Idempotent relations and factors of Jacobians. (English) Zbl 0652.14011
Let \(\pi_ i:\) \(C\to C_ i\), \(1\leq i\leq n\), be a collection of subcovers of a curve C. In this paper we investigate isogeny relations among the Jacobians \(J_ i=J_{C_ i}\), of the curves \(C_ i\). Denoting by \(\epsilon_ i=\epsilon_{\pi_ i}\in End^ 0(J_ C)\) the idempotent attached to the morphism \(\pi_ i\), we have the following result:
Theorem 1: \(J_ 1\times...\times J_ m\sim J_{m+1}\times...\times J_ n\Leftrightarrow \epsilon_ 1+...+\epsilon_ m\sim \epsilon_{m+1}+...+\epsilon_ n.\)
Here, the first \(\sim\) denotes isogeny of abelian varieties whereas the second \(\sim\) denotes “character equivalence” for elements \(a_ 1,a_ 2\in End^ 0(J):\) \(a_ 1\sim a_ 2\Leftrightarrow \chi (a_ 1)=\chi (a_ 2)\), all \({\mathbb{Q}}\)-characters \(\chi\) of \(End^ 0(J).\)
From theorem 1 follows in particular that every such idempotent relation induces the relation \(g_ 1+...+g_ m=g_{m+1}+...+g_ n\) among the genera \(g_ i=g_{C_ i}\) of the curves \(C_ i\), thereby generalizing a result of the reviewer [Can. Math. Bull. 28, 321-327 (1985; Zbl 0557.14017)], which in turn generalizes results of R. D. M. Accola [Proc. Am. Math. Soc. 21, 477-482 (1969; Zbl 0174.374) and 25, 598-602 (1970; Zbl 0212.425)].
Theorem 1 is, in fact a special case of a general theorem about idempotents \(\epsilon_ i\in End^ 0(A)\) on an abelian variety A and their corresponding factors \(\epsilon_ i(A):\)
Theorem 2: \(\epsilon_ 1(A)\times...\times \epsilon_ m(A)\sim \epsilon_{m+1}(A)\times...\times \epsilon_ n(A)\Leftrightarrow \epsilon_ 1+...+\epsilon_ m\sim \epsilon_{m+1}+...+\epsilon_ n.\)
The usefulness of theorem 1 rests ultimately on the ability to readily exhibit idempotent relations in \(End^ 0(J_ C)\). We present two such methods: viz. \((1)\quad via\) Galois theory (studying idempotent relations in \({\mathbb{Q}}[G])\); and \((2)\quad via\) intersection theory on the product surface \(C\times C\); and apply these to derive interesting isogeny relations for subcovers of \((a)\quad the\) Fermat curves; \((b)\quad the\) modular curves X(p); \((c)\quad the\) Drinfeld curves \(XY^ q-YX^ q=1\) and \((d)\quad the\) Humbert curves.
Theorem 1: \(J_ 1\times...\times J_ m\sim J_{m+1}\times...\times J_ n\Leftrightarrow \epsilon_ 1+...+\epsilon_ m\sim \epsilon_{m+1}+...+\epsilon_ n.\)
Here, the first \(\sim\) denotes isogeny of abelian varieties whereas the second \(\sim\) denotes “character equivalence” for elements \(a_ 1,a_ 2\in End^ 0(J):\) \(a_ 1\sim a_ 2\Leftrightarrow \chi (a_ 1)=\chi (a_ 2)\), all \({\mathbb{Q}}\)-characters \(\chi\) of \(End^ 0(J).\)
From theorem 1 follows in particular that every such idempotent relation induces the relation \(g_ 1+...+g_ m=g_{m+1}+...+g_ n\) among the genera \(g_ i=g_{C_ i}\) of the curves \(C_ i\), thereby generalizing a result of the reviewer [Can. Math. Bull. 28, 321-327 (1985; Zbl 0557.14017)], which in turn generalizes results of R. D. M. Accola [Proc. Am. Math. Soc. 21, 477-482 (1969; Zbl 0174.374) and 25, 598-602 (1970; Zbl 0212.425)].
Theorem 1 is, in fact a special case of a general theorem about idempotents \(\epsilon_ i\in End^ 0(A)\) on an abelian variety A and their corresponding factors \(\epsilon_ i(A):\)
Theorem 2: \(\epsilon_ 1(A)\times...\times \epsilon_ m(A)\sim \epsilon_{m+1}(A)\times...\times \epsilon_ n(A)\Leftrightarrow \epsilon_ 1+...+\epsilon_ m\sim \epsilon_{m+1}+...+\epsilon_ n.\)
The usefulness of theorem 1 rests ultimately on the ability to readily exhibit idempotent relations in \(End^ 0(J_ C)\). We present two such methods: viz. \((1)\quad via\) Galois theory (studying idempotent relations in \({\mathbb{Q}}[G])\); and \((2)\quad via\) intersection theory on the product surface \(C\times C\); and apply these to derive interesting isogeny relations for subcovers of \((a)\quad the\) Fermat curves; \((b)\quad the\) modular curves X(p); \((c)\quad the\) Drinfeld curves \(XY^ q-YX^ q=1\) and \((d)\quad the\) Humbert curves.
Reviewer: E.Kani
MSC:
| 14H40 | Jacobians, Prym varieties |
| 14H25 | Arithmetic ground fields for curves |
| 14K99 | Abelian varieties and schemes |
| 14H30 | Coverings of curves, fundamental group |
References:
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