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SUMMARY:Francesc Castella (University of California\, Santa Barbara)
DTSTART:20210422T180000Z
DTEND:20210422T190000Z
DTSTAMP:20260423T035750Z
UID:LATeN/42
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LATeN/42/">O
 n a conjecture of Darmon-Rotger in the adjoint CM case</a>\nby Francesc Ca
 stella (University of California\, Santa Barbara) as part of Coloquio Lati
 noamericano de Teoría de Números\n\n\nAbstract\nLet $E$ be an elliptic c
 urve over $\\mathbf{Q}$\, and suppose that $L(E\,s)$ has sign $+1$ in its 
 functional equation and vanishes at $s=1$. Let $p>3$ be a prime of good or
 dinary reduction for $E$.  \nA construction of Darmon-Rotger attaches to $
 E$ and an auxiliary weight one cuspidal eigenform $g$ a Selmer class $\\ka
 ppa_p(E\,g\,g^*)\\in\\mathrm{Sel}(\\mathbf{Q}\,V_pE)$. Assuming that $L(E\
 ,{\\rm ad}^0(g)\,1)\\neq 0$\, they conjectured that the following are equi
 valent: (1) $\\kappa_p(E\,g\,g^*)\\neq 0$\, (2) ${\\rm dim}_{\\mathbf{Q}_p
 }\\mathrm{Sel}(\\mathbf{Q}\,V_pE)=2$.\n\nIn this talk I will outline a pro
 of of Darmon-Rotger's conjecture when $g$ has CM and the Tate-Shafarevich 
 group of $E$ has finite $p$-primary part (and some mild additional hypothe
 ses). If time permits\, I'll also talk about the extension of these result
 s to the case of supersingular primes $p$. Based on joint work with Ming-L
 un Hsieh.\n
LOCATION:https://researchseminars.org/talk/LATeN/42/
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