Explicit bounds for split reductions of simple abelian varieties

Achter, Jeffrey D.

Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 1, pp. 41-55.

Soit $X/K$ une variété abélienne absolument simple, définie sur un corps de nombres ; nous étudions comment les réductions $X_{𝔭}$ tendent à être simples également. Nous montrons que si $End\left(X\right)$ est une algèbre de quaternions définie, alors la réduction $X_{𝔭}$ est géométriquement isogène au self-produit d’une variété abélienne absolument simple, ce pour $𝔭$ dans un ensemble de densité strictement positive, alors que si $X$ est de type Mumford, $X_{𝔭}$ est simple pour presque tout $𝔭$. Pour une large classe de variétés abéliennes avec anneau d’endomorphismes absolus commutatif, nous donnons une borne supérieure explicite pour la croissance de l’ensemble des premiers de réduction non-simple.

Let $X/K$ be an absolutely simple abelian variety over a number field; we study whether the reductions $X_{𝔭}$ tend to be simple, too. We show that if $End\left(X\right)$ is a definite quaternion algebra, then the reduction $X_{𝔭}$ is geometrically isogenous to the self-product of an absolutely simple abelian variety for $𝔭$ in a set of positive density, while if $X$ is of Mumford type, then $X_{𝔭}$ is simple for almost all $𝔭$. For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound for the growth of the set of primes of non-simple reduction.

Classification : 11G25

[1] Jeffrey D. Achter, Split reductions of simple abelian varieties, Math. Res. Lett. 16 (2009), no. 2, 199–213. | MR

[2] Grzegorz Banaszak, Wojciech Gajda, and Piotr Krasoń, On the image of Galois $l$-adic representations for abelian varieties of type III, Tohoku Math. J. (2) 62 (2010), no. 2, 163–189. MR2663452 | MR

[3] Ching-Li Chai and Frans Oort, A note on the existence of absolutely simple Jacobians, J. Pure Appl. Algebra 155 (2001), no. 2-3, 115–120. MRMR1801409 (2002a:14020) | MR

[4] Nick Chavdarov, The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy, Duke Math. J. 87 (1997), no. 1, 151–180. MR99d:11071 | MR | Zbl

[5] Gerd Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366. MRMR718935 (85g:11026a) | MR | Zbl

[6] Patrick X. Gallagher, The number of conjugacy classes in a finite group, Math. Z. 118 (1970), 175–179. MRMR0276318 (43 #2065) | MR | Zbl

[7] Nicholas M. Katz, Report on the irreducibility of $L$-functions, Number Theory, Analysis and Geometry, Springer-Verlag, New York, 2012, pp. 321–353.

[8] E. Kowalski, The large sieve and its applications, Cambridge Tracts in Mathematics, vol. 175, Cambridge University Press, Cambridge, 2008. MRMR2426239 (2009f:11123) | MR

[9] M. J. Larsen and R. Pink, Abelian varieties, $l$-adic representations, and $l$-independence, Math. Ann. 302 (1995), no. 3, 561–579. MRMR1339927 (97e:14057) | MR | Zbl

[10] Michael J. Larsen, Maximality of Galois actions for compatible systems, Duke Math. J. 80 (1995), no. 3, 601–630. MRMR1370110 (97a:11090) | MR | Zbl

[11] Martin W. Liebeck and László Pyber, Upper bounds for the number of conjugacy classes of a finite group, J. Algebra 198 (1997), no. 2, 538–562. MRMR1489911 (99c:20023) | MR | Zbl

[12] D. Mumford, A note of Shimura’s paper “Discontinuous groups and abelian varieties”, Math. Ann. 181 (1969), 345–351. MRMR0248146 (40 #1400) | MR | Zbl

[13] V. K. Murty and V. M. Patankar, Splitting of abelian varieties, IMRN 2008 (2008), Art. ID rnn033, 27 pages. | MR

[14] Rutger Noot, Abelian varieties—Galois representation and properties of ordinary reduction, Compositio Math. 97 (1995), no. 1-2, 161–171, Special issue in honour of Frans Oort. MRMR1355123 (97a:11093) | Numdam | MR | Zbl

[15] —, On Mumford’s families of abelian varieties, J. Pure Appl. Algebra 157 (2001), no. 1, 87–106. MRMR1809220 (2002a:14023) | MR

[16] Kenneth A. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98 (1976), no. 3, 751–804. MRMR0457455 (56 #15660) | MR | Zbl

[17] Jean-Pierre Serre, Lettre à Ken Ribet 1/1/1981, Œuvres. Collected papers. IV, Springer-Verlag, Berlin, 2000, pp. 1–16. MRMR1730973 (2001e:01037) | MR

[18] —, Lettre à Ken Ribet 7/3/1986, Œuvres. Collected papers. IV, Springer-Verlag, Berlin, 2000, pp. 56–65. MRMR1730973 (2001e:01037)

[19] —, Lettre à Marie-France Vignéras 10/2/1986, Œuvres. Collected papers. IV, Springer-Verlag, Berlin, 2000, pp. 38–55. MRMR1730973 (2001e:01037)

[20] Goro Shimura, On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math. (2) 78 (1963), 149–192. MRMR0156001 (27 #5934) | MR | Zbl

[21] A. Silverberg, Fields of definition for homomorphisms of abelian varieties, J. Pure Appl. Algebra 77 (1992), no. 3, 253–262. MRMR1154704 (93f:14022) | MR | Zbl

[22] A. Silverberg and Yu. G. Zarhin, Connectedness extensions for abelian varieties, Math. Z. 228 (1998), no. 2, 387–403. MRMR1630512 (99d:11065) | MR | Zbl

[23] John Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144. MRMR0206004 (34 #5829) | MR | Zbl

[24] Adrian Vasiu, Some cases of the Mumford-Tate conjecture and Shimura varieties, Indiana Univ. Math. J. 57 (2008), 1–76. | MR

[25] David Zywina, The large sieve and Galois representations, (2008). | MR