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Journal articles on the topic 'Abelian varietie'

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Published: 9 March 2023
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1

Zhao, Heer. "Degenerating abelian varieties via log abelian varieties." Asian Journal of Mathematics 22, no. 5 (2018): 811–40. http://dx.doi.org/10.4310/ajm.2018.v22.n5.a2.

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2

Silverberg, Alice. "abelian varieties." Duke Mathematical Journal 56, no. 1 (February 1988): 41–46. http://dx.doi.org/10.1215/s0012-7094-88-05603-7.

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3

Yu, Chia-Fu. "Abelian varieties over finite fields as basic abelian varieties." Forum Mathematicum 29, no. 2 (March 1, 2017): 489–500. http://dx.doi.org/10.1515/forum-2014-0141.

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AbstractIn this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic ${p>0}$ is isogenous to another one defined over a finite field. We also show that the category of abelian varieties over finite fields up to isogeny can be embedded into the category of basic abelian varieties with suitable endomorphism structures. Using this connection, we derive a new mass formula for a finite orbit of polarized abelian surfaces over a finite field.
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4

Kajiwara, Takeshi, Kazuya Kato, and Chikara Nakayama. "Logarithmic Abelian Varieties." Nagoya Mathematical Journal 189 (2008): 63–138. http://dx.doi.org/10.1017/s002776300000951x.

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AbstractWe develop the algebraic theory of log abelian varieties. This is Part II of our series of papers on log abelian varieties, and is an algebraic counterpart of the previous Part I ([6]), where we developed the analytic theory of log abelian varieties.
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5

Yamaki, Kazuhiko. "Trace of abelian varieties over function fields and the geometric Bogomolov conjecture." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 741 (August 1, 2018): 133–59. http://dx.doi.org/10.1515/crelle-2015-0086.

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Abstract We prove that the geometric Bogomolov conjecture for any abelian varieties is reduced to that for nowhere degenerate abelian varieties with trivial trace. In particular, the geometric Bogomolov conjecture holds for abelian varieties whose maximal nowhere degenerate abelian subvariety is isogenous to a constant abelian variety. To prove the results, we investigate closed subvarieties of abelian schemes over constant varieties, where constant varieties are varieties over a function field which can be defined over the constant field of the function field.
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6

Totaro, Burt. "Pseudo-abelian varieties." Annales scientifiques de l'École normale supérieure 46, no. 5 (2013): 693–721. http://dx.doi.org/10.24033/asens.2199.

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7

Ji, Shanyu. "on abelian varieties." Duke Mathematical Journal 58, no. 3 (June 1989): 657–67. http://dx.doi.org/10.1215/s0012-7094-89-05831-6.

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8

Bosch, Siegfried, and Werner Lütkebohmert. "Degenerating abelian varieties." Topology 30, no. 4 (1991): 653–98. http://dx.doi.org/10.1016/0040-9383(91)90045-6.

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9

Archinard, Natália. "Hypergeometric Abelian Varieties." Canadian Journal of Mathematics 55, no. 5 (October 1, 2003): 897–932. http://dx.doi.org/10.4153/cjm-2003-037-4.

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AbstractIn this paper, we construct abelian varieties associated to Gauss’ and Appell–Lauricella hypergeometric series. Abelian varieties of this kind and the algebraic curves we define to construct them were considered by several authors in settings ranging from monodromy groups (Deligne, Mostow), exceptional sets (Cohen, Wolfart, Wüstholz), modular embeddings (Cohen, Wolfart) to CM-type (Cohen, Shiga, Wolfart) and modularity (Darmon). Our contribution is to provide a complete, explicit and self-contained geometric construction.
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10

Chai, Ching-Li, and Frans Oort. "Hypersymmetric Abelian Varieties." Pure and Applied Mathematics Quarterly 2, no. 1 (2006): 1–27. http://dx.doi.org/10.4310/pamq.2006.v2.n1.a2.

