Specter of the Spheres: Prologue

It’s probably not fair to spend years on this blog teaching writing and prose techniques and critiquing other novels without ever showing my own. Here’s the prologue from my newest book for your enjoyment.

Specter of the Spheres


Priestess Elienne squinted toward the southern horizon. The blood moon hung low, and time was short until daylight burned again. The grand arch glinted in the moonlight, marking the entrance to the underground vault. They would make it in time.

Her defender swarm pattered a rhythmic beat across the ruined lands and lulled her into a trance as they pressed forward. Each defender looked exactly the same: smooth obsidian shells that came up to her knees and eight razor-sharp legs. They had no eyes or faces. They moved by sensing the priestess’s heat signature or potential enemies.

Elienne knew from her studies that people would have called them spiders in the old language, but this wasn’t quite right. They were larger, and they had an intelligence spiders didn’t have.

They also harbored the souls of dead people. This was how they got their name: specterlings. They weren’t ghosts, though, a common misunderstanding of how the necromantic arts worked. A priestess had incarnated the shells, merely giving the illusion of life.

Elienne’s head snapped toward the western hills as the lead defender cried out its warning signal.

The sun?

No. A lava nethermental rushed at them, and the first crawler darted forward to protect the priestess.

She shouted, “No! Get back!”

The specterlings swarmed forward anyway, not understanding the true danger. Their only trained goal was protection. In most circumstances their leg blades would shred any threat: the swarm a stampede of razors.

Molten rock oozed out of the nethermental’s body, leaving a trail of dark pumice. The huge body rose to twice her height. It swung its boulder arms viciously at the attackers. Bits of lava and fire splattered haphazardly.

Elienne had no time to figure out who would have breathed life into such a destructive golem—probably one of the Persuader’s minions. She watched in horror as the lava hardened in an instant, trapping every one of her defenders.

A bright trail of fire arced and crackled through the air as the beast swiped at the lead crawler again. The crawler’s incinerated body melted into poison, a last-resort line of defense should all the specterlings die.

But the lava nethermental stomped over it, unfazed. No one had predicted such forces would try to stop her. Elienne needed time she didn’t have. One of her remaining specterlings had to remain alive for a banishment enchantment.

She pulled a dark purple crystal from her pack and slammed it into the sandy ground. It poked out at an awkward angle but stayed upright.

A vibrant glow emanated outward, and she began to sing in the ancient language.

Swipe: two more specterlings down. Poison pooled outward, and Elienne’s breathing doubled as she realized she might be too close to it. Her flesh body was not immune.

There isn’t time to move away, she thought.

She continued to sing with intense focus. The crystal shook under the tension in the winding melody. The song carried Elienne away from the scene. She closed her eyes. The beauty hurt. Her body shook with the pain, and she took it all upon herself. She needed more pain than ever before if she was to kill the powerful beast.

The ground now shuddered under the weight of the nethermental as it trampled closer. She opened her eyes to see the damage. Only one specterling remained. Elienne looked at where the nethermental’s eyes should have been but only saw oozing rock.

It somehow knew her location and moved directly toward her.

She needed that last specterling as a sacrifice for the necromantic ritual to work. As the nethermental’s swipe came forward, she completed the song and pushed pain into the crystal.

The crystal converted the pain to energy, which shot into the specterling. The defender called out a dying shriek, and Elienne relaxed. The specterling died before it received the blow from the nethermental.

The energy from the sacrificed life pulsed into the blazing golem in an ascendant burst. It landed with a sharp crack, and the beast collapsed into a lifeless heap of black rock, still glimmering from the heat.

Elienne fell to the ground, panting from the effort. Her small army of defenders were dead, and now she was on her own. If any more danger appeared, she’d have to fight without necromancy. She looked to where the monster had been, and the world distorted in waves from the shimmering heat.

Elienne pushed back to her feet and returned the depleted crystal to her pack. The sands tumbled under her feet, making the long journey harder than it needed to be. A heavy weight pressed on her shoulders.

