GL+(2, ℝ)–orbits in Prym eigenform loci

Lanneau, Erwan; Nguyen, Duc-Manh

doi:10.2140/gt.2016.20.1359

Cited by 7 publications

(1 citation statement)

References 30 publications

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“…The topology of these curves is quite well-known. The Euler characteristic has been determined in [Bai07] for g = 2 and in [LN16b] for g = 3, 4. The number of cusps is calculated for g = 2 in [McM05] and in [LN14] for g = 3, 4.…”

Section: Introductionmentioning

confidence: 99%

Non-Existence and Finiteness Results for Teichmüller Curves in Prym Loci

Lanneau

Möller

2019

Experimental Mathematics

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The minimal stratum in Prym loci have been the first source of infinitely many primitive, but not algebraically primitive Teichmüller curves. We show that the stratum Prym(2,1,1) contains no such Teichmüller curve and the stratum Prym(2,2) at most 92 such Teichmüller curves.This complements the recent progress establishing general -but non-effective -methods to prove finiteness results for Teichmüller curves and serves as proof of concept how to use the torsion condition in the non-algebraically primitive case.

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Section: Introductionmentioning

confidence: 99%

Non-Existence and Finiteness Results for Teichmüller Curves in Prym Loci

Lanneau

Möller

2019

Experimental Mathematics

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Connected components of Prym eigenform loci in genus three

Lanneau¹,

Nguyen²

2017

Math. Ann.

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This paper is devoted to the classification of connected components of Prym eigenform loci in the strata H(2, 2) odd and H(1, 1, 2) of the Abelian differential bundle ΩM3 over M3. These loci, discovered by McMullen [Mc06], are GL + (2, R)-invariant submanifolds of complex dimension 3 of ΩMg that project to the locus of Riemann surfaces whose Jacobian variety has a factor admitting real multiplication by some quadratic order OD.It turns out that these subvarieties can be classified by the discriminant D of the corresponding quadratic orders. However there algebraic varieties are not necessarily irreducible. The main result we show is that for each discriminant D the corresponding locus has one component if D ≡ 0, 4 mod 8, two components if D ≡ 1 mod 8, and is empty if D ≡ 5 mod 8.Surprisingly our result contrasts with the case of Prym eigenform loci in the strata H(1, 1) (studied by McMullen [Mc07]) which is connected for every discriminant D.1.1. Brief facts summary in the genus 2 case. The locus E 2 = {(X, ω) ∈ ΩM 2 : Jac(X) admits real multiplication with ω as an eigenform}, plays an important role in the classification of SL(2, R)-orbit closures in ΩM 2 . Here A = Jac(X) ∈ A 2 , K is a real quadratic field, and the endomorphism ring is canonically isomorphic to the ring of homomorphisms of H 1 (X, Z) that preserve the Hodge decomposition. The polarization comes from the intersection form J 0 0 J on the homology. The locus E 2 is actually a (disjoint) union of subvarieties indexed by the discriminants of the orders O ⊂ End(Jac(X)). Since orders in quadratic fields (quadratic orders) are classified by their discriminant, the unique quadratic order with discriminant D is denoted by O D . We then define1 2 ERWAN LANNEAU AND DUC-MANH NGUYENThe subvarieties ΩE D are of interest since they are GL + (2, R)-invariant submanifolds of ΩM 2 (see [Mc07,Mc06]). We can further stratified ΩE D by defining ΩE D (κ) = ΩE D ∩ H(κ) for κ = (2) or κ = (1, 1). This defines complex submanifolds of dimension 2 and 3, respectively. Hence ΩE D (2) projects to a union of algebraic curves (Teichmüller curves) in the moduli space M 2 . Components of ΩED (1, 1) and ΩE D (2). It is well known that the set of Abelian varieties A ∈ A 2 admitting real multiplication by O D with a specified faithful representation i : O D → End(A) is parametrized by the Hilbert modular surface X D := (H × −H)/SL(O D ). In [Mc07], it has been shown that each ΩE D can be viewed as a C * -bundle over a Zariski open subset of X D , and we have E 2 = D≥4,D≡0,1 mod 4 ΩE D In particular ΩE D is a connected, complex suborbifold of ΩM 2 of dimension 3. The fact that there is only one (connected) eigenform locus for each D follows from the fact that there is only one faithful, proper, self-adjoint representation i : O D → M 4 (Z) up to conjugation by Sp(4, Z) (see [Mc07] Theorem 4.4). It follows that ΩE D (1, 1) is a Zariski open set in ΩE D . In particular ΩE D (1, 1) is connected for any quadratic discriminant D.The classification of components of ΩE D (2) has also bee...

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