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11

Carocca, Angel, Herbert Lange, and Rubí E. Rodríguez. "Abelian varieties with finite abelian group action." Archiv der Mathematik 112, no. 6 (April 20, 2019): 615–22. http://dx.doi.org/10.1007/s00013-018-1291-9.

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12

SILVERBERG, A., and YU G. ZARHIN. "Polarizations on abelian varieties." Mathematical Proceedings of the Cambridge Philosophical Society 133, no. 2 (September 2002): 223–33. http://dx.doi.org/10.1017/s0305004102005935.

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Every isogeny class over an algebraically closed field contains a principally polarized abelian variety ([10, corollary 1 to theorem 4 in section 23]). Howe ([3]; see also [4]) gave examples of isogeny classes of abelian varieties over finite fields with no principal polarizations (but not with the degrees of all the polarizations divisible by a given non-zero integer, as in Theorem 1·1 below). In [17] we obtained, for all odd primes [lscr ], isogeny classes of abelian varieties in positive characteristic, all of whose polarizations have degree divisible by [lscr ]2. We gave results in the more general context of invertible sheaves; see also Theorems 6·1 and 5·2 below. Our results gave the first examples for which all the polarizations of the abelian varieties in an isogeny class have degree divisible by a given prime. Inspired by our results in [17], Howe [5] recently obtained, for all odd primes [lscr ], examples of isogeny classes of abelian varieties over fields of arbitrary characteristic different from [lscr ] (including number fields), all of whose polarizations have degree divisible by [lscr ]2.
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13

Ullmo, Emmanuel, and Andrei Yafaev. "Algebraic flows on abelian varieties." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 741 (August 1, 2018): 47–66. http://dx.doi.org/10.1515/crelle-2015-0085.

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Abstract Let A be an abelian variety. The abelian Ax–Lindemann theorem shows that the Zariski closure of an algebraic flow in A is a translate of an abelian subvariety of A. The paper discusses some conjectures on the usual topological closure of an algebraic flow in A. The main result is a proof of these conjectures when the algebraic flow is given by an algebraic curve.
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14

Murty, V. Kumar, and Ying Zong. "Splitting of abelian varieties." Advances in Mathematics of Communications 8, no. 4 (November 2014): 511–19. http://dx.doi.org/10.3934/amc.2014.8.511.

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15

Kempf, George R. "Multiplication Over Abelian Varieties." American Journal of Mathematics 110, no. 4 (August 1988): 765. http://dx.doi.org/10.2307/2374649.

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16

Milne, J. S. "Periods of abelian varieties." Compositio Mathematica 140, no. 05 (September 2004): 1149–75. http://dx.doi.org/10.1112/s0010437x04000417.

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17

Pareschi, Giuseppe. "Syzygies of abelian varieties." Journal of the American Mathematical Society 13, no. 3 (April 10, 2000): 651–64. http://dx.doi.org/10.1090/s0894-0347-00-00335-0.

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18

Scholl, Travis. "Super-isolated abelian varieties." Journal of Number Theory 206 (January 2020): 138–68. http://dx.doi.org/10.1016/j.jnt.2019.06.008.

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19

Chen, Jungkai A., and Christopher D. Hacon. "Characterization of abelian varieties." Inventiones Mathematicae 143, no. 2 (February 1, 2001): 435–47. http://dx.doi.org/10.1007/s002220000111.

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20

Zarhin, Yuri G. "Almost isomorphic abelian varieties." European Journal of Mathematics 3, no. 1 (November 30, 2016): 22–33. http://dx.doi.org/10.1007/s40879-016-0122-4.

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21

Zarhin, Yuri G. "Abelian varieties without homotheties." Mathematical Research Letters 14, no. 1 (2007): 157–64. http://dx.doi.org/10.4310/mrl.2007.v14.n1.a13.

