CM values of higher automorphic Green functions for orthogonal groups

Gross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function $$G_s(z_1,z_2)$$ G s ( z 1 , z 2 ) for the elliptic modular group at positive integral spectral parameter s are given by logarithms of algebraic numbers in suitable class fields. We prove a partial average version of this conjecture, where we sum in the first variable $$z_1$$ z 1 over all CM points of a fixed discriminant $$d_1$$ d 1 (twisted by a genus character), and allow in the second variable the evaluation at individual CM points of discriminant $$d_2$$ d 2 . This result is deduced from more general statements for automorphic Green functions on Shimura varieties associated with the group $${\text {GSpin}}(n,2)$$ GSpin ( n , 2 ) . We also use our approach to prove a Gross–Kohnen–Zagier theorem for higher Heegner divisors on Kuga–Sato varieties over modular curves.

Localizing the Primary Motor Cortex of the Hand by the 10-5 and 10-20 Systems for Neurostimulation: An MRI Study

Background. The primary motor cortex of the hand (M1-Hand) is a target used in transcranial magnetic stimulation (TMS) and in transcranial direct current stimulation (tDCS) for the treatment and evaluation of motor neurological diseases. Magnetic resonance imaging–guided neuronavigation locates the M1-Hand with high precision, but at a high cost. Although less accurate, the C3/C4 points of the international 10-20 system (IS 10-20) are routinely used to locate the M1-Hand. The international 10-5 system (IS 10-5) was developed with additional points (C3h/C4h), which could make it more accurate, but has not yet been tested on the location of the M1-Hand. Objective. To analyze and compare the accuracy of C1/C2, C3h/C4h and C3/C4 points in locating the M1-Hand correspondence on the scalp. Methods. The authors comparatively analyzed the distances from points C1/C2, C3h/C4h, and C3/C4 to the correspondence of the M1-Hand on the scalp in 30 MRI head exams. Results. In most cases, the M1-Hand was located between C1-C3h and C2-C4h in the left and right hemispheres of the brain, respectively. The C3h (0.98 ± 0.49 cm) and C4h (0.98 ± 0.51 cm) points presented the shortest distances from the M1-Hand, with a significant difference when compared with C3/C4. The accuracy between C1/C2 and C3h/C4h was not statistically significant. Conclusion. The C3h/C4h and C1/C2 points were more accurate when compared with the C3 and C4 points in locating the M1-Hand correspondence on the scalp.

The Lambda Invariants at CM Points

In this paper, we show that $\lambda (z_1) -\lambda (z_2)$, $\lambda (z_1)$, and $1-\lambda (z_1)$ are all Borcherds products on $X(2) \times X(2)$. We then use the big CM value formula of Bruinier, Kudla, and Yang to give explicit factorization formulas for the norms of $\lambda (\frac{d+\sqrt d}2)$, $1-\lambda (\frac{d+\sqrt d}2)$, and $\lambda (\frac{d_1+\sqrt{d_1}}2) -\lambda (\frac{d_2+\sqrt{d_2}}2)$, with the latter under the condition $(d_1, d_2)=1$. Finally, we use these results to show that $\lambda (\frac{d+\sqrt d}2)$ is always an algebraic integer and can be easily used to construct units in the ray class field of ${\mathbb{Q}}(\sqrt{d})$ of modulus $2$. In the process, we also give explicit formulas for a whole family of local Whittaker functions, which are of independent interest.