She didn’t think she could continue; the ritual had taken too much out of her. Every muscle in her body drooped toward the ground, begging for rest. The singeing pain lingered from the ritual, but the sorrow at having failed her people hurt worse.

She was a priestess who had taken on vows. It had been over a millennium since the last failure, and that had wrecked the world. They almost hadn’t survived. With how things were now, all life would end this time.

Elienne glanced from her feet to the horizon once more. The arch marking the entrance to the vault grew as she approached, and she realized there was still hope. A quick flutter of energy titillated her chest, but the blood moon hung low. If the heat of the sun peaked the horizon, she’d be burned alive—darkness or moonlight were the only viable possibilities for survival.

She pressed on faster, not caring about her own life. It would be sacrificed at the coming ritual in hours anyway. Each one of the seven sects would send one priestess to complete the ritual and keep the world going.

They were seen as evil by most. Necromancy looked like an unnatural art to the rest of the world, and people had tried to squash them since the dawn of time.

But they weren’t evil. They protected life through sacrifice. Without death, there could be no life. Why couldn’t people understand that instead of sending these beasts to destroy them each blood moon?

Elienne shook herself free of these thoughts as she felt the sun burn her shoulders. She despaired that she had failed in her duty. She had been trained as a priestess for this one moment. The ritual kept the gods appeased. Without the ritual, during the full blood moon, the caverns would crack open and be exposed to the sun. They would all die, and without them, the humans would die as well.

Elienne ran with all her might. She couldn’t let that happen. A vicious scream came from her lips as the sun rounded the horizon. The lava nethermental had caused too much of a delay.

She reached the steps leading down to the vault, but the heat was too much; she collapsed. And with that, the end of the world began. The fools who had sent the monster knew not what they had done.

Six figures towered over Elienne, each adorning the black robes of a priestess. Every part of her body hurt. She could tell there were bruises all over. Flashes of the scene struck her memory: tumbling, crying out, and cracks of flesh on stone.

Elienne blinked several times as she tried to get her bearings.

One of the other priestesses said, “You made it. We didn’t think you were coming.”

She didn’t say anything. What was there to say? They’d all be dead the next night when they sacrificed themselves—no need to make friends. They had a job to do.

She looked around in fascination. Elienne had wondered what the inside of the vault looked like since she was a child. Now she had a single day to explore it. She cautiously pushed herself to her elbows. Her voice was shaky.

“What happened?”

Another priestess said, “We don’t know. We heard screams and rushed over. The vault sealed itself, so we’re safe now.”

Another said, “You should rest. You’ll need your strength for the ritual.”

Two of them swooped in with bizarre coordination, and Elienne felt too exhausted to resist. She gave in to the arms as she was carried to a different chamber. Seven beds were lined along the wall, and the two priestesses set Elienne on the first one.

She let the blackness overtake her.

The sound of whispered voices woke her, and she had no idea how much time had passed. They wouldn’t have let her sleep through the ritual. One of them noticed the stirring and called over.

“It is time.”

A pang of disappointment filled Elienne. She had dreamed of the vault her whole life, and now that she was here, she wouldn’t get the chance to see it before sacrificing herself on the altar.

She stood from the bed, still shaky from the pain song. A sudden fear filled her. What if she didn’t have the strength to go through with it? The upcoming ritual sacrifice would take much more concentration and intensity than the simple banishment spell she had done on the nethermental.

Elienne pulled herself from the bed and limped to the huddled group. Each step brought a sharp pain to her ankle, and she wondered if she had broken it while falling down the stairs.

One of the priestesses asked, “Can you make it to the altar?”

Elienne nodded. The group trudged along, up the stairs to the altar on top of the vault. She didn’t expect the simplicity of it. It appeared to be a stone statue with none of the intricate flair of the temple they worshiped at back home.

Seven circles rounded the altar, one for each priestess to stand on.

The group knew exactly what to do. They had all trained for this moment intensely. They waited for the proper alignment of the blood moon through the aperture in the statue. When the moment hit, a dark blood stain appeared across the altar symbolizing the first sacrifice that saved the world.