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22

Silverberg, A., and Yu G. Zarhin. "Isogenies of abelian varieties." Journal of Pure and Applied Algebra 90, no. 1 (November 1993): 23–37. http://dx.doi.org/10.1016/0022-4049(93)90133-e.

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23

Centeleghe, Tommaso, and Jakob Stix. "Categories of abelian varieties over finite fields, I: Abelian varieties over 𝔽p." Algebra & Number Theory 9, no. 1 (February 18, 2015): 225–65. http://dx.doi.org/10.2140/ant.2015.9.225.

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24

Zhao, Heer. "Extending finite-subgroup schemes of semistable abelian varieties via log-abelian varieties." Kyoto Journal of Mathematics 60, no. 3 (September 2020): 895–910. http://dx.doi.org/10.1215/21562261-2019-0049.

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25

ABDULALI, SALMAN. "ABELIAN VARIETIES OF TYPE III AND THE HODGE CONJECTURE." International Journal of Mathematics 10, no. 06 (September 1999): 667–75. http://dx.doi.org/10.1142/s0129167x99000264.

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We show that the algebraicity of Weil's Hodge cycles implies the usual Hodge conjecture for a general member of a PEL-family of abelian varieties of type III. We deduce the general Hodge conjecture for certain 6-dimensional abelian varieties of type III, and the usual Hodge and Tate conjectures for certain 4-dimensional abelian varieties of type III.
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26

Tang, Yunqing. "Cycles in the de Rham cohomology of abelian varieties over number fields." Compositio Mathematica 154, no. 4 (March 8, 2018): 850–82. http://dx.doi.org/10.1112/s0010437x17007679.

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In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of$\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus’ prediction for some families of abelian varieties. These families include geometrically simple abelian varieties of prime dimension that have non-trivial endomorphism ring. The proof uses a crystalline analogue of Faltings’ isogeny theorem due to Bost and the known cases of the Mumford–Tate conjecture.
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27

Jarden, Moshe. "Diamonds in Torsion of Abelian Varieties." Journal of the Institute of Mathematics of Jussieu 9, no. 3 (February 10, 2010): 477–80. http://dx.doi.org/10.1017/s1474748009000255.

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AbstractA theorem of Kuyk says that every Abelian extension of a Hilbertian field is Hilbertian. We conjecture that for an Abelian variety A defined over a Hilbertian field K every extension L of K in K(Ator) is Hilbertian. We prove our conjecture when K is a number field. The proof applies a result of Serre about l-torsion of Abelian varieties, information about l-adic analytic groups, and Haran's diamond theorem.
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28

Lombardo, Davide, and Andrea Maffei. "Abelian Varieties as Automorphism Groups of Smooth Projective Varieties." International Mathematics Research Notices 2020, no. 7 (April 19, 2018): 1942–56. http://dx.doi.org/10.1093/imrn/rny077.

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29

Odaka, Yuji. "Tropical Geometric Compactification of Moduli, II: A g Case and Holomorphic Limits." International Mathematics Research Notices 2019, no. 21 (January 31, 2018): 6614–60. http://dx.doi.org/10.1093/imrn/rnx293.

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Abstract We compactify the classical moduli variety Ag of principally polarized abelian varieties of complex dimension g, by attaching the moduli of flat tori of real dimensions at most g in an explicit manner. Equivalently, we explicitly determine the Gromov–Hausdorff limits of principally polarized abelian varieties. This work is analogous to [50], where we compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov–Hausdorff limits along holomorphic families of abelian varieties and show that these form special nontrivial subsets of the whole boundary. We also do the same for algebraic curves case and observe a crucial difference with the case of abelian varieties.
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30

Kajiwara, Takeshi, Kazuya Kato, and Chikara Nakayama. "Logarithmic abelian varieties, Part IV: Proper models." Nagoya Mathematical Journal 219 (September 2015): 9–63. http://dx.doi.org/10.1215/00277630-3140577.

