Chapter Eight. Local Heights of CM Points

In this paper we establish a Chowla–Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function ${\rm\Gamma}$ and an analogous function ${\rm\Gamma}_{2}$ at rational numbers. We combine this identity with work of Colmez to relate the CM values of the Hilbert modular function to Faltings heights of CM abelian varieties. We also give explicit formulas for products of exponentials of Faltings heights, allowing us to study some of their arithmetic properties using the Lang–Rohrlich conjecture.

Download Full-text

On two arithmetic theta lifts

Compositio Mathematica ◽

10.1112/s0010437x18007327 ◽

2018 ◽

Vol 154 (10) ◽

pp. 2090-2149 ◽

Cited By ~ 1

Author(s):

Stephan Ehlen ◽

Siddarth Sankaran

Keyword(s):

Modular Forms ◽

Theta Functions ◽

Chow Group ◽

Shimura Varieties ◽

Green Functions ◽

Theta Lift ◽

Moderate Growth ◽

Generating Series ◽

Cm Points ◽

The Difference

Our aim is to clarify the relationship between Kudla’s and Bruinier’s Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type, which play a key role in the arithmetic geometry of these cycles in the context of Kudla’s program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a non-holomorphic modular form and has trivial (cuspidal) holomorphic projection. Along the way, we construct a section of the Maaß lowering operator for moderate growth forms valued in the Weil representation using a regularized theta lift, which may be of independent interest, as it in particular has applications to mock modular forms. We also consider arithmetic-geometric applications to integral models of $U(n,1)$ Shimura varieties. Each family of Green functions gives rise to a formal arithmetic theta function, valued in an arithmetic Chow group, that is conjectured to be modular; our main result is the modularity of the difference of the two arithmetic theta functions. Finally, we relate the arithmetic heights of the special cycles to special derivatives of Eisenstein series, as predicted by Kudla’s conjecture, and describe a refinement of a theorem of Bruinier, Howard and Yang on arithmetic intersections against CM points.

Download Full-text

Equations with powers of singular moduli

International Journal of Number Theory ◽

10.1142/s1793042119500234 ◽

2019 ◽

Vol 15 (03) ◽

pp. 445-468 ◽

Cited By ~ 2

Author(s):

Antonin Riffaut

Keyword(s):

Rational Number ◽

The Other ◽

Other Hand ◽

Singular Moduli ◽

Cm Points ◽

The One ◽

Straight Lines ◽

Positive Integers

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli [Formula: see text] such that the numbers [Formula: see text], [Formula: see text] and [Formula: see text] are linearly dependent over [Formula: see text] for some positive integers [Formula: see text], must be of degree at most [Formula: see text]. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in [Formula: see text] defined over [Formula: see text]. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola [Formula: see text], where [Formula: see text].

Download Full-text

Special values of the Gamma function at CM points

The Ramanujan Journal ◽

10.1007/s11139-013-9531-x ◽

2014 ◽

Vol 36 (3) ◽

pp. 355-373 ◽

Cited By ~ 1

Author(s):

M. Ram Murty ◽

Chester Weatherby

Keyword(s):

Gamma Function ◽

Special Values ◽

Cm Points

Download Full-text

ALGEBRAIC INDEPENDENCE OF VALUES OF MODULAR FORMS

International Journal of Number Theory ◽

10.1142/s1793042111004769 ◽

2011 ◽

Vol 07 (04) ◽

pp. 1065-1074 ◽

Cited By ~ 5

Author(s):

SANOLI GUN ◽

M. RAM MURTY ◽

PURUSOTTAM RATH

Keyword(s):

Recent Work ◽

Modular Forms ◽

Fourier Coefficients ◽

Algebraic Independence ◽

Singular Values ◽

Critical Values ◽

Cm Points

We investigate values of modular forms with algebraic Fourier coefficients at algebraic arguments. As a consequence, we conclude about the nature of zeros of such modular forms. In particular, the singular values of modular forms (that is, values at CM points) are related to the recent work of Nesterenko. As an application, we deduce the transcendence of critical values of certain Hecke L-series. We also discuss how these investigations generalize to the case of quasi-modular forms with algebraic Fourier coefficients.

Download Full-text

Microsurgical anatomy of the distal anterior cerebral artery

Journal of Neurosurgery ◽

10.3171/jns.1978.49.2.0204 ◽

1978 ◽

Vol 49 (2) ◽

pp. 204-228 ◽

Cited By ~ 161

Author(s):

David Perlmutter ◽

Albert L. Rhoton

Keyword(s):

Corpus Callosum ◽

Cerebral Artery ◽

Anterior Cerebral Artery ◽

Internal Capsule ◽

Optic Chiasm ◽

Anterior Communicating Artery ◽

Microsurgical Anatomy ◽

Content Type ◽

Cm Points ◽

Distal Anterior Cerebral Artery

✓ The microsurgical anatomy of the distal anterior cerebral artery (ACA) has been defined in 50 cerebral hemispheres. The distal ACA, the portion beginning at the anterior communicating artery (ACoA), was divided into four segments (A2 through A5) according to Fischer. The distal ACA gave origin to central and cerebral branches. The central branches passed to the optic chiasm, suprachiasmatic area, and anterior forebrain below the corpus callosum. The cerebral branches were divided into cortical, subcortical, and callosal branches. The most frequent site of origin of the cortical branches was as follows: orbitofrontal and frontopolar arteries, A2; the anterior and middle internal frontal and callosomarginal arteries, A3; the paracentral artery, A4; and the superior and inferior parietal arteries, A5. The posterior internal frontal artery arose with approximately equal frequency from A3 and A4 and the callosomarginal artery. All the cortical branches arose more frequently from the pericallosal than the callosomarginal artery. Of the major cortical branches, the internal frontal and paracentral arteries arose most frequently from the callosomarginal artery. The distal ACA of one hemisphere sent branches to the contralateral hemisphere in 64% of brains. The anterior portions of the hemisphere between the 5-cm and 15-cm points on the circumferential line showed the most promise of revealing a recipient artery of sufficient size for an extracranial-intracranial artery anastomosis. The distal ACA was the principal artery supplying the corpus callosum. The recurrent artery, which arose from the A2 segment in 78% of hemispheres, sent branches into the subcortical area around the anterior limb of the internal capsule.

Download Full-text