They held hands, and the priestess from the Haiel faction began the song. Each faction had a separate part of the song, and none had heard any of the other parts. A thrill filled Elienne now that she would get to hear it in full. After a few measures of the melody, the second priestess joined in.

The harmony produced a dark, strange sensation inside of Elienne. It was nothing like the painful beauty she was used to. Her turn came third, and she started the song. She focused hard at producing a clean tone when the first notes came out raspy.

The sounds meshed, and her voice cleared. Pain entered the song with her voice, and she saw the others cringe who hadn’t started their part.

They needed to take on the pain. It was essential to the completion of the ritual.

The rest of the parts joined, one by one, and each brought its own emotions to it. The cacophony of the counterpoint almost made Elienne falter. It was hard to focus on her own part when so many strange sounds kept coming at her and making her feel intense sensations: rage, lust, and even joy.

As the song intensified to the point of no return, the blood stain brightened. Elienne feared they had sung all the way to morning, and now the sunlight would burn them before they succeeded.

The light brightened more and more. But it wasn’t the fire of sunlight. It had no heat, just an intense whiteness.

She was blinded by it but kept her focus on the song. A burst of ecstasy exploded in Elienne’s gut, and she couldn’t sing anymore. It didn’t matter, because the song ended, and all that remained was the light.

Specter of the Spheres - High Resolution

A note on genre and buying options if you’re curious:

The full novel isn’t quite as fantasy-oriented as it sounds. It fits better under the category of magical realism or slipstream (if you know what that is). Here’s the description:

The world ends each blood moon.

But a faction of priestesses sacrifice themselves to keep it going. What happens when Aceline wants more for her life and decides not to do it?

Wallace has chased his dream of becoming a poet for a lifetime. It leads him toward a mysterious aurora.

Robert just wanted to connect with other humans in a world dictated by screens, algorithms, and addiction.

These three become linked across worlds, and each must uphold their end of a quest to prevent catastrophe at the hands of a tyrant in a land full of necromancy.

It officially releases tomorrow as an e-book. It will be free to Kindle Unlimited subscribers and can be pre-ordered now. Also, the paper version can be ordered now, and you will get the e-book free if you get the hardcopy.

Amazon page here.


My 2018 Reading List

Two years ago I started to use a theme for my year of reading. This was an attempt to focus on one form or style for study. I did a year of giant novels and then a year of short fiction. This year I wanted to challenge myself to read something I have almost no familiarity with.

This will be The Year of Mystery Novels. I came to the realization that I’d never read any mystery novel, maybe ever. I haven’t even read classics like Sherlock Holmes or anything by Agatha ChristieYikes!

Sure, I’ve read the Dresden Files, but those are urban fantasy with mystery elements. I’ve read plenty of novels that have mysteries in them, but the mystery novel itself is different. It stands or falls on the ability to keep the reader surprised, guessing, and intrigued about a mystery.

This is an ability all novelists could use to up their writing. So, this year, I want to explore what makes a mystery novel work. My goal is to read ten, and these are the ones I’ve come up with so far:

  1. Hound of the Baskervilles by Arthur Conan Doyle
  2. The Big Sleep by Raymond Chandler
  3. Murder on the Orient Express by Agatha Christie
  4. The Intuitionist by Colson Whitehead
  5. The Name of the Rose by Umberto Eco
  6. Mystic River by Dennis Lehane
  7. Miss Smilla’s Feeling for Snow by Peter Høeg
  8. A Crime in the Neighborhood by Suzanne Bern

I need two more, so if you love mysteries and see something missing, let me know. I’m also not locked in to any of these. I tried to get a huge range from classic procedural to modern paranormal to experimental to literary.

As for the rest of my reading, I usually shoot for 52 books a year. That’s one book a week. It’s enough that I have to keep reading to do it, but it’s not stressful if I miss a few days for illness or travel or something.

Other than mysteries, I want to read about ten books written in 2018. At the end of last year, I realized I dropped the ball on current literature. I will also shoot for about ten nonfiction books. That leaves twelve books that I’ll probably fill in on a whim. I’m in the middle of a few fantasy series, so it will mostly be those (Malazan, The Stormlight Archives, The Wheel of Time, Dresden Files).

