Adjoint modular Galois representations and their Selmer groups

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(φ) of a two-dimensional modular Galois representation φ. We start with the p-adic Galois representation φ0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(φ0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(φ0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the one-variable case, we let φ denote the p-ordinary Galois representation with values in GL2(Zp[[T]]) lifting φ0, and the characteristic power series of the Selmer group Sel(ad(φ)) is given by a p-adic L-function interpolating L(1, ad(φk)) for weight k + 2 specialization φk of φ. In the two-variable case, we state a main conjecture on the characteristic power series in Zp[[T, S]] of Sel(ad(φ) ⊗ ν−1), where ν is the universal cyclotomic character with values in Zp[[S]]. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p-adic Siegel modular forms.

The affective component of pain in rodents: Direct evidence for a contribution of the anterior cingulate cortex

Numerous human and animal studies indirectly implicate neurons in the anterior cingulate cortex (ACC) in the encoding of the affective consequences of nociceptor stimulation. No causal evidence, however, has been put forth linking the ACC specifically to this function. Using a rodent pain assay that combines the hind-paw formalin model with the place-conditioning paradigm, we measured a learned behavior that directly reflects the affective component of pain in the rat (formalin-induced conditioned place avoidance) concomitantly with “acute” formalin-induced nociceptive behaviors (paw lifting, licking, and flinching) that reflect the intensity and localization of the nociceptive stimulus. Destruction of neurons originating from the rostral, but not caudal, ACC reduced formalin-induced conditioned place avoidance without reducing acute pain-related behaviors. These results provide evidence indicating that neurons in the ACC are necessary for the “aversiveness” of nociceptor stimulation.