endomorphism algebra of A . ◮ The set of morphisms of algebraic varieties A − → A that are also group homomorphisms form (after tensoring with Q ) a Q algebra End ( A ) ⊗ Q . ◮ The Abelian variety A is simple if and only if this algebra is a division algebra. V. Kumar Murty University of Toronto Splitting of Abelian varieties
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