The Prose of J.K. Rowling

I hate to be one of those people. But I can’t help it. From as far back as I can remember, I’ve always said that J.K. Rowling isn’t that great of a writer. I hope she’s humble enough to admit that her fame is mostly luck.

She wrote a story that resonated with the zeitgeist at exactly the right time, and that has very little to do with writing quality or marketing skills or anything under one’s control (if you disagree, consider Fifty Shades of Gray, then ask yourself how much your disagreement stems from me saying this about a beloved story from your childhood).

I’ve often pointed to the first Harry Potter book as an example of her low-quality writing. It reads like an early career first novel. And that’s fine, because it is.

People reply, “It’s YA! She’s a genius that gradually made her writing more sophisticated as the books went on, so that as readers aged, the reading level and maturity of the books grew with the audience.”

Okay. But let’s be honest. It’s much more likely that she just got better as a writer as she wrote more. I never read the final Harry Potter book to know if the prose style grew into something reasonable. In any case, it doesn’t matter. We have lots of books, published after that series, aimed at adults to examine.

I’m not a big mystery reader, and so I was considering doing a “Year of Mysteries” next year for the blog. I picked up The Cuckoo’s Calling by Robert Galbraith (J.K. Rowling) to read as the first book.

It’s kind of shocking to me that a major publisher would let this get through without serious edits. I was so distracted by the prose errors I couldn’t even focus on the content. I know this isn’t something most people notice, but it serves as a good reminder that J.K. Rowling is not a good writer. She’s famous. Those are different.

I was going to break down some of the prologue, but I thought people might consider that unfair. Prologues are often bad, even when handled by the greatest writers. So let’s start with the beginning of Chapter 1.

Though Robin Ellacott’s twenty-five years of life had seen their moments of drama and incident, she had never before woken up in the certain knowledge that she would remember the coming day for as long as she lived.

This is the opening sentence, and opening sentences tend to be more polished than the paragraphs that follow. This sentence reads like a first-year creative writing student attempting to impress a teacher by making things needlessly complicated. It reads like a student who hasn’t learned this is exactly how agents and publishers can tell you’re still an amateur. The real way to show maturity as a writer is to be precise and concise and readable and still get all the same information across.

Let’s break this down.

The first thing is the lack of precision in language.

She words it so the “years” are the one “seeing.” This is nitpicky but also confusing and imprecise (if you don’t understand why, it’s because only a conscious thing can “see” something).

Also, we begin with a subordinate clause. This is, by definition, beginning with inessential and/or unimportant information. The clause tries to cram in way too much information. There’s no need to force in her age to this mess of a sentence, because this will naturally become clear later.

I’ll concede there is wiggle room for personal style, but in this case, there’s too many “glue” words doing no work. For one, “that” can be eliminated without loss of information.

Then there’s the tense: “had seen,” “had never before,” “would remember the coming day.” Are we in past tense? Is the narrator omniscient or close third? I’ve read this sentence a dozen times, and I’m still not sure how she is certain about remembering something that hasn’t happened. Though the gist is obvious, it’s extraordinarily confusing if you take it as it is written. The sentence lacks clarity, precision, and readability.

Here’s my edited version:

Robin Ellacott woke with the certain knowledge that she would remember the coming day for as long as she lived.

Justification: One should draw the reader in as fast as possible. This straightforward edit does this by directly raising the question in the reader’s mind: why? Rowling’s version obscures this question by confusing the reader with tense switching, needless information, and excessive words. (I kept the awkward past/future thing, because I wanted the edit to be an actual edit and not a rewrite).

She has the whole rest of the book to let her prose get fancy (and confusing).

Moving on:

Shortly after midnight, her long-term boyfriend, Matthew, had proposed to her under the statue of Eros in the middle of Piccadilly Circus. In the giddy relief following her acceptance, he confessed that he had been planning to pop the question in the Thai restaurant where they just had eaten dinner, but that he had reckoned without the silent couple beside them, who had eavesdropped on their entire conversation.

