Abelian surfaces over totally real fields are potentially modular
Keyword(s):
Genus 2
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AbstractWe show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse–Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces $A$ A over ${\mathbf {Q}}$ Q with $\operatorname{End}_{ {\mathbf {C}}}A={\mathbf {Z}}$ End C A = Z . We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.
1972 ◽
Vol 78
(1)
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pp. 74-77
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2011 ◽
Vol 158
(2)
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pp. 247-305
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1973 ◽
Vol 175
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pp. 209-209
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2012 ◽
Vol 47
(2)
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pp. 143-148
Keyword(s):
2008 ◽
Vol 143
(2)
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pp. 225-279
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1998 ◽
Vol 41
(2)
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pp. 158-165
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Keyword(s):