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31

Kajiwara, Takeshi, Kazuya Kato, and Chikara Nakayama. "Logarithmic abelian varieties, Part IV: Proper models." Nagoya Mathematical Journal 219 (September 2015): 9–63. http://dx.doi.org/10.1017/s0027763000027082.

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32

Orr, Martin. "On compatibility between isogenies and polarizations of abelian varieties." International Journal of Number Theory 13, no. 03 (February 9, 2017): 673–704. http://dx.doi.org/10.1142/s1793042117500348.

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We discuss the notion of polarized isogenies of abelian varieties, that is, isogenies which are compatible with given principal polarizations. This is motivated by problems of unlikely intersections in Shimura varieties. Our aim is to show that certain questions about polarized isogenies can be reduced to questions about unpolarized isogenies or vice versa. Our main theorem concerns abelian varieties [Formula: see text] which are isogenous to a fixed abelian variety [Formula: see text]. It establishes the existence of a polarized isogeny [Formula: see text] whose degree is polynomially bounded in [Formula: see text], if there exist both an unpolarized isogeny [Formula: see text] of degree [Formula: see text] and a polarized isogeny [Formula: see text] of unknown degree. As a further result, we prove that given any two principally polarized abelian varieties related by an unpolarized isogeny, there exists a polarized isogeny between their fourth powers. The proofs of both theorems involve calculations in the endomorphism algebras of the abelian varieties, using the Albert classification of these endomorphism algebras and the classification of Hermitian forms over division algebras.
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33

KEARNES, KEITH A., and ROSS WILLARD. "FINITENESS PROPERTIES OF LOCALLY FINITE ABELIAN VARIETIES." International Journal of Algebra and Computation 09, no. 02 (April 1999): 157–68. http://dx.doi.org/10.1142/s0218196799000114.

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34

MCPHAIL, CAROLYN E., and SIDNEY A. MORRIS. "VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES." Bulletin of the Australian Mathematical Society 78, no. 3 (December 2008): 487–95. http://dx.doi.org/10.1017/s0004972708000877.

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AbstractThe variety of topological groups generated by the class of all abelian kω-groups has been shown to equal the variety of topological groups generated by the free abelian topological group on [0, 1]. In this paper it is proved that the free abelian topological group on a compact Hausdorff space X generates the same variety if and only if X is not scattered.
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35

Carocca, Angel, Herbert Lange, and Rubí E. Rodríguez. "RETRACTED ARTICLE: Abelian varieties with finite abelian group action." Archiv der Mathematik 112, no. 4 (October 8, 2018): 447–48. http://dx.doi.org/10.1007/s00013-018-1244-3.

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36

Benoist, Franck, and Françoise Delon. "Questions de corps de définition pour les variétés abéliennes en caractéristique positive." Journal of the Institute of Mathematics of Jussieu 7, no. 4 (October 2008): 623–39. http://dx.doi.org/10.1017/s1474748008000145.

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AbstractDichotomies in various conjectures from algebraic geometry are in fact occurrences of the dichotomy among Zariski structures. This is what Hrushovski showed and which enabled him to solve, positively, the geometric Mordell–Lang conjecture in positive characteristic. Are we able now to avoid this use of Zariski structures? Pillay and Ziegler have given a direct proof that works for semi-abelian varieties they called ‘very thin’, which include the ordinary abelian varieties. But it does not apply in all generality: we describe here an abelian variety which is not very thin. More generally, we consider from a model-theoretical point of view several questions about the fields of definition of semi-abelian varieties.
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37

Shin, Sug Woo. "Abelian varieties and Weil representations." Algebra & Number Theory 6, no. 8 (December 14, 2012): 1719–72. http://dx.doi.org/10.2140/ant.2012.6.1719.

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38

Masser, David, and Gisbert Wüstholz. "Polarization estimates for abelian varieties." Algebra & Number Theory 8, no. 5 (September 16, 2014): 1045–70. http://dx.doi.org/10.2140/ant.2014.8.1045.