At this point, I was a little concerned the entire book would be in past perfect (sometimes continuous) tense. She wrote the prologue this way and then several pages of the first chapter in this tense. She needs to switch to simple past already. It’s beyond tiring.

Because of the confusing nature of the first sentence and now awkward tense usage, it’s unclear to me if this proposal is the day she’ll remember forever or if it referred to the next day.

Everything is so wordy and passive: “that he had been,” “just had eaten dinner,” etc. For example, remove “just had eaten dinner” completely. This is doubled information. They were at a Thai restaurant shortly before midnight. It’s obvious they went there for dinner.

This might be a British idiom, but there seems to be a clause missing in the second sentence (despite it being more than double the length it should be). Again, the gist is there. He would have proposed if it weren’t for the silent couple eavesdropping on them. Honestly, the words, as written, don’t say that. Read the sentence carefully several times. I can’t make sense of it. I think it’s actually a sentence fragment.

Here’s my rewrite:

Shortly after midnight, her long-term boyfriend, Matthew, proposed to her under the statue of Eros in the middle of Piccadilly Circus. He confessed he would have popped the question in the Thai restaurant—if only the silent couple beside them weren’t eavesdropping on their entire conversation.

Obviously this new version isn’t perfect, but these simple readability changes show how far Rowling has to go to get from her “final draft” to solid prose style. Later, the narrator calls the proposal “the most perfect.” What does that mean? There aren’t levels of perfection. I think it’s supposed to be in Robin’s voice, but then the early narrative omniscience makes no sense.

If this is how Rowling writes for adults, no one can say she is a good writer. These first two chapters are confusing on a sentence-by-sentence level, all over the place in terms of tense and viewpoint, and messy in terms of prose style. She lacks the precision, clarity, and readability of any reasonably mature writer. Unless chapter two is vastly better, I don’t think I can read this book.

Year of Short Fiction Part 4: Breakfast at Tiffany’s

Breakfast at Tiffany’s is one of those weird cultural staples that literally everyone has heard of it. Most people over a certain age have probably seen the movie, but ask them what it’s about, and they probably have no idea. It’s kind of fascinating to think how a novella/film gets to such a point. I can’t even think of another cultural phenomenon of this type.

I was pretty excited going into this for a few reasons. I, too, had seen the movie enough years ago to not remember it. Oh, there’s the long cigarette, and a crazy cat, and a wacky party girl, and singing “Moon River,” but what was it about? What was the plot? The other reason I was excited was that Truman Capote’s In Cold Blood is one of two books that have ever made me cry. The way he writes is breathtaking.

The first thing to jump out at me was the vulgarity of the language. It was published in 1958, so we’ve moved past short fiction that hides indiscretions. But I still must imagine this novella pushed what was acceptable for the time. It openly talks about prostitution and homosexuality and a 14-year-old girl getting married to an adult man. Plus, Holly’s language is very direct and crude (I don’t recall if she swears, though).

Lolita came out a few years before Breakfast at Tiffany’s, and Tiffany’s doesn’t compare in disturbing imagery to that. So I guess I shouldn’t have been too surprised. It had more to do with tone than imagery, though.

The novella is basically a long character study, and it does an excellent job at this. Holly has to be one of the strangest characters of all time. Capote’s attention to detail is incredible. Almost every sentence that has Holly in it is crafted to expose some tiny piece of how her mind works. An early example is that the location on her business card is: traveling.

At first, it comes off as chaos. Nothing about the character makes sense, and the sentences she speaks come out in a stream-of-consciousness level of confusion. But then, by about halfway or so, she’ll do something weird, and you find yourself thinking: that’s so Holly. There appears to be a deep internal logic to it. Holly feels very real and knowable.

The plot itself is fairly melodramatic. It goes by at rapid-fire pace. This short novella has Holly being in love with and engaged to several people. She travels to probably a dozen places, often not even in the U.S. There’s parties. She’s involved with a scheme to smuggle drugs orchestrated by a man in prison. She gets pregnant and miscarries. It’s almost impossible to take stock of all that happens in this, and there’s almost no real emotion behind any of it.

