Introduction Endomorphisms and Cohomology Connections to Monodromy Representation Theory formulation Tate Cycles Splitting of Abelian varieties V. Kumar Murty University of Toronto Workshop on Curves and Applications University of Calgary August 21, 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 simple question about polynomials ◮ Question: Given an irreducible polynomial f ( T ) ∈ Z [ T ], and a prime p , does it necessarily remain irreducible modulo p ? ◮ Answer: Obviously not. ◮ For example, T 2 + 1 ≡ ( T + 1) 2 (mod 2) . 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 simple question about polynomials ◮ But sometimes, it does remain irreducible. For example T 2 + 1 (mod 7) is irreducible. ◮ 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! 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 not so simple question about polynomials ◮ Question: Given an irreducible polynomial f ( T ) ∈ Z [ T ], is there a prime p such that f ( T ) (mod p ) is irreducible? ◮ Answer: No. A simple example is f ( T ) = T 4 + 1 . ◮ Another simple example is f ( T ) = T 4 − 2 T 2 + 9 . 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 ) ◮ T 4 + 1 = ( T + 1) 4 (mod 2) ◮ T 4 + 1 = ( T 2 + T − 1)( T 2 − T − 1) (mod 3) ◮ T 4 + 1 = ( T 2 − 2)( T 2 + 2) (mod 5) ◮ T 4 + 1 = ( T 2 + 3 T + 1)( T 2 − 3 T + 1) (mod 7) 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 Another example ◮ Similarly, we see that f ( T ) = T 4 − 2 T 2 + 9 is irreducible, ◮ but f ( T ) ≡ ( T + 1) 4 (mod 2) ◮ f ( T ) ≡ T 2 ( T 2 − 2) (mod 3) ◮ f ( T ) ≡ ( T 2 + T + 2)( T 2 − T + 2) (mod 5) , · · · 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 Failure of a local-global principle ◮ Expressed another way, this means that an irreducible polynomial f ( T ) ∈ Z [ T ] may become reducible (mod p ) for every prime p . ◮ This is a failure of a local-global principle: reducibility locally everywhere does not imply global reducibility. 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 Failure of a local-global principle ◮ On the other hand, there are limits to this failure. ◮ If there is a prime p such that f ( T ) (mod p ) is irreducible, are there infinitely many such primes? ◮ Answer: Yes, 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 An example ◮ Let R denote the ring Z [ √− 1]. ◮ The factorization T 2 + 1 ≡ ( T + 1) 2 (mod 2) corresponds to the factorization 2 R = I 2 where I = ((1 + √− 1) R ) 2 . ◮ The irreducibility of T 2 + 1 (mod 7) means that 7 R is a prime ideal in 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. 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 examples revisited ◮ In particular, consider again the examples given earlier T 4 − 2 T 2 + 9 and T 4 + 1 ◮ They have splitting field √ √ Q ( − 1 , 2) and Q ( ζ 8 ) (respectively). ◮ Both have 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
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