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39

Kimura, Shun-ichi. "Correspondences to abelian varieties II." Hiroshima Mathematical Journal 35, no. 2 (July 2005): 251–61. http://dx.doi.org/10.32917/hmj/1150998274.

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40

Kempf, George R. "Linear Systems on Abelian Varieties." American Journal of Mathematics 111, no. 1 (February 1989): 65. http://dx.doi.org/10.2307/2374480.

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41

ABE, Yukitaka. "GEOMETRICALLY SIMPLE QUASI-ABELIAN VARIETIES." Kyushu Journal of Mathematics 72, no. 2 (2018): 269–75. http://dx.doi.org/10.2206/kyushujm.72.269.

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42

Faltings, Gerd. "Diophantine Approximation on Abelian Varieties." Annals of Mathematics 133, no. 3 (May 1991): 549. http://dx.doi.org/10.2307/2944319.

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43

Bo-Hae Im and Michael Larsen. "Abelian varieties over cyclic fields." American Journal of Mathematics 130, no. 5 (2008): 1195–210. http://dx.doi.org/10.1353/ajm.0.0020.

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44

Nakamaye, Michael. "Seshadri Constants on Abelian Varieties." American Journal of Mathematics 118, no. 3 (1996): 621–35. http://dx.doi.org/10.1353/ajm.1996.0028.

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45

Kimura, Shun-Ichi. "Correspondences to abelian varieties I." Duke Mathematical Journal 73, no. 3 (March 1994): 583–91. http://dx.doi.org/10.1215/s0012-7094-94-07324-9.

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46

Milne, J. S. "Lefschetz classes on abelian varieties." Duke Mathematical Journal 96, no. 3 (February 1999): 639–75. http://dx.doi.org/10.1215/s0012-7094-99-09620-5.

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47

Moonen, B. J. J., and Yu G. Zarhin. "Weil classes on abelian varieties." Journal für die reine und angewandte Mathematik (Crelles Journal) 1998, no. 496 (March 2, 1998): 83–92. http://dx.doi.org/10.1515/crll.1998.034.

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48

Kamienny, Sheldon, and Joseph L. Wetherell. "On torsion in abelian varieties." Communications in Algebra 26, no. 5 (January 1998): 1675–78. http://dx.doi.org/10.1080/00927879808826230.

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49

Lubicz, David, and Damien Robert. "Computing isogenies between abelian varieties." Compositio Mathematica 148, no. 5 (July 10, 2012): 1483–515. http://dx.doi.org/10.1112/s0010437x12000243.

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AbstractWe describe an efficient algorithm for the computation of separable isogenies between abelian varieties represented in the coordinate system given by algebraic theta functions. Let A be an abelian variety of dimension g defined over a field of odd characteristic. Our algorithm comprises two principal steps. First, given a theta null point for A and a subgroup K isotropic for the Weil pairing, we explain how to compute the theta null point corresponding to the quotient abelian variety A/K. Then, from the knowledge of a theta null point of A/K, we present an algorithm to obtain a rational expression for an isogeny from A to A/K. The algorithm that results from combining these two steps can be viewed as a higher-dimensional analog of the well-known algorithm of Vélu for computing isogenies between elliptic curves. In the case where K is isomorphic to (ℤ/ℓℤ)g for ℓ∈ℕ*, the overall time complexity of this algorithm is equivalent to O(log ℓ) additions in A and a constant number of ℓth root extractions in the base field of A. In order to improve the efficiency of our algorithms, we introduce a compressed representation that allows us to encode a point of level 4ℓ of a g-dimensional abelian variety using only g(g+1)/2⋅4g coordinates. We also give formulas for computing the Weil and commutator pairings given input points in theta coordinates.
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50

Rubei, Elena. "On syzygies of abelian varieties." Transactions of the American Mathematical Society 352, no. 6 (March 7, 2000): 2569–79. http://dx.doi.org/10.1090/s0002-9947-00-02398-9.

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