Capote clearly did this on purpose. Holly’s character is flighty, and she often jumps into things without any thought. If we think of the novella as a character study, then all these crazy events occurring is part of the brilliance of the novella. The plot doesn’t have weight for the main character, so it would be a mistake to have the events play a significant role to the reader. Holly moves on, and so should the reader.

And now we come full circle. No one remembers the plot to Breakfast at Tiffany’s by design. We’re only meant to remember Holly. Even her last name is “Golightly.”

The only moments of emotional poignancy are when the narrator reflects on it all, and when we see beneath Holly’s shell. He falls in love with Holly for real (this is a bit of a theme to the book: what is love?). This is quite well done, because it contrasts so starkly with Holly’s indifference and shows how devastating her indifference can be as she tears through people’s lives.

Capote gives Holly one piece of depth that prevents her from being some caricature of a socialite. She cares deeply about her brother, and it is probably the only real human connection she’s ever had. A lot of her carefree attitude stems from a disturbing fact dropped subtly in tiny details. She runs from human connection because of the psychological trauma of being a child bride.

Overall, the novella was way better than I expected in terms of character development. It was also sort of disappointing in a way. I went in expecting it to be a romance between the narrator and Holly done in a brilliant literary Capote-esque way. It’s not that at all. But once you get over the initial shock (and genre confusion), it’s brilliant.

Year of Short Fiction Part 3: The Red Pony

The Red Pony is in one sense a novella published by John Steinbeck, but in another sense it is a collection of four short stories, originally published episodically in magazines. This makes it slightly difficult to pin down exactly what date to slap on this. The first story was published in 1933, so it came out before all his most famous works.

I was pretty excited to read this, because Steinbeck is one of the best long-form “family epic” writers. His masterpiece, in my eyes, is East of Eden, which chronicles several generations in great detail. It is true that The Pearl and Of Mice and Men and other of his short works pack a punch, but nothing compares to the deep characterization he pulls of in his longer works.

I’m torn on this one. It’s certainly my least favorite of the short fiction series so far. I can intellectualize it’s strong points, but I didn’t connect with any part of it. And the end is super weird, but we’ll get to that. Obviously there will be “spoilers,” but I haven’t really been saying that considering these stories are a hundred years old and only take an hour or so to read.

All four stories form key moments in Jody’s maturation from childhood to adulthood. Steinbeck does a great job of establishing his innocence in a small amount of space by dropping small details throughout the beginning. One of the most interesting was that Jody had a rifle, but he wasn’t allowed bullets until he demonstrated maturity with it for a full year.

This is Steinbeck establishing family dynamics and rituals. It shows that Jody hasn’t undergone one of the key rituals on the path to adulthood. The first story is about how Jody trusts Billy to take care of a horse that eventually dies. Steinbeck cleverly foreshadows this by mentioning the buzzards at the start that eventually deal the horse’s final blow.

I see the novella as a sequence of four deaths and how Jody matures in reaction to each as he ages. In reaction to the death in the first story, Jody lashes out in anger and can’t understand how the adults in his life didn’t protect him from it.

The second death is stranger. A man comes to the family’s house and wants to live out his last days there because it is where he was born. This brings another perspective to Jody. The man rides off to the mountains with an old, dying horse. Again, the horse and human presumably die, but Jody learns of a more mature way to accept the inevitability of death through this stranger.

The third death is of a pregnant horse. This horse must die to save the pony inside of it. This teaches Jody of the circle of life. Death can bring about new life, which itself will eventually die.

Though Jody doesn’t realize it, this is a redemption story for Billy. Billy had promised not to let the horse die in the first story, and he feels guilt for letting it happen. Here, he promises Jody the colt inside the horse, and he has to kill the horse to deliver it. He succeeds in his promise this time. He gives Jody a horse to make up for the one that died and can let go of his guilt.

