Adjoint L-value formula and its relation to Tate conjecture

Abstract: For a Hecke eigenform $f$, we state an adjoint $L$-value formula relative to each quaternion algebra $D$ over $\mathbf{Q}$ with discriminant $d$ and reduced norm $N$. A key to prove the formula is the theta correspondence for the quadratic $\mathbf{Q}$-space $(D,N)$. Under the $R=T$-theorem, the $p$-part of the Bloch-Kato conjecture is known; so, the formula is an adjoint Selmer class number formula. We also describe how to relate the formula to a consequence of the Tate conjecture for quaternionic Shimura varieties.

algebraic geometrynumber theory

Audience: researchers in the topic

Séminaire de géométrie arithmétique et motivique (Paris Nord)