The final story doesn’t actually have a death in it, but Jody’s grandfather comes to live with the family. The grandfather participated in traveling west across the country. It was a grand adventure, but the grandfather makes it very clear such adventure is over. This kills Jody’s dreams of doing the same.

This is the final straw in Jody’s maturation. He learned of death, life, violence, the fallibility of adults, and now his boyhood dreams are gone. He must learn to live pragmatically in the real world.

The ending was extremely strange at first.

Jody ran into the kitchen where his mother was wiping the last of the breakfast dishes. “Can I have a lemon to make a lemonade for Grandfather?”

His mother mimicked—“And another lemon to make a lemonade for you.”

“No, ma’am. I don’t want one.”

“Jody! You’re sick!” Then she stopped suddenly. “Take a lemon out of the cooler,” she said softly. “Here, I’ll reach the squeezer down to you.”

Lemonade hadn’t made an appearance for the whole novella. What on Earth could this ending be?

It solidifies the idea that Jody has fully matured. His youthful self merely would have feigned interest in helping his grandfather to get himself a lemonade. The only reason the mother can think of that he wouldn’t want one is that he’s sick. But then she realizes he has matured. He’s acting selflessly, and so she encourages it.

I get what Steinbeck was doing. I just didn’t find it very compelling. I dreaded picking it up when I stopped between stories. There is pretty much no narrative momentum. Part of this comes from the stories being early in Steinbeck’s career, but I think when you look at it broken down in the way I did, it becomes clear that this is first and foremost a carefully constructed exercise. It’s obviously well done. I just didn’t like it much.

Local Class Field Theory

Today will probably be our last class field theory post. I want to end with a brief description of local class field theory. Let {K} be a global field, and {v\in M_K} a place. We have our standard inclusion {i_v: K_v^*\hookrightarrow \mathcal{J}_K} by putting the element in the {v} component and {1}‘s everywhere else. Suppose {L/K} is abelian. We have the Artin map {\psi_{L/K}: C_K/N_{L/K}(C_L)\stackrel{\sim}{\rightarrow} G=Gal(L/K)}.

What we learned two posts ago is that the image upon composing the maps {K_v\rightarrow \mathcal{J}_K\twoheadrightarrow C_K/N_{L/K}(C_L)\rightarrow G} is exactly the decomposition group {G_w} where {w} lies over {v}. The image of the units {\mathcal{O}_v^*} under the map is the inertia group {I_w}. This gives us two exact sequences that fit together:

\displaystyle \begin{matrix} 1 & \rightarrow & N_{L_w/K_v}(L_w^*) & \rightarrow & K_v^* & \rightarrow & G_w & \rightarrow & 1 \\ & & \cup & & \cup & & & & \\ 1 & \rightarrow & N_{L_w/K_v}(\mathcal{O}_w^*) & \rightarrow & \mathcal{O}_v^* & \rightarrow & I_w & \rightarrow & 1\end{matrix}

This gives us a local Artin map {\psi_{w/v}: K_v^*\rightarrow G_w}. The main theorem of local class field theory is that the map is surjective with kernel {N_{L_w/K_v}(L_w^*)} and moreover the map can be defined independently of localizing the global fields. Just like global class field theory there is an “existence” part of the theorem as well.
This part says that every finite abelian extension of local fields arises as the localization of an extension of global fields and the local Artin maps {\psi_{w/v}} give a bijection between

\displaystyle \left\{ \text{finite index open subgroups in} \ K_v^*\right\} \leftrightarrow \left\{\text{finite abelian ext} \ L_w/K_v \right\}

where the correspondence is {U \leftrightarrow \ker \psi_{w/v}}.
Let’s do the simplest example. Let’s think about the quadratic extensions {\mathbb{Q}_p} by looking at the bijection. The standard construction is that {\mathbb{Q}_p(\sqrt{d})} are the quadratic extensions where {d} is not a square, and {\mathbb{Q}_p(\sqrt{d'})} is the same extension if {d/d'} is a square. Thus there is a nice bijection between the quadratic extensions and the non-trivial elements of {\mathbb{Q}_p^*/(\mathbb{Q}_p^*)^2}.

Local class field theory tells us that the quadradic extensions of {\mathbb{Q}_p} are in bijection with the open index {2} subgroups {U\subset \mathbb{Q}_p^*} via {L/\mathbb{Q}_p\mapsto N_{L/\mathbb{Q}_p}(L^*)}. It turns out that any index {2} subgroup at all must be open because it will contain {(\mathbb{Z}_p^*)^2}.

Now it will be useful to think in different terms, which is actually a more standard modern reformulation of class field theory since it generalizes to “higher dimensions.” There is a bijection between the open index {2} subgroups {U} of {\mathbb{Q}_p^*} and surjective characters {\chi: \mathbb{Q}_p^*\rightarrow \{\pm 1\}} just by taking the kernel of the character. Thus we have reformulated the problem of counting these subgroups to counting characters.

The unit groups have the form (if {p} odd) {\mathbb{Q}_p^*\simeq p^\mathbb{Z}\times \mathbb{Z}_p^*} and {\mathbb{Z}_p^*\simeq \mu_{p-1}\times (1 + p\mathbb{Z}_p)} where {\mu_{p-1}} is thought of via the Teichmuller lift. If {\chi} has order {2}, then it is trivial on {(\mathbb{Q}_p^*)^2\simeq p^{2\mathbb{Z}}\times (\mathbb{Z}_p^*)^2\simeq p^{2\mathbb{Z}}\times (\mu_{p-1})^2\times (1+p\mathbb{Z}_p)}.

Thus we get the result that {\chi} factors through {p^\mathbb{Z}/p^{2\mathbb{Z}}\times \mu_{p-1}/(\mu_{p-1})^2} which is a finite abelian {2}-group of order {4}. Thus the number of non-trivial characters is {3}. This gives us a nice alternate description to the classical Kummer description. It tells us there are exactly {3} quadratic extensions up to isomorphism. These aren’t hard to figure out explicitly either. Just take some {a\in \mathbb{Z}_p^*} which is not a square. The three quadratic extensions are {\mathbb{Q}_p(\sqrt{p})}, {\mathbb{Q}_p(\sqrt{a})}, and {\mathbb{Q}_p(\sqrt{ap})}. A similar description can be computed when {p=2}, but you get {7} in that case.

That is all for class field theory for now. We’ll move on to complex multiplication, and I think we’ve done enough of the basics that we can probably do what we need as we need it now.

Irreducible Character Basis

I’d just like to expand a little on the topic of the irreducible characters being a basis for the class functions of a group cf(G) from two times ago.

Let’s put an inner product on cf(G). Suppose \alpha, \beta \in cf(G). Then define \displaystyle \langle \alpha, \beta \rangle =\frac{1}{|G|}\sum_{g\in G} \alpha(g)\overline{\beta(g)}.

The proof of the day is that the irreducible characters actually form an orthonormal basis of cf(G) with respect to this inner product.

Let e_i=\sum_{g\in G} a_{ig}g. Then we have that a_{ig}=\frac{n_i\chi_i(g^{-1})}{|G|} (although just a straightforward calculation, it is not all that short, so we’ll skip it for now). Thus e_j=\frac{1}{|G|}\sum n_j\chi_j(g^{-1})g.

So now examine \frac{\chi_i(e_j)}{n_j}=\frac{1}{|G|}\sum \chi_j(g^{-1})\chi_i(g)
=\frac{1}{|G|}\sum \chi_i(g)\overline{\chi_j(g)}
= \langle \chi_i, \chi_j \rangle.

Where we note that since \chi_j is a character \chi_j(g^{-1})=\overline{\chi_j(g)}. Thus we have that \langle \chi_i, \chi_j \rangle = \delta_{ij}.

This fact can be used to get some neat results about the character table of a group, and as consequences of those we get new ways to prove lots of familiar things, like |G|=\sum n_i^2 where the n_i are the degrees of the characters. You also get a new proof of Burnside’s Lemma. I’m not very interested in any of these things, though.

I may move on to induced representations and induced characters. I may think of something entirely new to start in on. I haven’t decided yet.